Math In Focus Year 11 2 unit Ch5functions and Graphs

8122019 Math In Focus Year 11 2 unit Ch5functions and Graphs

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TERMINOLOGY

5

Arc of a curve Part or a section of a curve between twopoints

Asymptote A line towards which a curve approaches butnever touches

Cartesian coordinates Named after Descartes A system oflocating points ( x y ) on a number plane Point ( x y ) hasCartesian coordinates x and y

Curve Another word for arc When a function consistsof all values of x on an interval the graph of y f x =

] g is

called a curve y f x = ] gDependent variable A variable is a symbol that canrepresent any value in a set of values A dependentvariable is a variable whose value depends on the valuechosen for the independent variable

Direct relationship Occurs when one variable variesdirectly with another ie as one variable increases sodoes the other or as one variable decreases so doesthe other

Discrete Separate values of a variable rather than acontinuum The values are distinct and unrelated

Domain The set of possible values of x in a given domainfor which a function is de1047297ned

Even function An even function has line symmetry(re1047298ection) about the y -axis and f x f x =- -] ]g gFunction For each value of the independent variable x there is exactly one value of y the dependent variableA vertical line test can be used to determine if arelationship is a function

Independent variable A variable is independent if it maybe chosen freely within the domain of the function

Odd function An odd function has rotational symmetry

about the origin (0 0) and where f x f x =- -] ]g gOrdered pair A pair of variables one independent andone dependent that together make up a single point inthe number plane usually written in the form ( x y )

Ordinates The vertical or y coordinates of a point arecalled ordinates

Range The set of real numbers that the dependentvariable y can take over the domain (sometimes calledthe image of the function)

Vertical line test A vertical line will only cut the graph ofa function in at most one point If the vertical line cuts

the graph in more than one point it is not a function

Functions andGraphs



201Chapter 5 Functions and Graphs

INTRODUCTION

FUNCTIONS AND THEIR GRAPHS are used in many areas such as mathematics

science and economics In this chapter you will study functions function

notation and how to sketch graphs Some of these graphs will be studied inmore detail in later chapters

DID YOU KNOW

The number plane is called the Cartesian plane after Rene

Descartes (1596ndash1650) He was known as one of the 1047297rst

modern mathematicians along with Pierre de Fermat

(1601ndash1665) Descartes used the number plane to develop

analytical geometry He discovered that any equation

with two unknown variables can be represented by a lineThe points in the number plane can be called Cartesian

coordinates

Descartes used letters at the beginning of the

alphabet to stand for numbers that are known and letters

near the end of the alphabet for unknown numbers This is

why we still use x and y so often

Do a search on Descartes to 1047297nd out more details of

his life and work

Descartes

Functions

Definition of a function

Many examples of functions exist both in mathematics and in real life These

occur when we compare two different quantities These quantities are called

variables since they vary or take on different values according to some pattern

We put these two variables into a grouping called an ordered pair



202 Maths In Focus Mathematics Preliminary Course

EXAMPLES

1 Eye colour

Name Anne Jacquie Donna Hien Marco Russell Trang

Colour Blue Brown Grey Brown Green Brown Brown

Ordered pairs are (Anne Blue) (Jacquie Brown) (Donna Grey) (Hien

Brown) (Marco Green) (Russell Brown) and (Trang Brown)

2 y x 1= +

x 1 2 3 4

y 2 3 4 5

The ordered pairs are (1 2) (2 3) (3 4) and (4 5)

3

A

B

C

D

E

1

2

3

4

The ordered pairs are (A 1) (B 1) (C 4) (D 3) and (E 2)

Notice that in all the examples there was only one ordered pair for each

variable For example it would not make sense for Anne to have both blueand brown eyes (Although in rare cases some people have one eye thatrsquos a

different colour from the other)

A relation is a set of ordered points (x y ) where the variables x and y are

related according to some rule

A function is a special type of relation It is like a machine where for

every INPUT there is only one OUTPUT

INPUT PROCESS OUTPUT

The first variable (INPUT) is called the independent variable and the

second (OUTPUT) the dependent variable The process is a rule or pattern




For example in y x 1= + we can use any number for x (the independent

variable) say x 3=

When x

y

3

3 1

4

=

= +

=

As this value of y depends on the number we choose for x y is called thedependent variable

A function is a relationship between two variables where for

every independent variable there is only one dependent variable

This means that for every x value there is only one y value

While we often call the

independent variable

x and the dependent

variable y there are other

pronumerals we could

use You will meet some

of these in this course

Investigation

When we graph functions in mathematics the independent variable

(usually the x -value) is on the horizontal axis while the dependent

variable (usually the y -value) is on the vertical axis

In other areas the dependent variable goes on the horizontal axis Find

out in which subjects this happens at school by surveying teachers or

students in different subjects Research different types of graphs on the

Internet to find some examples

Here is an example of a relationship that is NOT a function Can you see the

difference between this example and the previous ones

A

B

C

D

E

1

2

3

4

In this example the ordered pairs are (A 1) (A 2) (B 1) (C 4) (D 3)

and (E 2)

Notice that A has two dependent variables 1 and 2 This means that it is

NOT a function




Here are two examples of graphs on a number plane

1

x

y

2

x

y

There is a very simple test to see if these graphs are functions Notice that

in the first example there are two values of y when x 0= The y -axis passes

through both these points

x

y




If a vertical line cuts a graph only once anywhere along the graph the

graph is a function

y

x

If a vertical line cuts a graph in more than one place anywhere along the

graph the graph is not a function

x

y

There are also other x values that give two y values around the curve If

we drew a vertical line anywhere along the curve it would cross the curve in

two places everywhere except one point Can you see where this is

In the second graph a vertical line would only ever cross the curve in one

place

So when a vertical line cuts a graph in more than one place it shows thatit is not a function




EXAMPLES

1 Is this graph a function

Solution

A vertical line only cuts the graph once So the graph is a function

2 Is this circle a function

Solution

A vertical line can cut the curve in more than one place So the circle is

not a function

You will learn how to sketch these

graphs later in this chapter




3 Does this set of ordered pairs represent a function

2 3 1 4 0 5 1 3 2 4- -^ ^ ^ ^ ^h h h h h Solution

For each x value there is only one y value so this set of ordered pairs is a

function

4 Is this a function

y

x

3

Solution

y

x

3

Although it looks like this is not a function the open circle at x 3= on

the top line means that x 3= is not included while the closed circle on

the bottom line means that x 3= is included on this line

So a vertical line only touches the graph once at x 3= The graph is

a function




1

2

3

4

5

6

7

8

9 1 3 2 1 3 3 4 0-^ ^ ^ ^h h h h 10 1 3 2 1 2 7 4 0-^ ^ ^ ^h h h h 11

1

2

3

4

5

1

2

3

4

5

12 1

2

3

4

5

1

2

3

4

5

131

2

3

4

5

1

2

3

4

5

51 Exercises

Which of these curves are functions




14 Name Ben Paul Pierre Hamish Jacob Lee Pierre Lien

Sport Tennis Football Tennis Football Football Badminton Football Badminton

15 A 3

B 4

C 7

D 3

E 5

F 7

G 4

Function notation

If y depends on what value we give x in a function then we can say that y is afunction of x We can write this as y f x= ] g

Notice that these two examples are asking for the same value and f (3) is

the value of the function when x 3=

EXAMPLES

1 Find the value of y when x 3= in the equation y x 1= +

Solution

When x

y x

3

1

3 1

4

=

= +

= +

=

2 If f x x 1= +] g evaluate f (3)

Solution

f x x

f

1

3 3 1

4

= +

= +

=

]] gg

If y f x= ] g then f (a ) is the value of y at the point on the function where x a=




EXAMPLES

1 If f x x x3 12= + +] g find f 2-] g

Solution

( ) ( )f 2 2 3 2 1

4 6 1

1

2- = - + - +

= - +

= -

] g

2 If f x x x3 2= -] g find the value of f 1-] g

Solution

( )

( )

f x x x

f 1 1 1

1 12

3 2

3

= -

- = - - -

= - -

= -

2] ]g g

3 Find the values of x for which f x 0=] g given that f x x x3 102= + -] g

Solution

( )

ie

( ) ( )

f x

x x

x x

x x

x x

0

3 10 0

5 2 0

5 0 2 0

5 2

2

=

+ - =

+ - =

+ = - =

= - =

4 Find f f f 3 2 0] ] ]g g g and if f f x4-] ]g g is defined as

when

when f x

x x

x x

3 4 2

2 21

$=

+

-

] g )

Solution

since 4 21-

( ) ( ) since

( ) ( ) since

( ) ( ) since

( ) ( )

f

f

f

f

3 3 3 4 3 2

13

2 3 2 4 2 2

10

0 2 0 0 2

0

4 2 4

8

1

$

$

= +

=

= +

=

= -

=

- = - -

=

5 Find the value of g g g 1 2 3+ - -] ] ]g g g ifwhen

when

when

x

x

x

2

1 2

1

2

1

-

-

g x

x

x2 1

5

2

= -] g

This is the same as 1047297nding y

when 2x -=

Putting (x) 0=f is different

from 1047297nding (0) f Follow

this example carefully

Use f (x) 3x 4= + when

x is 2 or more and use

f (x) 2x = - when x is less

than 2




Solution

( ) ( )

( )

( )

g

g

g

1 2 1 1 1 1 2

1

2 5 2 1

3 3 3 2

9

since

since

since2

1

2

= - -

=

- = - -

=

=

( ) ( ) ( ) g g g 1 2 3 1 5 9

3

So + - - = + -

= -

DID YOU KNOW

Leonhard Euler (1707ndash83) from Switzerland studied functions and invented the termf(x) for function notation He studied theology astronomy medicine physics and oriental

languages as well as mathematics and wrote more than 500 books and articles on

mathematics He found time between books to marry and have 13 children and even when

he went blind he kept on having books published

1 Given f x x 3= +] g find f 1] g and

f 3-] g

2 If h x x 22= -] g find h h0 2] ]g g

and h 4-] g

3 If f x x2= -] g find f f f 5 1 3-] ] ]g g g

and f 2-] g

4 Find the value of f f 0 2+ -] ]g g iff x x x 14 2

= - +] g

5 Find f 3-] g if f x x x2 5 43= - +] g

6 If f x x2 5= -] g find x when

f x 13=] g

7 Given f x x 32= +] g find any

values of x for which f x 28=] g

8 If f x 3x=] g find x when

f x271

=] g

9 Find values of z for which

f z 5=] g given f z z2 3= +] g

10 If f x x2 9= -] g find f p^ h and

f x h+] g

11 Find g x 1-] g when

g x x x2 32= + +] g

12 If f x x 13= -] g find f k] g as a

product of factors

13 Given f t t t 2 12= + +] g find

t when f t 0=] g Also find any

values of t for which f t 9=] g

14 Given f t t t 54 2

= + -] g find thevalue of f b f b- -] ]g g

15 f x x x

x x

1

1

for

for

32

=] g )

Find f f 5 1] ]g g and 1-] g

16 f x

x x

x x

x x

2 4 1

3 1 1

1

if

if

if 2

1 1

$

=

-

+ -

-

] gZ

[

]]

]]

Find the values of

f f f 2 2 1- - + -] ] ]g g g

52 Exercises

We can use pronumerals

other than f for functions






EXAMPLE

Find the x - and y -intercepts of the function f x x x7 82= + -] g

Solution

For x -intercept y 0=

x x

x x

x x

x x

0 7 8

8 1

8 0 1 0

8 1

2= + -

= + -

+ = - =

= - =

] ]g g

For y -intercept x 0=

y 0 7 0 8

8

2= + -

= -

] ]g g

This is the same as y x x 7 82= + -

You will use the intercepts

to draw graphs in the next

section in this chapter

Domain and range

You have already seen that the x -coordinate is called the independent variable

and the y -coordinate is the dependent variable

The set of all real numbers x for which a function is defined is called the

domain

The set of real values for y or f (x ) as x varies is called the range (or

image) of f

EXAMPLE

Find the domain and range of f x x2=] g

Solution

You can see the domain and range from the graph which is the parabola y x2=

x

y

CONTINUED




Notice that the parabola curves outwards gradually and will take on any

real value for x However it is always on or above the x -axis

Domain all real x

Range y y 0$

You can also find the domain and range from the equation y x2= Notice

that you can substitute any value for x and you will find a value of y

However all the y -values are positive or zero since squaring any number

will give a positive answer (except zero)

Odd and even functions

When you draw a graph it can help to know some of its properties for

example whether it is increasing or decreasing on an interval or arc of thecurve (part of the curve lying between two points)

If a curve is increasing as x increases so does y and the curve is moving

upwards looking from left to right

If a curve is decreasing then as x increases y decreases and the curve

moves downwards from left to right




EXAMPLES

1 State the domain over which each curve is increasing and decreasing

x x 3 x 2 x 1

y

Solution

The left-hand side of the parabola is decreasing and the right side is

increasing

So the curve is increasing for x 2 x 2 and the curve is decreasing when

x 1 x 2

2

x x 3

x 2 x 1

y

Solution

The left-hand side of the curve is increasing until it reaches the y -axis

(where x 0= ) It then turns around and decreases until x 3 and then

increases again

So the curve is increasing for x x x03

1 2 and the curve is

decreasing for x x03

1 1

The curve isnrsquot increasing or

decreasing at x2 We say that it is

stationary at that point You will

study stationary points and further

curve sketching in the HSC Course

Notice that the curve is

stationary at x 0= and x x 3

=




Functions are odd if they have point symmetry about the origin A graph

rotated 180deg about the origin gives the original graph

This is an odd function

x

y

For even functions f x f x= -] ]g g for all values of x

For odd functions f x f x- = -] ]g g for all values of x in the domain

As well as looking at where the curve is increasing and decreasing we can

see if the curve is symmetrical in some way You have already seen that the

parabola is symmetrical in earlier stages of mathematics and you have learned

how to find the axis of symmetry Other types of graphs can also be symmetrical

Functions are even if they are symmetrical about the y -axis They have

line symmetry (reflection) about the y -axisThis is an even function

x

y




EXAMPLES

1 Show that f x x 32= +] g is an even function

Solution

f x x

x

f x

f x x

3

3

3 is an even function

2

2

2`

- = - +

= +

=

= +

] ]]

]

g gg

g

2 Show that f x x x3= -] g is an odd function

Solution

f x x x

x x

x x

f x

f x x x is an odd function

3

3

3

3`

- = - - -

= - +

= - -

= -

= -

] ] ]^]

]

g g gh

gg

Investigation

Explore the family of graphs of f x xn=] g

For what values of n is the function even

For what values of n is the function odd

Which families of functions are still even or odd given k Let k take on

different values both positive and negative

1 f x kxn=] g

2 f x x kn= +] g

3 f x x k n= +

] ]g g

k is called a parameter

Some graphics calculators

and computer programs use

parameters to show how

changing values of k change the

shape of graphs

1 Find the x - and y -intercept of

each function

(a) y x3 2= -

(b) x y 2 5 20 0- + =

(c) x y 3 12 0+ - =

(d) f x x x32= +] g

(e) f x x 42= -] g

(f) p x x x5 62= + +] g

(g) y x x8 152= - +

(h) p x x 53

= +

] g

53 Exercises




(i) y xx

x3

0= + ] g

(j) g x x9 2= -] g

2 Show that f x f x= -] ]g g where

f x x 22= -

] g What type of

function is it

3 If f x x 13= +] g find

(a) f x2^ h (b) ( )f x 26

(c) f x-] g Is it an even or odd function(d)

4 Show that g x x x x3 28 4 2= + -] g is

an even function

5 Show that f (x ) is odd where

f x x=] g

6 Show that f x x 12= -] g is an even

function

7 Show that f x x x4 3= -] g is an

odd function

8 Prove that f x x x4 2= +] g is an

even function and hence find

f x f x- -] ]g g

9 Are these functions even odd or

neither

(a) y x x

x4 2

3

=

-

(b) y x 1

13

=

-

(c) f xx 4

32

=

-

] g

(d) y x

x

33

=+

-

(e) f x x x

x5 2

3

=-] g

10 If n is a positive integer for

what values of n is the function

f x xn=] g

even(a)

odd(b)

11 Can the function f x x xn= +] g

ever be

even(a)

odd(b)

12 For the functions below state

(i) the domain over which the

graph is increasing

(ii) the domain over which

the graph is decreasing

(iii) whether the graph is oddeven or neither

x

y(a)

x

4

y(b)

2-2

x

y(c)




Investigation

Use a graphics calculator or a computer with graphing software to sketchgraphs and explore what effect different constants have on each type of

graph

If your calculator or computer does not have the ability to use parameters

(this may be called dynamic graphing) simply draw different graphs by

choosing several values for k Make sure you include positive and negative

numbers and fractions for k

Alternatively you may sketch these by hand

Sketch the families of graphs for these graphs with parameter1 k

y kx

y kx

y kx

y kx

y xk

(a)

(b)

(c)

(d)

(e)

2

3

4

=

=

=

=

=

What effect does the parameter k have on these graphs Could you give a

general comment about y k f x= ] g Sketch the families of graphs for these graphs with parameter2 k

y x k

y x k

y x k

y x k

y x k1

(a)

(b)

(c)

(d)

(e)

2

2

3

4

= +

= +

= +

= +

= +

] g


general comment about y f x k= +] g

-2

1 2

-4

-1-2

2

4

y

x

(d) y

x

(e)

CONTINUED




Gradient form

y mx b= + has gradient m and y -intercept b

General formax by c 0+ + =

Investigation

Are straight line graphs always functions Can you find an example of a

straight line that is not a function

Are there any odd or even straight lines What are their equations

For the family of functions y k f x= ] g as k varies the function changes

its slope or steepness

For the family of functions y f x k= +] g as k varies the graph moves up

or down (vertical translation)For the family of functions y f x k= +] g as k varies the graph moves left

or right (horizontal translation)


y x k

y x k

y x k

y x k y

x k

1

(a)

(b)

(c)

(d)(e)

2

3

4

= +

= +

= +

= +

=+

]]

]

gg

g



When 0 k 2 the graphmoves to the left and when

0 k 1 the graph moves to

the right

Notice that the shape of most graphs is generally the same regardless of the

parameter k For example the parabola still has the same shape even though it

may be narrower or wider or upside down

This means that if you know the shape of a graph by looking at its

equation you can sketch it easily by using some of the graphing techniques in

this chapter rather than a time-consuming table of values It also helps you to

understand graphs more and makes it easier to find the domain and rangeYou have already sketched some of these graphs in previous years

Linear Function

A linear function is a function whose graph is a straight line




EXAMPLE

Sketch the function f x x3 5= -

] g and state its domain and range

Solution

This is a linear function It could be written as y x3 5= -

Find the intercepts


0 3 5

5 3

1

x

x

x32

=

=

=

-


3 5

5

y 0=

= -

-] g

-1

-2

y

5

4

3

2

1 1 23

6

-3

-4

-5

1 4-1-2 32-3-4

x

Notice that the line extends over the whole of the number plane so that

it covers all real numbers for both the domain and rangeDomain all real x

Range all real y

Notice too that you can

substitute any real number

into the equation of the

function for x and any real

number is possible for y

The linear function ax by c 0+ + = has domain all real x

and range all real y where a and b are non-zero

Special lines

Horizontal and vertical lines have special equations

Use a graphics calculator or a computer with dynamic graphing capability

to explore the effect of a parameter on a linear function or choose

different values of k (both positive and negative)

Sketch the families of graphs for these graphs with parameter k

1 y kx=

2 y x k= +

3 y mx b= + where m and b are both parameters

What effect do the parameters m and b have on these graphs




EXAMPLES

1 Sketch y 2= on a number plane What is its domain and range

Solution

x can be any value and y is always 2

Some of the points on the line will be (0 2) (1 2) and (2 2)

This gives a horizontal line with y -intercept 2

-1

-3

y

4

3

2

1

5

-2

-4

-5

1 4-1-2

x

32-3-4

Domain xall real Range 2 y y =

2 Sketch x 1= -

on a number plane and state its domain and range

Solution

y can be any value and x is always 1-

Some of the points on the line will be 1 0 1 1- -^ ^h h and 1 2-^ h

This gives a vertical line with x -intercept 1-

Domain 1x x = - Range y all real

-

-

4

3

2

5

-2

-4

-5

1 4-1-2-4

y

x




x a= is a vertical line with x -intercept a

Domain x x a= + Range all real y

y b= is a horizontal line with y -intercept b

Domain all real x

Range y y b=

54 Exercises

1 Find the x - and y -intercepts of

each function

(a) y x 2= -

(b) f x x2 3= +] g (c) x y 2 1 0+ =-

(d) x y 3 0+ =-

(e) x y 3 6 2 0=- -

2 Draw the graph of each straight

line

(a) x 4=

(b) x 3 0=-

(c) y 5=

(d) y 1 0+ =

(e) f x x2 1= -] g (f) y x 4= +

(g) f x x3 2= +] g

(h) x y 3+ =

(i) x y 1 0=- -

(j) x y 2 3 0+ =-

3 Find the domain and range of

(a) x y 3 2 7 0+ =-

(b) y 2=

(c) x 4= - (d) x 2 0=-

(e) y 3 0=-

4 Which of these linear functions

are even or odd

(a) y x2=

(b) y 3=

(c) x 4=

(d) y x= -

(e) y x=

5 By sketching x y 4 0=- - and

x y 2 3 3 0+ =- on the same set

of axes find the point where they

meet




Applications

The parabola shape is used in many different applications as it has specialproperties that are very useful For example if a light is placed inside the parabola

at a special place (called the focus) then all light rays coming from this light and

bouncing off the parabola shape will radiate out parallel to each other giving a

strong light This is how car headlights work Satellite dishes also use this property

of the parabola as sound coming in to the dish will bounce back to the focus

The pronumeral

a is called the

coef1047297cient of x 2

Quadratic Function

The quadratic function gives the graph of a parabola

f x ax bx c 2= + +] g is the general equation of a parabola

If a 02 the parabola is concave upwards

If a 01 the parabola is concave downwards






EXAMPLES

1 (a) Sketch the graph of y x 12= - showing intercepts

(b) State the domain and range

Solution

This is the graph of a parabola Since(a) a 02 it is concave upward


x

x

x

0 1

1

1

2

2

= -

=

=


0 1

1

y 2= -

= -

From the graph the curve is moving outwards and will extend(b)

to all real x values The minimum y value is 1-

Domain xall real

Range y y 1$ -

2 Sketch f x x 1 2= +] ]g g

Solution

This is a quadratic function We find the intercepts to see where the

parabola will lie

Alternatively you may know from your work on parameters that

f x x a 2= +] ]g g will move the function f x x2

=] g horizontally a units to the

left

So f x x 1 2= +] ]g g moves the parabola f x x2=] g 1 unit to the left


0

1 0

1

x

x

x

1 2= +

+ =

= -

] g


1

y 0 1 2= +

=

] g

-1

-

4

3

2

1

5

-2

-4

-5

-

1 41-2 5-4

y

x




3 For the quadratic function f x x x 62= + -] g

Find the(a) x - and y -intercepts

Find the minimum value of the function(b)

State the domain and range(c)

For what values of(d) x is the curve decreasing

Solution

For(a) x -intercept y 0=

This means f x 0=] g

x x

x xx x

x x

0 6

3 23 0 2 0

3 2

2= + -

= + -

+ = - =

= - =

] ]g g


f 0 0 0 66

2= + -

= -

] ] ]g g g

Since(b) a 02 the quadratic function has a minimum value

Since the parabola is symmetrical this will lie halfway between the

x -intercepts

Halfway between 3x = - and 2x =

23 2

21- +

= -

Minimum value is f 21

-c m

f 21

21

21

6

41

21

6

641

2

- = - + - -

= - -

= -

c c cm m m

So the minimum value is 641

-

CONTINUED

You will learn more

about this in Chapter 9

-1

-3

4

3

2

1

5

-2

-4

-5

1 4-1-2 32-3-4

y

x




Sketching the quadratic function gives a concave upward parabola(c)

From the graph notice that the parabola is gradually going outwards and

will include all real x values

Since the minimum value is 641

- all y values are greater than this

Domain xall real Range 6 y y

41

$ - 1 The curve decreases down to the minimum point and then(d)

increases So the curve is decreasing for all x

2

11 -

4 (a) Find the x - and y -intercepts and the maximum value of the

quadratic function f x x x4 52= - + +] g

(b) Sketch the function and state the domain and range

(c) For what values of x is the curve increasing

Solution

For(a) x -intercept 0 y =

So f x 0=] g

0 4 54 5 0

0

x xx x

x x5 1

2

2

= - + +

=

+ =

- -

-] ]g g

x x

x x

5 0 1 0

5 1

- = + =

= = -

For y -intercept 0x =

f 0 0 4 0 5

5

2= - + +

=

] ] ]g g g

-1

-3

4

2

5

-2

-4

-5

-

y

1 41-2 5-4

x

-2 4




Since a 01 the quadratic function is concave downwards and has a

maximum value halfway between the x -intercepts 1x = - and x 5=

21 5

2- +

=

f 2 2 4 2 59

= - + +

=

2] ] ]g g g

So the maximum value is 9

Sketching the quadratic function gives a concave downward parabola(b)

From the graph the function can take on all real numbers for x but the

maximum value for y is 9


From the graph the function is increasing on the left of the(c)

maximum point and decreasing on the right

So the function is increasing when x 21


each function

(a) 2 y x x2= +

(b) 3 y x x2= - +

(c) f x x 12= -] g

(d) y x x 22= - -

(e) y x x9 82= +-

2 Sketch

(a) 2 y x2= +

(b) y x 12= - +

(c) f x x 42= -] g

(d) 2 y x x2= +

(e) y x x2= - -

(f) f x x 3= -2] ]g g

55 Exercises

-1

9

8

7

5

4

3

2

6

1

-2

-3

-4

-5

y

2 51 643-1-2-3-4

x




EXAMPLES

1 Sketch f x x 1= -] g and state its domain and range

Solution

Method 1 Table of values

When sketching any new graph for the first time you can use a table of

values A good selection of values is x3 3 - but if these donrsquot give

enough information you can find other values

Absolute Value Function

You may not have seen the graphs of absolute functions before If you are not

sure about what they look like you can use a table of values or look at the

definition of absolute value

(g) f x x 1 2= +] ]g g

(h) y x x3 42= + -

(i) y x x2 5 32= - +

(j) f x x x3 22= - + -] g

3 For each parabola findthe(i) x - and y -intercepts

the domain and range(ii)

(a) ndash y x x7 122= +

(b) f x x x42= +] g

(c) y x x2 82= - -

(d) y x x6 92= +-

(e) f t t 4 2= -] g


(a) y x 52= -

(b) f x x x6

2= -

] g (c) f x x x 22= - -] g

(d) y x2= -

(e) f x x 7 2= -] ]g g

5 Find the range of each function

over the given domain

(a) y x2= for x0 3

(b) y x 42= - + for x1 2 -

(c) f x x 12= -] g for x2 5 -

(d) y x x2 32= + - for x2 4 -

(e) y x x

22= - +- for

x0 4

6 Find the domain over which each

function is

increasing(i)

decreasing(ii)

(a) y x2=

(b) y x2

= - (c) f x x 92

= -] g

(d) y x x42= - +

(e) f x x 5 2= +] ]g g

7 Show that f x x2= -] g is an even

function

8 State whether these functions are

even or odd or neither

(a) y x 12= +

(b) f x x 32= -

] g

(c) y x2 2= -

(d) f x x x32= -] g

(e) f x x x2= +] g

(f) y x 42= -

(g) y x x2 32= - -

(h) y x x5 42= +-

(i) p x x 1 2= +] ]g g

(j) y x 2= -2] g




CONTINUED

eg When x 3= -

| | y 3 13 12

= - -

= -

=

x -3 -2 -1 0 1 2 3

y 2 1 0 -1 0 1 2

This gives a v-shaped graph

y

-2

4

3

2

1

5

-1

-3

-4

-5

1 4-1-2 32-3-4

x

Method 2 Use the definition of absolute value

| | y x

x x

x x1

1 0

1 0

when

when 1

$= - =

-

- -amp This gives 2 straight line graphs

y x x1 0$= - ] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x - 1




y x 1= - - x 01] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y =- x - 1

Draw these on the same number plane and then disregard the dotted

lines to get the graph shown in method 1

-3

4

3

2

1

5

-2

-1

-4

-5

y y

3-1-2 421-3-4

x

y = - x - 1

y = x - 1

Method 3 If you know the shape of the absolute value functions find the

intercepts

For x -intercept 0 y =

So f x 0=

] g

| |

| |

x

x

x

0 1

1

1`

= -

=

=


( ) | |f 0 0 11

= -

= -




The graph is V -shaped passing through these intercepts

-3

4

3

2

1

5

-2

-1

-4

-5

y

4-2 5321-1-3-4

x

From the graph notice that x values can be any real number while the

minimum value of y is 1-

Domain all real x

Range y y 1$ -

2 Sketch | | y x 2= +

Solution


| | ( ) y x x xx x2 2 2 0

2 2 0whenwhen 1

$= + = + +

- + +

This gives 2 straight lines

2 y x= + when x 2 0$+

x 2$ -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x + 2

If you already know how

to sketch the graph of

y | x |= translate the

graph of y | x | 1= -

down 1 unit giving it a

y-intercept of 1-

CONTINUED




2 y x= - +] g when x 2 01+

ie y x 2= - - when x 21 -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2

Draw these on the same number plane and then disregard the dotted lines

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2

y = x + 2

Method 2 Find intercepts


So 0f x =] g

0 | 2 |

0 2

2

x

x

x

= +

= +

- =


(0) | 0 2 |

2

f = +

=

There is only one

solution for the

equation | x 2 | 0+ =

Can you see why





-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

If you know how to

sketch the graph of

y | x |= translate it 2

places to the left for the

graph of y | x 2 |= +

Investigation

Are graphs that involve absolute value always functions Can you find an

example of one that is not a function

Can you find any odd or even functions involving absolute values What

are their equations


to explore the effect of a parameter on an absolute value function or

choose different values of k (both positive and negative)


1 | |f x k x=] g

2 | |f x x k= +] g

3 | |f x x k= +] g

What effect does the parameter k have on these graphs

The equations and inequations involving absolute values that you studied in

Chapter 3 can be solved graphically




EXAMPLES

Solve

1 |2 1 | 3x - =

Solution

Sketch | 2 1 | y x= - and 3 y = on the same number plane

The solution of |2 1 | 3x - = occurs at the intersection of the graphs that

is x 1 2= -

2 |2 1 | 3 2x x= -+

Solution

Sketch | 2 1 | y x= + and 3 2 y x= - on the same number plane

The solution is 3x =

3 | 1 | 2x 1+

Solution

Sketch | 1 | y x= + and 2 y = on the same number plane

The graph shows that

there is only one solution

Algebraically you need to

1047297nd the 2 possible solutions

and then check them




The solution of | 1 | 2x 1+ is where the graph | 1 | y x= + is below the

graph 2 y = that is x3 11 1-


each function

(a) | | y x=

(b) | |f x x 7= +] g

(c) | |f x x 2= -] g

(d) 5 | | y x=

(e) | |f x x 3= - +] g

(f) | 6 | y x= +

(g) | |f x x3 2= -] g

(h) | 5 4 | y x= +

(i) | 7 1 | y x= -

(j) | |f x x2 9= +] g

2 Sketch each graph on a number

plane

(a) | | y x=

(b) | |f x x 1= +] g

(c) | |f x x 3= -] g

(d) 2 | | y x=

(e) | |f x x= -] g

(f) | 1 | y x= +

(g) | |f x x 1= - -] g

(h) | 2 3 | y x= -

(i) | 4 2 | y x= +

(j) | |f x x3 1= +] g


each function

(a) | 1 | y x= -

(b) | |f x x 8= -] g

(c) | |f x x2 5= +] g

(d) 2 | | 3 y x= -

(e) | |f x x 3= - -] g


function is

increasing(i)

decreasing(ii)

(a) | 2 | y x= -

(b) | |f x x 2= +

] g

(c) | |f x x2 3= -] g

(d) 4 | | 1 y x= -

(e) | |f x x= -] g

5 For each domain find the range

of each function

(a) | | y x= for x2 2 -

(b) | |f x x 4= - -] g for

x4 3 -

(c) | |f x x 4= +] g for x7 2 -

(d) | 2 5 | y x= - for x3 3 -

(e) | |f x x= -] g for x1 1 -

6 For what values of x is each

function increasing

(a) | 3 | y x= +

(b) | |f x x 4= - +] g

(c) | |f x x 9= -] g

(d) | | y x 2 1= - -

(e) | |f x x 2= - +] g

56 Exercises




7 Solve graphically

(a) | | 3x =

(b) | |x 12

(c) | |x 2

(d) | 2 | 1x + =

(e) | 3 | 0x- =

(f) |2 3 | 1x - =

(g) | |x 1 41-

(h) | |x 1 3+

(i) | |x 2 22-

(j) | |x 3 1$-

(k) | |x2 3 5+

(l) | |x2 1 1$-

(m) |3 1 | 3x x- = +

(n) |3 2 | 4x x- = -

(o) | 1 | 1x x- = +

(p) | 3 | 2 2x x+ = + (q) |2 1 | 1x x+ = -

(r) |2 5 | 3x x- = -

(s) | 1 | 2x x- =

(t) |2 3 | 3x x- = +

The Hyperbola

A hyperbola is a function with its equation in the form xy a y xa

or= =

EXAMPLE

Sketch1

y x=

Solution

1 y x= is a discontinuous curve since the function is undefined at x 0=

Drawing up a table of values gives

x -3 -2 -121-

41- 0

41

21 1 2 3

y 3

1-

2

1- -1 -2 -4 mdash 4 2 1

2

1

3

1

Class Discussion

What happens to the graph as x becomes closer to 0 What happens as x

becomes very large in both positive and negative directions The value of

y is never 0 Why




To sketch the graph of a more general hyperbola we can use the domain and

range to help find the asymptotes (lines towards which the curve approaches

but never touches)

The hyperbola is an example of a discontinuous graph since it has a gap

in it and is in two separate parts

Investigation

Is the hyperbola always a function Can you find an example of a

hyperbola that is not a function

Are there any families of odd or even hyperbolas What are their

equations


to explore the effect of a parameter on a hyperbola or choose differentvalues of k (both positive and negative)


1 y xk

=

21

y x k= +

31

y x k

=+


EXAMPLES

1 (a) Find the domain and range of f xx 3

3=

-

] g

Hence sketch the graph of the function(b)

Solution

This is the equation of a hyperbola

To find the domain we notice that x 3 0-

So x 3

Also y cannot be zero (see example on page 238)

Domain all real x x 3

Range all real y y 0

The lines 3x = and 0 y = (the x -axis) are called asymptotes

The denominator cannot

be zero

CONTINUED




To make the graph more accurate we can find another point or two The

easiest one to find is the y -intercept


1

y 0 3

3=

-

= -

-3

4

3

2

1

5

-2

-1

-4

-5

y

-1-2 4 521-3-4

x

x = 3

y = 0

Asymptotes

3

2 Sketch y x2 4

1= -

+

Solution

This is the equation of a hyperbola The negative sign turns the hyperbolaaround so that it will be in the opposite quadrants If you are not sure

where it will be you can find two or three points on the curve

To find the domain we notice that x2 4 0+

x

x

2 4

2

-

-

For the range y can never be zero

Domain all real x x 2 -


So there are asymptotes at x 2= - and y 0= (the x -axis)

To make the graph more accurate we can find the y -interceptFor y -intercept x 0=

( ) y

2 0 41

41

= -+

= -

Notice that this graph is

a translation of3

y x

=

three units to the right




y

-2

x

-

1

4

The function f xbx c

a=

+

] g is a hyperbola with

domain x xb

c all real -amp 0 and

range all real y y 0

1 For each graph

State the domain and range(i)

Find the(ii) y -intercept if it

exists

Sketch the graph(iii)

(a)2

y x=

(b)1

y x= -

(c) f xx 1

1=

+] g

(d) f xx 2

3=

-

] g

(e)3 6

1 y

x=

+

(f) f xx 3

2= -

-

] g

(g) f xx 1

4=

-

] g

(h)1

2 y

x= -

+

(i) f xx6 3

2=

-

] g

(j)2

6 y

x= -

+

2 Show that f x x2

=] g is an odd

function



(a) f xx2 5

1=

+

] g for x2 2 -

(b)3

1 y

x=

+ for x2 0 -

(c) f xx2 4

5=

-

] g for x3 1 -

57 Exercises




(d) f xx 4

3= -

-

] g for x3 3 -

(e)3 1

2 y

x= -

+ for x0 5

4 Find the domain of each function

over the given range

(a)3

y x= for y 1 3

(b)2

y x= - for y 221

- -

(c) f xx 1

1=

-

] g for y 171

- -

(d) f xx2 1

3= -

+

] g for

y 131

- -

(e)3 2

6 y

x=

- for y 1

21

6

Circles and Semi-circles

The circle is used in many applications including building and design

Circle gate

A graph whose equation is in the form 0x ax y by c 2 2

+ + + + = has theshape of a circle

There is a special case of this formula

The graph of x y r 2 2 2+ = is a circle centre 0 0^ h and radius r

Proof

r y

x

( x y)

y

x




Given the circle with centre (0 0) and radius r

Let (x y ) be a general point on the circle with distances from the origin x

on the x -axis and y on the y -axis as shown

By Pythagorasrsquo theorem

c a b

r x y

2 2 2

2 2 2

`

= +

= +

EXAMPLE

Sketch the graph of(a) 4x y 2 2+ = Is it a function

State its domain and range(b)

Solution

This is a circle with radius 2 and centre (0 0)(a)

y

x

-2

-2 2

2

The circle is not a function since a vertical line will cut it in more than

one place

y

x

2

2

2

-2

The radius is 4

CONTINUED




Notice that the(b) x -values for this graph lie between 2- and 2 and

the y -values also lie between 2- and 2

Domain 2 2 x x -

Range 2 2 y y -

The circle x y r 2 2 2+ = has domain x r x r - + and

range y r y r -

The equation of a circle centre (a b ) and radius r is ndash ndashx a y b r 2 2 2+ =] ^g h

We can use Pythagorasrsquo theorem to find the equation of a more general circle

Proof

Take a general point on the circle (x y ) and draw a right-angled triangle as

shown

y

x

(a b)

x

y

r

( x y)

a

b x - a

y - b

Notice that the small sides of the triangle are ndashx a and ndash y b and the

hypotenuse is r the radius


ndash ndash

c a b

r x a y b

2 2 2

2 2 2

= +

= +] ^g h




EXAMPLES

1 (a) Sketch the graph of x y 812 2+ =

(b) State its domain and range

Solution

The equation is in the form(a) x y r 2 2 2+ =

This is a circle centre (0 0) and radius 9

y

x 9

9

-9

-9

From the graph we can see all the values that are possible for(b) x

and y for the circle

Domain 9 9 x x -

Range 9 9 y y -

2 (a) Sketch the circle ndash x y 1 2 42 2+ + =] ^g h


Solution

The equation is in the form(a) ndash ndash x a y b r 2 2 2+ =] ^g h

ndash

ndash ndash

x y

x y

1 2 4

1 2 2

2

2 2

+ + =

+ - =

2

2

] ^

] ]_

g h

g gi

So 1 2a b= = - and 2r =

CONTINUED




This is a circle with centre 1 2-^ h and radius 2

To draw the circle plot the centre point 1 2-^ h and count 2 units up

down left and right to find points on the circle

y

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

(1 -2)



Domain 1 3 x x -

Range 4 0 y y -

3 Find the equation of a circle with radius 3 and centre 2 1-^ h inexpanded form

Solution

This is a general circle with equation ndash ndashx a y b r 2 2 2+ =] ^g h where

a b2 1= - = and r 3=

Substituting

ndash ndash

ndash

ndash

x a y b r

x y

x y

2 1 3

2 1 9

2 2 2

2 2 2

2 2

+ =

- - + =

+ + =

] ^]] ^

] ^

g hg g hg h

Remove the grouping symbols

ndash

ndash

a b a ab b

x x x

x x

a b a ab b

y y y

y y

2

2 2 2 2

4 4

2

1 2 1 1

2 1

So

So

2 2 2

2 2 2

2

2 2 2

2 2 2

2

+ = + +

+ = + +

= + +

= - +

= - +

= - +

]] ] ]]^ ^ ]

gg g ggh h g

The equation of the circle is

ndash

x x y y

x x y y

x x y y

x x y y

4 4 2 1 9

4 2 5 9

4 2 5 9

4 2 4 0

9 9

2

2

2

2

+ + + - + =

+ + - + =

+ + + =

+ + - - =

- -

You may need to revise this

in Chapter 2




Investigation

The circle is not a function Could you break the circle up into

two functions

Change the subject of this equation to y

What do you notice when you change the subject to y Do you get two

functions What are their domains and ranges

If you have a graphics calculator how could you draw the graph of a

circle

The equation of the semi-circle above the x -axis with centre (0 0)

and radius r is y r x2 2= -

The equation of the semi-circle below the x -axis with centre (0 0)

and radius r is y r x2 2= - -

y r x2 2= - is the semi-circle above the x -axis since its range is y $ 0

for all values

y

x r

r

-r

The domain is x r x r - and the range is y y r 0

Proof

ndash

x y r

y r x y r x

2 2 2

2 2 2

2 2

+ =

=

= -

This gives two functions

By rearranging the equation of a circle we can also find the equations of

semi-circles




y r x2 2= - - is the semi-circle above the x -axis since its range is

y 0 for all values

y

x r

-

r

-r

The domain is x r x r - and the range is y r y 0 -

EXAMPLES

Sketch each function and state the domain and range

1 f x x92

= -] g

Solution

This is in the form f x r x2 2= -] g where r 3=

It is a semi-circle above the x -axis with centre (0 0) and radius 3

y

x 3

3

-3

Domain 3 3 x x -

Range 0 3 y y




2 y x4 2= - -

Solution

This is in the form y r x2 2= - - where r 2=

It is a semi-circle below the x -axis with centre (0 0) and radius 2

y

x 2

-2

-2

Domain 2 2 x x -

Range 2 0 y y -

1 For each of the following

sketch each graph(i)

state the domain and(ii)

range

(a) 9x y 2 2+ =

(b) x y 16 02 2+ =-

(c) ndash ndashx y 2 1 42 2+ =] ^g h

(d) 1 9x y 2 2+ + =

] g

(e) ndashx y 2 1 12 2+ + =] ^g h

2 For each semi-circle

state whether it is above or(i)

below the x -axis

sketch the function(ii)

state the domain and(iii)

range

(a) 25 y x2= - -

(b) 1 y x2= -

(c) 36 y x2= -

(d) 64 y x2= - -

(e) 7 y x2= - -

3 Find the length of the radius and

the coordinates of the centre of

each circle

(a) 100x y 2 2+ =

(b) 5x y 2 2+ =


(d) ndashx y 5 6 492 2+ + =] ^g h

(e) ndashx y 3 812 2+ =^ h

58 Exercises




4 Find the equation of each circle

in expanded form (without

grouping symbols)

Centre (0 0) and radius 4(a)

Centre (3 2) and radius 5(b)

Centre(c) 1 5-

^ h and radius 3Centre (2 3) and radius 6(d)

Centre(e) 4 2-^ h and radius 5

Centre(f) 0 2-^ h and radius 1

Centre (4 2) and radius 7(g)

Centre(h) 3 4- -^ h and radius 9

Centre(i) 2 0-^ h and radius 5

Centre(j) 4 7- -

^ h and radius 3

Other Graphs

There are many other different types of graphs We will look at some of these

graphs and explore their domain and range

Exponential and logarithmic functions

EXAMPLES

1 Sketch the graph of f x 3x=] g and state its domain and range

Solution

If you do not know what this graph looks like draw up a table of values

You may need to revise the indices that you studied in Chapter 1

eg When 0x =

y 3

1

c=

=

x

y

1

3

3

1

31

When1

1

= -

=

=

=

-

x 3- 2- 1- 0 1 2 3

y 271

91

31

1 3 9 27

If you already know what the shape of the graph is you can draw it

just using 2 or 3 points to make it more accurate

You will meet these

graphs again in the

HSC Course




This is an exponential function with y -intercept 1 We can find one

other point

When

x

y

1

3

3

1

=

=

=

y

x

1

2

1

3

From the graph x can be any real value (the equation shows this as well

since any x value substituted into the equation will give a value for y )

From the graph y is always positive which can be confirmed by

substituting different values of x into the equation

Domain xall real Range y y 02

2 Sketch logf x x=] g and state the domain and range

Solution

Use the LOG key on your calculator to complete the table of values

Notice that you canrsquot find the log of 0 or a negative number

x minus2 minus1 0 05 1 2 3 4

y minus03 0 03 05 06

y

x

1

2

1 2 3 4

-1

From the graph and by trying different values on the calculator y can be

any real number while x is always positive

Domain x x 02 + Range y all real

You learned about

exponential graphs in earlier

stages of maths




The exponential function y ax= has domain all real x and

range y y 02

The logarithmic function log y xa

= has domain x x 02 + and

range all real y

Cubic function

A cubic function has an equation where the highest power of x is x3

EXAMPLE

1 Sketch the function f x x 23= +


Solution

Draw up a table of values

x minus3 minus2 minus1 0 1 2 3

y minus25 minus6 1 2 3 10 29

y

x

1

1

-2 2 3 4

-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

The function can have any real x or y value

Domain xall real Range y all real

If you already

know the shape of

( ) y x f x x 2 3= = + 3 hasthe same shape as ( )f x x =

3

but it is translated 2 units up

(this gives a y -intercept of 2)




Domain and range

Sometimes there is a restricted domain that affects the range of a function

EXAMPLE

1 Find the range of f x x 23= +] g over the given domain of x1 4 -

Solution

The graph of f x x 23= +] g is the cubic function in the previous example

From the graph the range is all real y However with a restricted

domain of x1 4 - we need to see where the endpoints of this

function are

f

f

1 1 2

1 21

4 4 2

64 2

66

3

3

- = - +

= - +

=

= +

= +

=

] ]

] ]

g g

g g

Sketching the graph we can see that the values of y all lie between

these points

y

x

(-1 1)

(4 66)

Range 1 66 y y




You may not know what a function looks like on a graph but you can still

find its domain and range by looking at its equation

When finding the domain we look for values of x that are impossible

For example with the hyperbola you have already seen that the denominator

of a fraction cannot be zero

For the range we look for the results when different values of x aresubstituted into the equation For example x 2 will always give zero or a

positive number

EXAMPLE

Find the domain and range of f x x 4= -] g

Solution

We can only find the square root of a positive number or zerondash 4 0x

x 4So $

$

When you take the square root of a number the answer is always positive

(or zero) So y 0$

Domain x x 4$ + Range y y 0$

59 Exercises


(a) 4 3 y x= +

(b) f x 4= -] g

(c) 3x =

(d) ndashf x x4 12=] g

(e) ndash p x x 23=] g

(f) f x xx 12 2= - -] g

(g) 64x y 2 2+ =

(h) f t t 4

3=

-] g

(i) ( ) g 2

5zz

= +

(j) | |f x x=] g


(a) y x=

(b) 2 y x= -

(c) | |f x x2 3= -] g

(d) | | 2 y x= -

(e) f x x2 5= - +] g

(f) | | y x5= -

(g) 2 y x=

(h) y 5x= -

(i) f x xx 1

= +] g

(j)2

4 3 y x

x= -

3 Find the x -intercepts of

(a) y x x 5 2= -] g

(b) ndash ndashf x x x x1 2 3= +] ] ] ]g g g g (c) y x x x6 83 2

= +-

(d) g x x x164 2= -] g

(e) 49x y 2 2+ =

You may like to

simplify the function

by dividing by x




4 (a) Solve x1 02$-

(b) Find the domain of

f x x1 2= -] g

5 Find the domain of

(a) 2 y x x2= - -

(b) g t t t 62= +] g

6 Each of the graphs has a

restricted domain Find the range

in each case

(a) y x2 3= - in the domain

x3 3 -

(b) y x2= in the domain

x2 3 -

(c) f x x3=] g in the domain

x2 1 - (d)

1 y x= in the domain

x1 5

(e) | | y x= in the domain

0 4x

(f) y x x22= - in the domain

x3 3 -

(g) y x2= - in the domain

x1 1 -

(h) y x 12= - in the domain

x2 3 -

(i) y x x2 32= - - in the domain

x4 4 -

(j) y x x7 62= - + - in the

domain 0 7x

7 (a) Find the domain for the

function y x 1

3=

+

Explain why there is no(b)

x - intercept for the function

State the range of the(c)function

8 Given the function f x x

x=] g

find the domain of the(a)

function

find its range(b)

9 Draw each graph on a number

plane

(a) f x x4=] g

(b) y x3= -

(c) y x 34= -

(d) 2 p x x3=] g

(e) 1 g x x3= +] g

(f) 100x y 2 2+ =

(g) 2 1 y x= +

10 (a) Find the domain and range of

y x 1= -

(b) Sketch the graph of y x 1= -

11 Sketch the graph of y 5x=

12 For each function state

its domain and range(i)

the domain over which the(ii)

function is increasing

the domain over which the(iii)

function is decreasing(a) y x2 9= -

(b) f x x 22= -] g

(c)1

y x=

(d) f x x3=] g

(e) f x 3x=] g

13 (a) Solve x4 02$-

(b) Find the domain and range of

(i) 4 y x2= -

(ii) y x4

2= - -




DID YOU KNOW

A lampshade can produce a hyperbola

where the light meets the 1047298at wall

bull Can you 1047297nd any other shapes made by

a light

Lamp casting its light

Limits and Continuity

Limits

The exponential function and the hyperbola are examples of functions that

approach a limit The curve y ax= approaches the x -axis when x approaches

very large negative numbers but never touches it

That is when x a 0x 3-

Putting a 3- into index form gives

a a1

1

03

Z

=

=

3

3

-

We say that the limit of ax as x approaches 3- is 0 In symbols we write

lim a 0x=

x 3-

A line that a graph approaches

but never touches is called an

asymptote

EXAMPLES

1 Find lim x

x x5x 0

2+

Solution

Substituting 0x = into the function gives00

which is undefined

Factorising and cancelling help us find the limit

( )

lim lim

lim

xx x

x

x x

x

5 5

5

5

x x

x

0

2

0 1

1

0

+=

+

= +

=

] g




2 Find limx

x

4

22

-

-

x 2

Solution

Substituting 2x = into the function gives 00 which is undefined

lim lim

lim

x

x

x x

x

x

4

2

2 2

2

21

41

2 1

1

-

-=

+ -

-

=+

=

x x2 2

x 2

^ _h i

3 Find limh

h x hx h2 72 2+ -

h 0

Solution

lim lim

lim

h

h x hx h

h

h hx x

hx x

x

2 7 2 7

2 7

7

2 2 2

2

2

+ -=

+ -

= + -

= -

h 0

h h0 0

^ h

Continuity

Many functions are continuous That is they have a smooth unbroken curve(or line) However there are some discontinuous functions that have gaps in

their graphs The hyperbola is an example

If a curve is discontinuous at a certain point we can use limits to find the

value that the curve approaches at that point

EXAMPLES

1 Find lim

x

x

1

12

-

-

x 1

and hence describe the domain and range of the curve

11

y x

x2

=-

- Sketch the curve

Solution

Substituting 1x = into11

xx2

-

- gives

00

CONTINUED




( )

lim lim

limx

x

x

x x

x11

1

1 1

1

2

x x

x

1

2

1

1

-

-=

-

+ -

= +

=

-

] ]g g

11

y xx2

=-

- is discontinuous at 1x = since y is undefined at that point

This leaves a gap in the curve The limit tells us that y 2 as 1x so

the gap is at 1 2^ h

Domain 1x x xall real Range 2 y y y all real

y xx

x

x x

x

11

1

1 1

1

2

=-

-

=

+

= +

-

-^ ^h h

` the graph is y x 1= + where x 1

2 Find limx

x x2

2x 2

2

+

+ -

-

and hence sketch the curve y x

x x2

22

= + -

+

Solution

Substituting x 2= - intox

x x2

22

+

+ - gives

00

lim lim

lim

xx x

x

x x

x

22

2

1 2

1

3

x x

x

2

2

2

2

+

+ -=

+

- +

=

= -

-

- -

-

^^ ^

^

hh h

h2 y

x

x xx

y x

x

x

x

22

2

2

1

1

is discontinuous at2

=+

+ -= -

=+

= -

+ -^ ^h h

So the function is y x 1= - where x 2 - It is discontinuous at 2 3- -^ h

Remember that x 1




1 Find(a) lim x 52

+x 4

(b) lim t 7-t 3 -

(c) lim x x2 43+ -

x 2

(d) lim xx x32

+

x 0

(e) limh

h h

2

22

-

- -

h 2

(f) lim y

y

5

1253

-

-

y 5

(g) limx

x x

12 12

+

+ +

x 1-

(h) limx

x x4

2 82

+

+ -

x 4 -

(i) limc

c

4

22

-

-

c 2

(j) limx x

x 12

-

-

x 1

(k) lim h

h h h2 73 2+ -

h 0

(l) limh

hx hx h32 2- +

h 0

(m) limh

hx h x hx h2 3 53 2 2- + -

h 0

(n) lim x c x c 3 3

-

-

x c

2 Determine which of thesefunctions are discontinuous and

find x values for which they are

discontinuous

(a) 3 y x2= -

(b)1

1 y

x=

+

(c) f x x 1= -] g

(d)4

1 y

x2=

+

(e)4

1

y x2=

-

3 Sketch these functions showing

any points of discontinuity

(a)3

y xx x2

= +

(b)33

y x

x x2

=+

+

(c)1

5 4 y

xx x2

=+

+ +

510 Exercises




Regions

Class Investigation

How many solutions are there for y x 2$ + How would you record

them all

Inequalities can be shown as regions in the Cartesian plane

You can shade regions on a number plane that involve either linear or

non-linear graphs This means that we can have regions bounded by a circle or

a parabola or any of the other graphs you have drawn in this chapter

Regions can be bounded or unbounded

A bounded region means that the line or curve is included in the region

EXAMPLE

Sketch the region x 3

Solution

x 3 includes both 3x = and x 31 in the regionSketch 3x = as an unbroken or filled in line as it will be included in the

region Shade in all points where x 31 as shown

y

x =3

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

Remember that x 3= is a

vertical line with x-intercept 3




An unbounded region means that the line or curve is not included in the

region

EXAMPLE

Sketch the region y 12 -

Solution

y 12 - doesnrsquot include y 1= - When this happens it is an unbounded

region and we draw the line y 1= - as a broken line to show it is not

included

Sketch y 1= - as a broken line and shade in all points where y 12 - as

shown

y

x

1

1

-2 2 3 4

-3

-4

-5

2

3

4

5

-1

-2

-3-4

-1 y = -1

Remember that y 1= - is a

horizontal line with

y -intercept 1-

For lines that are not horizontal or vertical or for curves we need to

check a point to see if it lies in the region




EXAMPLES

Find the region defined by

1 y x 2$ +

Solution

First sketch 2 y x= + as an unbroken line

On one side of the line y x 22 + and on the other side y x 21 +

To find which side gives y x 22 + test a point on one side of the

line (not on the line)

For example choose 0 0^ h and substitute into

y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0

^ h does not lie in the region y x 2$ + The

region is on the other side of the line

2 x y 2 3 61-

Solution

First sketch 2 3 6x y - = as a broken line as it is not included in the

region

To find which side of the line gives x y 2 3 61- test a point on one

side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g

Any point in the region will

make the inequality true

Test one to see this




This means that 0 1^ h lies in the region x y 2 3 61-

3 x y 12 22+

Solution

The equation 1x y 2 2+ = is a circle radius 1 and centre 0 0^ h

Draw 1x y 2 2+ = as a broken line since the region does not include

the curve

Choose a point inside the circle say 0 0^ h x y 1

0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+

So the region lies outside the circle

4 y x2$

SolutionThe equation y x2

= is a parabola Sketch this as an unbroken line as it is

included in the region

2 x -3 y = 6

CONTINUED




Choose a point inside the parabola say 1 3^ h

3 1

y x

3 1

(true)

2

22

2

$

So 1 3^ h lies in the region

Sometimes a region includes two or more inequalities When this

happens sketch each region on the number plane and the final region is

where they overlap (intersect)

EXAMPLE

Sketch the region x y 4 22 - and y x2

Solution

Draw the three regions either separately or on the same set of axes and

see where they overlap

y = x 2




If you are given a region you should also be able to describe it

algebraically

Put the three regions together

EXAMPLES

Describe each region 1 y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

SolutionThe shaded area is below and including 6 y = so can be described as

y 6

It is also to the left of but not including the line 4x = which can be

described as x 41

The region is the intersection of these two regions

y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution

The shaded area is the interior of the circle centre (0 0) and radius 2 but

it does not include the circle

The equation of the circle is 2 4x y x y or2 2 2 2 2+ = + =

You may know (or guess) the inequality for the inside of the circle

If you are unsure choose a point inside the circle and substitute into the

equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g

So the region is x y 42 21+

511 Exercises

1 Shade the region defined by

(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-




2 Write an inequation to describe

each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x

3 Shade each region described

(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31

4 Describe as an inequality

the set of points that lie(a)

below the line y x3 2= -

the set of points that lie(b)inside the parabola 2 y x2

= +

the interior of a circle with(c)

radius 7 and centre (0 0)

the exterior of a circle with(d)


the set of points that lie to(e)

the left of the line 5x = and

above the line 2 y =




5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h

6 Shade the intersection of these

regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -

7 Shade the region bounded by

the curve(a) y x2= the x -axis

and the lines 1x = and 3x =

the curve(b) y x3= the y -axis

and the lines 0 y = and 1 y =

the curve(c) 4x y 2 2+ = the

x -axis and the lines 0x = and

1x = in the first quadrant

the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =

8 Shade the regions bounded by

the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $

The 1047297rst quadrant is

where x and y values

are both positive

Application

Regions are used in business applications to 1047297nd optimum pro1047297t Two (or more)

equations are graphed together and the region where a pro1047297t is made is shaded

The optimum pro1047297t occurs at the endpoints (or vertices) of the region

EXAMPLE

A company makes both roller skates ( X ) and ice skates (Y ) Roller skates make a

$25 pro1047297t while ice skates make a pro1047297t of $21 Each pair of roller skates spends

2 hours on machine A (available 12 hours per day) and 2 hours on machine B

(available 8 hours per day) Each pair of ice skates spends 3 hours on machine A

and 1 hour on machine B

How many skates of each type should be made each day to give the greatest

pro1047297t while making the most ef1047297cient use of the machines




SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=

Sketch the regions and 1047297nd the

point of intersection of the lines

The shaded area shows all possible ways of making a pro1047297t Optimum pro1047297t

occurs at one of the endpoints of the regions

(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g

3 2^ h gives the greatest pro1047297t so 3 pairs of roller skates and 2 pairs of iceskates each day gives optimum pro1047297t




Test Yourself 5

1 If f x x x3 42= - -] g find

(a) f 2-] g (b) f a] g (c) x when f x 0=] g

2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +

3 Find the domain and range of each graph

in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find

(a) f 2] g (b) f 3-] g (c) f 3] g (d) f 5] g (e) f 0] g

6 Shade the region y x2 1$ +

7 Shade the region where x y 3 1and1 $ -

8 Shade the region given by x y 12 2$+

9 Shade the region given by

2 3 6 0 2x y xand $+ - -

10 Shade the region y x 12 + and

x y 2+

11 Describe each region

(a)

(b)

(c)

12 (a) Write down the domain and range of

the curve3

2 y

x=

-

(b) Sketch the graph of3

2 y

x=

-




13 (a) Sketch the graph | 1 | y x= +

From the graph solve(b)(i) | 1 | 3x + =

(ii) | |x 1 31+

(iii) | |x 1 32+

14 If f x x3 4= -] g find

(a) f 2] g (b) x when f x 7=] g

(c) x when f x 0=] g


(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -

16 State which functions are (i) even

(ii) odd (iii) neither even nor odd

(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+

18 Sketch 10 log y y x y xandx= = = on the

same number plane

19 (a) State the domain and range of

2 4 y x= -

Sketch the graph of(b) 2 4 y x= -

20 Show that

(a) f x x x3 14 2= + -] g is even

(b) f x x x3= -] g is odd

Challenge Exercise 5

1 Find the values of b if f x x x3 7 12= - +] g

and f b 7=] g

2 Sketch 2 1 y x 2= + -] g in the domain

x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant

4 Draw the graph of | | 3 4 y x x= + -

5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]

Find f f f 3 4 0-] ] ]g g g and sketch the

curve


1

1 y

x2=

-

7 Sketch the region x y x y 2 61 1+

2 4 0x y $+ -

8 Find the domain and range of x y 2

= inthe first quadrant

9 If f x x x x2 2 123 2= - -] g find x when

f x 0=] g

10 Sketch the region defined by y x 2

12

+

in the first quadrant




11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )

find the value of h h h2 1 0-+ -] ] ]g g g and

sketch the curve

12 Sketch 1 y x2= - in the first quadrant

13 Sketch the region y x y x x5 21$ - +

14 If f x x2 1= -] g show that2( )f a f a2

= -^ _h i for all real a

15 Find the values of x for which f x 0=

] g when f x x x2 52

= - -] g (give exact

answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+

(c) Hence sketch the graph of

32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g

20 What is the domain of4

1 y x2

=

-

21 Sketch f xx

11

2= -] g






INTRODUCTION

FUNCTIONS AND THEIR GRAPHS are used in many areas such as mathematics

science and economics In this chapter you will study functions function

notation and how to sketch graphs Some of these graphs will be studied inmore detail in later chapters

DID YOU KNOW

The number plane is called the Cartesian plane after Rene

Descartes (1596ndash1650) He was known as one of the 1047297rst

modern mathematicians along with Pierre de Fermat

(1601ndash1665) Descartes used the number plane to develop

analytical geometry He discovered that any equation

with two unknown variables can be represented by a lineThe points in the number plane can be called Cartesian

coordinates

Descartes used letters at the beginning of the

alphabet to stand for numbers that are known and letters

near the end of the alphabet for unknown numbers This is

why we still use x and y so often

Do a search on Descartes to 1047297nd out more details of

his life and work

Descartes

Functions

Definition of a function

Many examples of functions exist both in mathematics and in real life These

occur when we compare two different quantities These quantities are called

variables since they vary or take on different values according to some pattern

We put these two variables into a grouping called an ordered pair




EXAMPLES

1 Eye colour





2 y x 1= +

x 1 2 3 4

y 2 3 4 5


3

A

B

C

D

E

1

2

3

4
















variable) say x 3=

When x

y

3

3 1

4

=

= +

=







x and the dependent





Investigation










A

B

C

D

E

1

2

3

4


and (E 2)


NOT a function





1

x

y

2

x

y




x

y





graph is a function

y

x



x

y





place





EXAMPLES


Solution



Solution


not a function







2 3 1 4 0 5 1 3 2 4- -^ ^ ^ ^ ^h h h h h Solution


function


y

x

3

Solution

y

x

3





a function




1

2

3

4

5

6

7

8

9 1 3 2 1 3 3 4 0-^ ^ ^ ^h h h h 10 1 3 2 1 2 7 4 0-^ ^ ^ ^h h h h 11

1

2

3

4

5

1

2

3

4

5

12 1

2

3

4

5

1

2

3

4

5

131

2

3

4

5

1

2

3

4

5

51 Exercises







15 A 3

B 4

C 7

D 3

E 5

F 7

G 4

Function notation




EXAMPLES


Solution

When x

y x

3

1

3 1

4

=

= +

= +

=


Solution

f x x

f

1

3 3 1

4

= +

= +

=

]] gg





EXAMPLES

1 If f x x x3 12= + +] g find f 2-] g

Solution

( ) ( )f 2 2 3 2 1

4 6 1

1

2- = - + - +

= - +

= -

] g


Solution

( )

( )

f x x x

f 1 1 1

1 12

3 2

3

= -

- = - - -

= - -

= -

2] ]g g


Solution

( )

ie

( ) ( )

f x

x x

x x

x x

x x

0

3 10 0

5 2 0

5 0 2 0

5 2

2

=

+ - =

+ - =

+ = - =

= - =


when

when f x

x x

x x

3 4 2

2 21

$=

+

-

] g )

Solution

since 4 21-

( ) ( ) since

( ) ( ) since

( ) ( ) since

( ) ( )

f

f

f

f

3 3 3 4 3 2

13

2 3 2 4 2 2

10

0 2 0 0 2

0

4 2 4

8

1

$

$

= +

=

= +

=

= -

=

- = - -

=


when

when

x

x

x

2

1 2

1

2

1

-

-

g x

x

x2 1

5

2

= -] g


when 2x -=







than 2




Solution

( ) ( )

( )

( )

g

g

g

1 2 1 1 1 1 2

1

2 5 2 1

3 3 3 2

9

since

since

since2

1

2

= - -

=

- = - -

=

=

( ) ( ) ( ) g g g 1 2 3 1 5 9

3

So + - - = + -

= -

DID YOU KNOW






f 3-] g

2 If h x x 22= -] g find h h0 2] ]g g

and h 4-] g


and f 2-] g


= - +] g

5 Find f 3-] g if f x x x2 5 43= - +] g


f x 13=] g




f x271

=] g




f x h+] g


g x x x2 32= + +] g


product of factors






15 f x x x

x x

1

1

for

for

32

=] g )


16 f x

x x

x x

x x

2 4 1

3 1 1

1

if

if

if 2

1 1

$

=

-

+ -

-

] gZ

[

]]

]]

Find the values of

f f f 2 2 1- - + -] ] ]g g g

52 Exercises








EXAMPLE


Solution


x x

x x

x x

x x

0 7 8

8 1

8 0 1 0

8 1

2= + -

= + -

+ = - =

= - =

] ]g g


y 0 7 0 8

8

2= + -

= -

] ]g g





Domain and range




domain


image) of f

EXAMPLE


Solution


x

y

CONTINUED






Domain all real x

Range y y 0$















EXAMPLES


x x 3 x 2 x 1

y

Solution


increasing


x 1 x 2

2

x x 3

x 2 x 1

y

Solution



increases again




1 1








=







x

y









x

y




EXAMPLES


Solution

f x x

x

f x

f x x

3

3


2

2

2`

- = - +

= +

=

= +

] ]]

]

g gg

g


Solution

f x x x

x x

x x

f x


3

3

3

3`

- = - - -

= - +

= - -

= -

= -

] ] ]^]

]

g g gh

gg

Investigation






1 f x kxn=] g

2 f x x kn= +] g

3 f x x k n= +

] ]g g






shape of graphs


each function

(a) y x3 2= -

(b) x y 2 5 20 0- + =

(c) x y 3 12 0+ - =

(d) f x x x32= +] g

(e) f x x 42= -] g

(f) p x x x5 62= + +] g

(g) y x x8 152= - +

(h) p x x 53

= +

] g

53 Exercises




(i) y xx

x3

0= + ] g

(j) g x x9 2= -] g


f x x 22= -

] g What type of

function is it


(a) f x2^ h (b) ( )f x 26



an even function


f x x=] g


function


odd function



f x f x- -] ]g g


neither

(a) y x x

x4 2

3

=

-

(b) y x 1

13

=

-

(c) f xx 4

32

=

-

] g

(d) y x

x

33

=+

-

(e) f x x x

x5 2

3

=-] g



f x xn=] g

even(a)

odd(b)


ever be

even(a)

odd(b)



graph is increasing




x

y(a)

x

4

y(b)

2-2

x

y(c)




Investigation


graph







y kx

y kx

y kx

y kx

y xk

(a)

(b)

(c)

(d)

(e)

2

3

4

=

=

=

=

=



y x k

y x k

y x k

y x k

y x k1

(a)

(b)

(c)

(d)

(e)

2

2

3

4

= +

= +

= +

= +

= +

] g



-2

1 2

-4

-1-2

2

4

y

x

(d) y

x

(e)

CONTINUED




Gradient form



Investigation










y x k

y x k

y x k

y x k y

x k

1

(a)

(b)

(c)

(d)(e)

2

3

4

= +

= +

= +

= +

=+

]]

]

gg

g





the right








Linear Function





EXAMPLE



Solution


Find the intercepts


0 3 5

5 3

1

x

x

x32

=

=

=

-


3 5

5

y 0=

= -

-] g

-1

-2

y

5

4

3

2

1 1 23

6

-3

-4

-5

1 4-1-2 32-3-4

x



Range all real y








Special lines






1 y kx=

2 y x k= +






EXAMPLES


Solution




-1

-3

y

4

3

2

1

5

-2

-4

-5

1 4-1-2

x

32-3-4


2 Sketch x 1= -


Solution





-

-

4

3

2

5

-2

-4

-5

1 4-1-2-4

y

x







Domain all real x

Range y y b=

54 Exercises


each function

(a) y x 2= -

(b) f x x2 3= +] g (c) x y 2 1 0+ =-

(d) x y 3 0+ =-

(e) x y 3 6 2 0=- -


line

(a) x 4=

(b) x 3 0=-

(c) y 5=

(d) y 1 0+ =

(e) f x x2 1= -] g (f) y x 4= +

(g) f x x3 2= +] g

(h) x y 3+ =

(i) x y 1 0=- -

(j) x y 2 3 0+ =-


(a) x y 3 2 7 0+ =-

(b) y 2=

(c) x 4= - (d) x 2 0=-

(e) y 3 0=-


are even or odd

(a) y x2=

(b) y 3=

(c) x 4=

(d) y x= -

(e) y x=




meet




Applications






The pronumeral

a is called the


Quadratic Function










EXAMPLES



Solution



x

x

x

0 1

1

1

2

2

= -

=

=


0 1

1

y 2= -

= -



Domain xall real

Range y y 1$ -


Solution


parabola will lie




left



0

1 0

1

x

x

x

1 2= +

+ =

= -

] g


1

y 0 1 2= +

=

] g

-1

-

4

3

2

1

5

-2

-4

-5

-

1 41-2 5-4

y

x









Solution



x x

x xx x

x x

0 6

3 23 0 2 0

3 2

2= + -

= + -

+ = - =

= - =

] ]g g


f 0 0 0 66

2= + -

= -

] ] ]g g g



x -intercepts


23 2

21- +

= -


-c m

f 21

21

21

6

41

21

6

641

2

- = - + - -

= - -

= -

c c cm m m


-

CONTINUED

You will learn more


-1

-3

4

3

2

1

5

-2

-4

-5

1 4-1-2 32-3-4

y

x










41



2

11 -





Solution


So f x 0=] g

0 4 54 5 0

0

x xx x

x x5 1

2

2

= - + +

=

+ =

- -

-] ]g g

x x

x x

5 0 1 0

5 1

- = + =

= = -


f 0 0 4 0 5

5

2= - + +

=

] ] ]g g g

-1

-3

4

2

5

-2

-4

-5

-

y

1 41-2 5-4

x

-2 4






21 5

2- +

=

f 2 2 4 2 59

= - + +

=

2] ] ]g g g










each function

(a) 2 y x x2= +

(b) 3 y x x2= - +

(c) f x x 12= -] g

(d) y x x 22= - -

(e) y x x9 82= +-

2 Sketch

(a) 2 y x2= +

(b) y x 12= - +

(c) f x x 42= -] g

(d) 2 y x x2= +

(e) y x x2= - -

(f) f x x 3= -2] ]g g

55 Exercises

-1

9

8

7

5

4

3

2

6

1

-2

-3

-4

-5

y

2 51 643-1-2-3-4

x




EXAMPLES


Solution









(g) f x x 1 2= +] ]g g

(h) y x x3 42= + -

(i) y x x2 5 32= - +

(j) f x x x3 22= - + -] g



(a) ndash y x x7 122= +

(b) f x x x42= +] g

(c) y x x2 82= - -

(d) y x x6 92= +-

(e) f t t 4 2= -] g


(a) y x 52= -

(b) f x x x6

2= -

] g (c) f x x x 22= - -] g

(d) y x2= -

(e) f x x 7 2= -] ]g g



(a) y x2= for x0 3

(b) y x 42= - + for x1 2 -

(c) f x x 12= -] g for x2 5 -

(d) y x x2 32= + - for x2 4 -

(e) y x x

22= - +- for

x0 4


function is

increasing(i)

decreasing(ii)

(a) y x2=

(b) y x2

= - (c) f x x 92

= -] g

(d) y x x42= - +

(e) f x x 5 2= +] ]g g


function



(a) y x 12= +

(b) f x x 32= -

] g

(c) y x2 2= -

(d) f x x x32= -] g

(e) f x x x2= +] g

(f) y x 42= -

(g) y x x2 32= - -

(h) y x x5 42= +-

(i) p x x 1 2= +] ]g g

(j) y x 2= -2] g




CONTINUED

eg When x 3= -

| | y 3 13 12

= - -

= -

=

x -3 -2 -1 0 1 2 3

y 2 1 0 -1 0 1 2


y

-2

4

3

2

1

5

-1

-3

-4

-5

1 4-1-2 32-3-4

x


| | y x

x x

x x1

1 0

1 0

when

when 1

$= - =

-


y x x1 0$= - ] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x - 1




y x 1= - - x 01] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y =- x - 1



-3

4

3

2

1

5

-2

-1

-4

-5

y y

3-1-2 421-3-4

x

y = - x - 1

y = x - 1


intercepts


So f x 0=

] g

| |

| |

x

x

x

0 1

1

1`

= -

=

=


( ) | |f 0 0 11

= -

= -





-3

4

3

2

1

5

-2

-1

-4

-5

y

4-2 5321-1-3-4

x



Domain all real x

Range y y 1$ -

2 Sketch | | y x 2= +

Solution


| | ( ) y x x xx x2 2 2 0

2 2 0whenwhen 1

$= + = + +

- + +


2 y x= + when x 2 0$+

x 2$ -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x + 2






y-intercept of 1-

CONTINUED




2 y x= - +] g when x 2 01+

ie y x 2= - - when x 21 -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2


-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2

y = x + 2



So 0f x =] g

0 | 2 |

0 2

2

x

x

x

= +

= +

- =


(0) | 0 2 |

2

f = +

=

There is only one

solution for the


Can you see why





-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

If you know how to

sketch the graph of




Investigation




are their equations





1 | |f x k x=] g

2 | |f x x k= +] g

3 | |f x x k= +] g







EXAMPLES

Solve

1 |2 1 | 3x - =

Solution



is x 1 2= -

2 |2 1 | 3 2x x= -+

Solution



3 | 1 | 2x 1+

Solution






and then check them







each function

(a) | | y x=

(b) | |f x x 7= +] g

(c) | |f x x 2= -] g

(d) 5 | | y x=

(e) | |f x x 3= - +] g

(f) | 6 | y x= +

(g) | |f x x3 2= -] g

(h) | 5 4 | y x= +

(i) | 7 1 | y x= -

(j) | |f x x2 9= +] g


plane

(a) | | y x=

(b) | |f x x 1= +] g

(c) | |f x x 3= -] g

(d) 2 | | y x=

(e) | |f x x= -] g

(f) | 1 | y x= +

(g) | |f x x 1= - -] g

(h) | 2 3 | y x= -

(i) | 4 2 | y x= +

(j) | |f x x3 1= +] g


each function

(a) | 1 | y x= -

(b) | |f x x 8= -] g

(c) | |f x x2 5= +] g

(d) 2 | | 3 y x= -

(e) | |f x x 3= - -] g


function is

increasing(i)

decreasing(ii)

(a) | 2 | y x= -

(b) | |f x x 2= +

] g

(c) | |f x x2 3= -] g

(d) 4 | | 1 y x= -

(e) | |f x x= -] g


of each function

(a) | | y x= for x2 2 -

(b) | |f x x 4= - -] g for

x4 3 -

(c) | |f x x 4= +] g for x7 2 -

(d) | 2 5 | y x= - for x3 3 -

(e) | |f x x= -] g for x1 1 -


function increasing

(a) | 3 | y x= +

(b) | |f x x 4= - +] g

(c) | |f x x 9= -] g

(d) | | y x 2 1= - -

(e) | |f x x 2= - +] g

56 Exercises




7 Solve graphically

(a) | | 3x =

(b) | |x 12

(c) | |x 2

(d) | 2 | 1x + =

(e) | 3 | 0x- =

(f) |2 3 | 1x - =

(g) | |x 1 41-

(h) | |x 1 3+

(i) | |x 2 22-

(j) | |x 3 1$-

(k) | |x2 3 5+

(l) | |x2 1 1$-

(m) |3 1 | 3x x- = +

(n) |3 2 | 4x x- = -

(o) | 1 | 1x x- = +

(p) | 3 | 2 2x x+ = + (q) |2 1 | 1x x+ = -

(r) |2 5 | 3x x- = -

(s) | 1 | 2x x- =

(t) |2 3 | 3x x- = +

The Hyperbola


or= =

EXAMPLE

Sketch1

y x=

Solution



x -3 -2 -121-

41- 0

41

21 1 2 3

y 3

1-

2

1- -1 -2 -4 mdash 4 2 1

2

1

3

1

Class Discussion



y is never 0 Why






but never touches)



Investigation




equations




1 y xk

=

21

y x k= +

31

y x k

=+


EXAMPLES


3=

-

] g


Solution



So x 3






be zero

CONTINUED







1

y 0 3

3=

-

= -

-3

4

3

2

1

5

-2

-1

-4

-5

y

-1-2 4 521-3-4

x

x = 3

y = 0

Asymptotes

3

2 Sketch y x2 4

1= -

+

Solution




x

x

2 4

2

-

-






( ) y

2 0 41

41

= -+

= -


a translation of3

y x

=





y

-2

x

-

1

4


a=

+


domain x xb



1 For each graph



exists


(a)2

y x=

(b)1

y x= -

(c) f xx 1

1=

+] g

(d) f xx 2

3=

-

] g

(e)3 6

1 y

x=

+

(f) f xx 3

2= -

-

] g

(g) f xx 1

4=

-

] g

(h)1

2 y

x= -

+

(i) f xx6 3

2=

-

] g

(j)2

6 y

x= -

+

2 Show that f x x2

=] g is an odd

function



(a) f xx2 5

1=

+

] g for x2 2 -

(b)3

1 y

x=

+ for x2 0 -

(c) f xx2 4

5=

-

] g for x3 1 -

57 Exercises




(d) f xx 4

3= -

-

] g for x3 3 -

(e)3 1

2 y

x= -

+ for x0 5



(a)3

y x= for y 1 3

(b)2

y x= - for y 221

- -

(c) f xx 1

1=

-

] g for y 171

- -

(d) f xx2 1

3= -

+

] g for

y 131

- -

(e)3 2

6 y

x=

- for y 1

21

6



Circle gate





Proof

r y

x

( x y)

y

x








c a b

r x y

2 2 2

2 2 2

`

= +

= +

EXAMPLE



Solution


y

x

-2

-2 2

2


one place

y

x

2

2

2

-2

The radius is 4

CONTINUED






Domain 2 2 x x -

Range 2 2 y y -


range y r y r -



Proof


shown

y

x

(a b)

x

y

r

( x y)

a

b x - a

y - b




ndash ndash

c a b

r x a y b

2 2 2

2 2 2

= +

= +] ^g h




EXAMPLES



Solution



y

x 9

9

-9

-9



Domain 9 9 x x -

Range 9 9 y y -



Solution


ndash

ndash ndash

x y

x y

1 2 4

1 2 2

2

2 2

+ + =

+ - =

2

2

] ^

] ]_

g h

g gi

So 1 2a b= = - and 2r =

CONTINUED







y

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

(1 -2)



Domain 1 3 x x -

Range 4 0 y y -


Solution


a b2 1= - = and r 3=

Substituting

ndash ndash

ndash

ndash

x a y b r

x y

x y

2 1 3

2 1 9

2 2 2

2 2 2

2 2

+ =

- - + =

+ + =

] ^]] ^

] ^

g hg g hg h


ndash

ndash

a b a ab b

x x x

x x

a b a ab b

y y y

y y

2

2 2 2 2

4 4

2

1 2 1 1

2 1

So

So

2 2 2

2 2 2

2

2 2 2

2 2 2

2

+ = + +

+ = + +

= + +

= - +

= - +

= - +

]] ] ]]^ ^ ]

gg g ggh h g


ndash

x x y y

x x y y

x x y y

x x y y

4 4 2 1 9

4 2 5 9

4 2 5 9

4 2 4 0

9 9

2

2

2

2

+ + + - + =

+ + - + =

+ + + =

+ + - - =

- -


in Chapter 2




Investigation


two functions





circle






for all values

y

x r

r

-r


Proof

ndash

x y r

y r x y r x

2 2 2

2 2 2

2 2

+ =

=

= -



semi-circles





y 0 for all values

y

x r

-

r

-r


EXAMPLES


1 f x x92

= -] g

Solution



y

x 3

3

-3

Domain 3 3 x x -

Range 0 3 y y




2 y x4 2= - -

Solution



y

x 2

-2

-2

Domain 2 2 x x -

Range 2 0 y y -




range

(a) 9x y 2 2+ =

(b) x y 16 02 2+ =-


(d) 1 9x y 2 2+ + =

] g

(e) ndashx y 2 1 12 2+ + =] ^g h



below the x -axis



range

(a) 25 y x2= - -

(b) 1 y x2= -

(c) 36 y x2= -

(d) 64 y x2= - -

(e) 7 y x2= - -



each circle

(a) 100x y 2 2+ =

(b) 5x y 2 2+ =


(d) ndashx y 5 6 492 2+ + =] ^g h

(e) ndashx y 3 812 2+ =^ h

58 Exercises






grouping symbols)



Centre(c) 1 5-







Centre(j) 4 7- -

^ h and radius 3

Other Graphs




EXAMPLES


Solution



eg When 0x =

y 3

1

c=

=

x

y

1

3

3

1

31

When1

1

= -

=

=

=

-

x 3- 2- 1- 0 1 2 3

y 271

91

31

1 3 9 27



You will meet these

graphs again in the

HSC Course





other point

When

x

y

1

3

3

1

=

=

=

y

x

1

2

1

3







Solution



x minus2 minus1 0 05 1 2 3 4

y minus03 0 03 05 06

y

x

1

2

1 2 3 4

-1




You learned about


stages of maths





range y y 02



range all real y

Cubic function


EXAMPLE



Solution



y minus25 minus6 1 2 3 10 29

y

x

1

1

-2 2 3 4

-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4



If you already

know the shape of


3






Domain and range


EXAMPLE


Solution




function are

f

f

1 1 2

1 21

4 4 2

64 2

66

3

3

- = - +

= - +

=

= +

= +

=

] ]

] ]

g g

g g


these points

y

x

(-1 1)

(4 66)

Range 1 66 y y










positive number

EXAMPLE


Solution


x 4So $

$


(or zero) So y 0$


59 Exercises


(a) 4 3 y x= +

(b) f x 4= -] g

(c) 3x =



(f) f x xx 12 2= - -] g

(g) 64x y 2 2+ =

(h) f t t 4

3=

-] g

(i) ( ) g 2

5zz

= +

(j) | |f x x=] g


(a) y x=

(b) 2 y x= -

(c) | |f x x2 3= -] g

(d) | | 2 y x= -

(e) f x x2 5= - +] g

(f) | | y x5= -

(g) 2 y x=

(h) y 5x= -

(i) f x xx 1

= +] g

(j)2

4 3 y x

x= -


(a) y x x 5 2= -] g


= +-

(d) g x x x164 2= -] g

(e) 49x y 2 2+ =

You may like to


by dividing by x




4 (a) Solve x1 02$-


f x x1 2= -] g


(a) 2 y x x2= - -

(b) g t t t 62= +] g



in each case


x3 3 -


x2 3 -


x2 1 - (d)


x1 5


0 4x


x3 3 -


x1 1 -


x2 3 -


x4 4 -

(j) y x x7 62= - + - in the

domain 0 7x


function y x 1

3=

+





x=] g


function

find its range(b)


plane

(a) f x x4=] g

(b) y x3= -

(c) y x 34= -

(d) 2 p x x3=] g

(e) 1 g x x3= +] g

(f) 100x y 2 2+ =

(g) 2 1 y x= +


y x 1= -









(b) f x x 22= -] g

(c)1

y x=

(d) f x x3=] g

(e) f x 3x=] g

13 (a) Solve x4 02$-


(i) 4 y x2= -

(ii) y x4

2= - -




DID YOU KNOW




a light



Limits






a a1

1

03

Z

=

=

3

3

-


lim a 0x=

x 3-



asymptote

EXAMPLES

1 Find lim x

x x5x 0

2+

Solution


which is undefined


( )

lim lim

lim

xx x

x

x x

x

5 5

5

5

x x

x

0

2

0 1

1

0

+=

+

= +

=

] g




2 Find limx

x

4

22

-

-

x 2

Solution


lim lim

lim

x

x

x x

x

x

4

2

2 2

2

21

41

2 1

1

-

-=

+ -

-

=+

=

x x2 2

x 2

^ _h i

3 Find limh

h x hx h2 72 2+ -

h 0

Solution

lim lim

lim

h

h x hx h

h

h hx x

hx x

x

2 7 2 7

2 7

7

2 2 2

2

2

+ -=

+ -

= + -

= -

h 0

h h0 0

^ h

Continuity





EXAMPLES

1 Find lim

x

x

1

12

-

-

x 1


11

y x

x2

=-

- Sketch the curve

Solution


xx2

-

- gives

00

CONTINUED




( )

lim lim

limx

x

x

x x

x11

1

1 1

1

2

x x

x

1

2

1

1

-

-=

-

+ -

= +

=

-

] ]g g

11

y xx2

=-





y xx

x

x x

x

11

1

1 1

1

2

=-

-

=

+

= +

-

-^ ^h h


2 Find limx

x x2

2x 2

2

+

+ -

-


x x2

22

= + -

+

Solution


x x2

22

+

+ - gives

00

lim lim

lim

xx x

x

x x

x

22

2

1 2

1

3

x x

x

2

2

2

2

+

+ -=

+

- +

=

= -

-

- -

-

^^ ^

^

hh h

h2 y

x

x xx

y x

x

x

x

22

2

2

1

1


=+

+ -= -

=+

= -

+ -^ ^h h


Remember that x 1




1 Find(a) lim x 52

+x 4

(b) lim t 7-t 3 -

(c) lim x x2 43+ -

x 2

(d) lim xx x32

+

x 0

(e) limh

h h

2

22

-

- -

h 2

(f) lim y

y

5

1253

-

-

y 5

(g) limx

x x

12 12

+

+ +

x 1-

(h) limx

x x4

2 82

+

+ -

x 4 -

(i) limc

c

4

22

-

-

c 2

(j) limx x

x 12

-

-

x 1

(k) lim h

h h h2 73 2+ -

h 0

(l) limh

hx hx h32 2- +

h 0

(m) limh

hx h x hx h2 3 53 2 2- + -

h 0

(n) lim x c x c 3 3

-

-

x c



discontinuous

(a) 3 y x2= -

(b)1

1 y

x=

+

(c) f x x 1= -] g

(d)4

1 y

x2=

+

(e)4

1

y x2=

-



(a)3

y xx x2

= +

(b)33

y x

x x2

=+

+

(c)1

5 4 y

xx x2

=+

+ +

510 Exercises




Regions

Class Investigation


them all







EXAMPLE


Solution



y

x =3

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4







region

EXAMPLE


Solution



included


shown

y

x

1

1

-2 2 3 4

-3

-4

-5

2

3

4

5

-1

-2

-3-4

-1 y = -1



y -intercept 1-






EXAMPLES


1 y x 2$ +

Solution






y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0



2 x y 2 3 61-

Solution


region


side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g








3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g






EXAMPLES

1 Eye colour





2 y x 1= +

x 1 2 3 4

y 2 3 4 5


3

A

B

C

D

E

1

2

3

4
















variable) say x 3=

When x

y

3

3 1

4

=

= +

=







x and the dependent





Investigation










A

B

C

D

E

1

2

3

4


and (E 2)


NOT a function





1

x

y

2

x

y




x

y





graph is a function

y

x



x

y





place





EXAMPLES


Solution



Solution


not a function







2 3 1 4 0 5 1 3 2 4- -^ ^ ^ ^ ^h h h h h Solution


function


y

x

3

Solution

y

x

3





a function




1

2

3

4

5

6

7

8

9 1 3 2 1 3 3 4 0-^ ^ ^ ^h h h h 10 1 3 2 1 2 7 4 0-^ ^ ^ ^h h h h 11

1

2

3

4

5

1

2

3

4

5

12 1

2

3

4

5

1

2

3

4

5

131

2

3

4

5

1

2

3

4

5

51 Exercises







15 A 3

B 4

C 7

D 3

E 5

F 7

G 4

Function notation




EXAMPLES


Solution

When x

y x

3

1

3 1

4

=

= +

= +

=


Solution

f x x

f

1

3 3 1

4

= +

= +

=

]] gg





EXAMPLES

1 If f x x x3 12= + +] g find f 2-] g

Solution

( ) ( )f 2 2 3 2 1

4 6 1

1

2- = - + - +

= - +

= -

] g


Solution

( )

( )

f x x x

f 1 1 1

1 12

3 2

3

= -

- = - - -

= - -

= -

2] ]g g


Solution

( )

ie

( ) ( )

f x

x x

x x

x x

x x

0

3 10 0

5 2 0

5 0 2 0

5 2

2

=

+ - =

+ - =

+ = - =

= - =


when

when f x

x x

x x

3 4 2

2 21

$=

+

-

] g )

Solution

since 4 21-

( ) ( ) since

( ) ( ) since

( ) ( ) since

( ) ( )

f

f

f

f

3 3 3 4 3 2

13

2 3 2 4 2 2

10

0 2 0 0 2

0

4 2 4

8

1

$

$

= +

=

= +

=

= -

=

- = - -

=


when

when

x

x

x

2

1 2

1

2

1

-

-

g x

x

x2 1

5

2

= -] g


when 2x -=







than 2




Solution

( ) ( )

( )

( )

g

g

g

1 2 1 1 1 1 2

1

2 5 2 1

3 3 3 2

9

since

since

since2

1

2

= - -

=

- = - -

=

=

( ) ( ) ( ) g g g 1 2 3 1 5 9

3

So + - - = + -

= -

DID YOU KNOW






f 3-] g

2 If h x x 22= -] g find h h0 2] ]g g

and h 4-] g


and f 2-] g


= - +] g

5 Find f 3-] g if f x x x2 5 43= - +] g


f x 13=] g




f x271

=] g




f x h+] g


g x x x2 32= + +] g


product of factors






15 f x x x

x x

1

1

for

for

32

=] g )


16 f x

x x

x x

x x

2 4 1

3 1 1

1

if

if

if 2

1 1

$

=

-

+ -

-

] gZ

[

]]

]]

Find the values of

f f f 2 2 1- - + -] ] ]g g g

52 Exercises








EXAMPLE


Solution


x x

x x

x x

x x

0 7 8

8 1

8 0 1 0

8 1

2= + -

= + -

+ = - =

= - =

] ]g g


y 0 7 0 8

8

2= + -

= -

] ]g g





Domain and range




domain


image) of f

EXAMPLE


Solution


x

y

CONTINUED






Domain all real x

Range y y 0$















EXAMPLES


x x 3 x 2 x 1

y

Solution


increasing


x 1 x 2

2

x x 3

x 2 x 1

y

Solution



increases again




1 1








=







x

y









x

y




EXAMPLES


Solution

f x x

x

f x

f x x

3

3


2

2

2`

- = - +

= +

=

= +

] ]]

]

g gg

g


Solution

f x x x

x x

x x

f x


3

3

3

3`

- = - - -

= - +

= - -

= -

= -

] ] ]^]

]

g g gh

gg

Investigation






1 f x kxn=] g

2 f x x kn= +] g

3 f x x k n= +

] ]g g






shape of graphs


each function

(a) y x3 2= -

(b) x y 2 5 20 0- + =

(c) x y 3 12 0+ - =

(d) f x x x32= +] g

(e) f x x 42= -] g

(f) p x x x5 62= + +] g

(g) y x x8 152= - +

(h) p x x 53

= +

] g

53 Exercises




(i) y xx

x3

0= + ] g

(j) g x x9 2= -] g


f x x 22= -

] g What type of

function is it


(a) f x2^ h (b) ( )f x 26



an even function


f x x=] g


function


odd function



f x f x- -] ]g g


neither

(a) y x x

x4 2

3

=

-

(b) y x 1

13

=

-

(c) f xx 4

32

=

-

] g

(d) y x

x

33

=+

-

(e) f x x x

x5 2

3

=-] g



f x xn=] g

even(a)

odd(b)


ever be

even(a)

odd(b)



graph is increasing




x

y(a)

x

4

y(b)

2-2

x

y(c)




Investigation


graph







y kx

y kx

y kx

y kx

y xk

(a)

(b)

(c)

(d)

(e)

2

3

4

=

=

=

=

=



y x k

y x k

y x k

y x k

y x k1

(a)

(b)

(c)

(d)

(e)

2

2

3

4

= +

= +

= +

= +

= +

] g



-2

1 2

-4

-1-2

2

4

y

x

(d) y

x

(e)

CONTINUED




Gradient form



Investigation










y x k

y x k

y x k

y x k y

x k

1

(a)

(b)

(c)

(d)(e)

2

3

4

= +

= +

= +

= +

=+

]]

]

gg

g





the right








Linear Function





EXAMPLE



Solution


Find the intercepts


0 3 5

5 3

1

x

x

x32

=

=

=

-


3 5

5

y 0=

= -

-] g

-1

-2

y

5

4

3

2

1 1 23

6

-3

-4

-5

1 4-1-2 32-3-4

x



Range all real y








Special lines






1 y kx=

2 y x k= +






EXAMPLES


Solution




-1

-3

y

4

3

2

1

5

-2

-4

-5

1 4-1-2

x

32-3-4


2 Sketch x 1= -


Solution





-

-

4

3

2

5

-2

-4

-5

1 4-1-2-4

y

x







Domain all real x

Range y y b=

54 Exercises


each function

(a) y x 2= -

(b) f x x2 3= +] g (c) x y 2 1 0+ =-

(d) x y 3 0+ =-

(e) x y 3 6 2 0=- -


line

(a) x 4=

(b) x 3 0=-

(c) y 5=

(d) y 1 0+ =

(e) f x x2 1= -] g (f) y x 4= +

(g) f x x3 2= +] g

(h) x y 3+ =

(i) x y 1 0=- -

(j) x y 2 3 0+ =-


(a) x y 3 2 7 0+ =-

(b) y 2=

(c) x 4= - (d) x 2 0=-

(e) y 3 0=-


are even or odd

(a) y x2=

(b) y 3=

(c) x 4=

(d) y x= -

(e) y x=




meet




Applications






The pronumeral

a is called the


Quadratic Function










EXAMPLES



Solution



x

x

x

0 1

1

1

2

2

= -

=

=


0 1

1

y 2= -

= -



Domain xall real

Range y y 1$ -


Solution


parabola will lie




left



0

1 0

1

x

x

x

1 2= +

+ =

= -

] g


1

y 0 1 2= +

=

] g

-1

-

4

3

2

1

5

-2

-4

-5

-

1 41-2 5-4

y

x









Solution



x x

x xx x

x x

0 6

3 23 0 2 0

3 2

2= + -

= + -

+ = - =

= - =

] ]g g


f 0 0 0 66

2= + -

= -

] ] ]g g g



x -intercepts


23 2

21- +

= -


-c m

f 21

21

21

6

41

21

6

641

2

- = - + - -

= - -

= -

c c cm m m


-

CONTINUED

You will learn more


-1

-3

4

3

2

1

5

-2

-4

-5

1 4-1-2 32-3-4

y

x










41



2

11 -





Solution


So f x 0=] g

0 4 54 5 0

0

x xx x

x x5 1

2

2

= - + +

=

+ =

- -

-] ]g g

x x

x x

5 0 1 0

5 1

- = + =

= = -


f 0 0 4 0 5

5

2= - + +

=

] ] ]g g g

-1

-3

4

2

5

-2

-4

-5

-

y

1 41-2 5-4

x

-2 4






21 5

2- +

=

f 2 2 4 2 59

= - + +

=

2] ] ]g g g










each function

(a) 2 y x x2= +

(b) 3 y x x2= - +

(c) f x x 12= -] g

(d) y x x 22= - -

(e) y x x9 82= +-

2 Sketch

(a) 2 y x2= +

(b) y x 12= - +

(c) f x x 42= -] g

(d) 2 y x x2= +

(e) y x x2= - -

(f) f x x 3= -2] ]g g

55 Exercises

-1

9

8

7

5

4

3

2

6

1

-2

-3

-4

-5

y

2 51 643-1-2-3-4

x




EXAMPLES


Solution









(g) f x x 1 2= +] ]g g

(h) y x x3 42= + -

(i) y x x2 5 32= - +

(j) f x x x3 22= - + -] g



(a) ndash y x x7 122= +

(b) f x x x42= +] g

(c) y x x2 82= - -

(d) y x x6 92= +-

(e) f t t 4 2= -] g


(a) y x 52= -

(b) f x x x6

2= -

] g (c) f x x x 22= - -] g

(d) y x2= -

(e) f x x 7 2= -] ]g g



(a) y x2= for x0 3

(b) y x 42= - + for x1 2 -

(c) f x x 12= -] g for x2 5 -

(d) y x x2 32= + - for x2 4 -

(e) y x x

22= - +- for

x0 4


function is

increasing(i)

decreasing(ii)

(a) y x2=

(b) y x2

= - (c) f x x 92

= -] g

(d) y x x42= - +

(e) f x x 5 2= +] ]g g


function



(a) y x 12= +

(b) f x x 32= -

] g

(c) y x2 2= -

(d) f x x x32= -] g

(e) f x x x2= +] g

(f) y x 42= -

(g) y x x2 32= - -

(h) y x x5 42= +-

(i) p x x 1 2= +] ]g g

(j) y x 2= -2] g




CONTINUED

eg When x 3= -

| | y 3 13 12

= - -

= -

=

x -3 -2 -1 0 1 2 3

y 2 1 0 -1 0 1 2


y

-2

4

3

2

1

5

-1

-3

-4

-5

1 4-1-2 32-3-4

x


| | y x

x x

x x1

1 0

1 0

when

when 1

$= - =

-


y x x1 0$= - ] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x - 1




y x 1= - - x 01] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y =- x - 1



-3

4

3

2

1

5

-2

-1

-4

-5

y y

3-1-2 421-3-4

x

y = - x - 1

y = x - 1


intercepts


So f x 0=

] g

| |

| |

x

x

x

0 1

1

1`

= -

=

=


( ) | |f 0 0 11

= -

= -





-3

4

3

2

1

5

-2

-1

-4

-5

y

4-2 5321-1-3-4

x



Domain all real x

Range y y 1$ -

2 Sketch | | y x 2= +

Solution


| | ( ) y x x xx x2 2 2 0

2 2 0whenwhen 1

$= + = + +

- + +


2 y x= + when x 2 0$+

x 2$ -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x + 2






y-intercept of 1-

CONTINUED




2 y x= - +] g when x 2 01+

ie y x 2= - - when x 21 -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2


-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2

y = x + 2



So 0f x =] g

0 | 2 |

0 2

2

x

x

x

= +

= +

- =


(0) | 0 2 |

2

f = +

=

There is only one

solution for the


Can you see why





-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

If you know how to

sketch the graph of




Investigation




are their equations





1 | |f x k x=] g

2 | |f x x k= +] g

3 | |f x x k= +] g







EXAMPLES

Solve

1 |2 1 | 3x - =

Solution



is x 1 2= -

2 |2 1 | 3 2x x= -+

Solution



3 | 1 | 2x 1+

Solution






and then check them







each function

(a) | | y x=

(b) | |f x x 7= +] g

(c) | |f x x 2= -] g

(d) 5 | | y x=

(e) | |f x x 3= - +] g

(f) | 6 | y x= +

(g) | |f x x3 2= -] g

(h) | 5 4 | y x= +

(i) | 7 1 | y x= -

(j) | |f x x2 9= +] g


plane

(a) | | y x=

(b) | |f x x 1= +] g

(c) | |f x x 3= -] g

(d) 2 | | y x=

(e) | |f x x= -] g

(f) | 1 | y x= +

(g) | |f x x 1= - -] g

(h) | 2 3 | y x= -

(i) | 4 2 | y x= +

(j) | |f x x3 1= +] g


each function

(a) | 1 | y x= -

(b) | |f x x 8= -] g

(c) | |f x x2 5= +] g

(d) 2 | | 3 y x= -

(e) | |f x x 3= - -] g


function is

increasing(i)

decreasing(ii)

(a) | 2 | y x= -

(b) | |f x x 2= +

] g

(c) | |f x x2 3= -] g

(d) 4 | | 1 y x= -

(e) | |f x x= -] g


of each function

(a) | | y x= for x2 2 -

(b) | |f x x 4= - -] g for

x4 3 -

(c) | |f x x 4= +] g for x7 2 -

(d) | 2 5 | y x= - for x3 3 -

(e) | |f x x= -] g for x1 1 -


function increasing

(a) | 3 | y x= +

(b) | |f x x 4= - +] g

(c) | |f x x 9= -] g

(d) | | y x 2 1= - -

(e) | |f x x 2= - +] g

56 Exercises




7 Solve graphically

(a) | | 3x =

(b) | |x 12

(c) | |x 2

(d) | 2 | 1x + =

(e) | 3 | 0x- =

(f) |2 3 | 1x - =

(g) | |x 1 41-

(h) | |x 1 3+

(i) | |x 2 22-

(j) | |x 3 1$-

(k) | |x2 3 5+

(l) | |x2 1 1$-

(m) |3 1 | 3x x- = +

(n) |3 2 | 4x x- = -

(o) | 1 | 1x x- = +

(p) | 3 | 2 2x x+ = + (q) |2 1 | 1x x+ = -

(r) |2 5 | 3x x- = -

(s) | 1 | 2x x- =

(t) |2 3 | 3x x- = +

The Hyperbola


or= =

EXAMPLE

Sketch1

y x=

Solution



x -3 -2 -121-

41- 0

41

21 1 2 3

y 3

1-

2

1- -1 -2 -4 mdash 4 2 1

2

1

3

1

Class Discussion



y is never 0 Why






but never touches)



Investigation




equations




1 y xk

=

21

y x k= +

31

y x k

=+


EXAMPLES


3=

-

] g


Solution



So x 3






be zero

CONTINUED







1

y 0 3

3=

-

= -

-3

4

3

2

1

5

-2

-1

-4

-5

y

-1-2 4 521-3-4

x

x = 3

y = 0

Asymptotes

3

2 Sketch y x2 4

1= -

+

Solution




x

x

2 4

2

-

-






( ) y

2 0 41

41

= -+

= -


a translation of3

y x

=





y

-2

x

-

1

4


a=

+


domain x xb



1 For each graph



exists


(a)2

y x=

(b)1

y x= -

(c) f xx 1

1=

+] g

(d) f xx 2

3=

-

] g

(e)3 6

1 y

x=

+

(f) f xx 3

2= -

-

] g

(g) f xx 1

4=

-

] g

(h)1

2 y

x= -

+

(i) f xx6 3

2=

-

] g

(j)2

6 y

x= -

+

2 Show that f x x2

=] g is an odd

function



(a) f xx2 5

1=

+

] g for x2 2 -

(b)3

1 y

x=

+ for x2 0 -

(c) f xx2 4

5=

-

] g for x3 1 -

57 Exercises




(d) f xx 4

3= -

-

] g for x3 3 -

(e)3 1

2 y

x= -

+ for x0 5



(a)3

y x= for y 1 3

(b)2

y x= - for y 221

- -

(c) f xx 1

1=

-

] g for y 171

- -

(d) f xx2 1

3= -

+

] g for

y 131

- -

(e)3 2

6 y

x=

- for y 1

21

6



Circle gate





Proof

r y

x

( x y)

y

x








c a b

r x y

2 2 2

2 2 2

`

= +

= +

EXAMPLE



Solution


y

x

-2

-2 2

2


one place

y

x

2

2

2

-2

The radius is 4

CONTINUED






Domain 2 2 x x -

Range 2 2 y y -


range y r y r -



Proof


shown

y

x

(a b)

x

y

r

( x y)

a

b x - a

y - b




ndash ndash

c a b

r x a y b

2 2 2

2 2 2

= +

= +] ^g h




EXAMPLES



Solution



y

x 9

9

-9

-9



Domain 9 9 x x -

Range 9 9 y y -



Solution


ndash

ndash ndash

x y

x y

1 2 4

1 2 2

2

2 2

+ + =

+ - =

2

2

] ^

] ]_

g h

g gi

So 1 2a b= = - and 2r =

CONTINUED







y

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

(1 -2)



Domain 1 3 x x -

Range 4 0 y y -


Solution


a b2 1= - = and r 3=

Substituting

ndash ndash

ndash

ndash

x a y b r

x y

x y

2 1 3

2 1 9

2 2 2

2 2 2

2 2

+ =

- - + =

+ + =

] ^]] ^

] ^

g hg g hg h


ndash

ndash

a b a ab b

x x x

x x

a b a ab b

y y y

y y

2

2 2 2 2

4 4

2

1 2 1 1

2 1

So

So

2 2 2

2 2 2

2

2 2 2

2 2 2

2

+ = + +

+ = + +

= + +

= - +

= - +

= - +

]] ] ]]^ ^ ]

gg g ggh h g


ndash

x x y y

x x y y

x x y y

x x y y

4 4 2 1 9

4 2 5 9

4 2 5 9

4 2 4 0

9 9

2

2

2

2

+ + + - + =

+ + - + =

+ + + =

+ + - - =

- -


in Chapter 2




Investigation


two functions





circle






for all values

y

x r

r

-r


Proof

ndash

x y r

y r x y r x

2 2 2

2 2 2

2 2

+ =

=

= -



semi-circles





y 0 for all values

y

x r

-

r

-r


EXAMPLES


1 f x x92

= -] g

Solution



y

x 3

3

-3

Domain 3 3 x x -

Range 0 3 y y




2 y x4 2= - -

Solution



y

x 2

-2

-2

Domain 2 2 x x -

Range 2 0 y y -




range

(a) 9x y 2 2+ =

(b) x y 16 02 2+ =-


(d) 1 9x y 2 2+ + =

] g

(e) ndashx y 2 1 12 2+ + =] ^g h



below the x -axis



range

(a) 25 y x2= - -

(b) 1 y x2= -

(c) 36 y x2= -

(d) 64 y x2= - -

(e) 7 y x2= - -



each circle

(a) 100x y 2 2+ =

(b) 5x y 2 2+ =


(d) ndashx y 5 6 492 2+ + =] ^g h

(e) ndashx y 3 812 2+ =^ h

58 Exercises






grouping symbols)



Centre(c) 1 5-







Centre(j) 4 7- -

^ h and radius 3

Other Graphs




EXAMPLES


Solution



eg When 0x =

y 3

1

c=

=

x

y

1

3

3

1

31

When1

1

= -

=

=

=

-

x 3- 2- 1- 0 1 2 3

y 271

91

31

1 3 9 27



You will meet these

graphs again in the

HSC Course





other point

When

x

y

1

3

3

1

=

=

=

y

x

1

2

1

3







Solution



x minus2 minus1 0 05 1 2 3 4

y minus03 0 03 05 06

y

x

1

2

1 2 3 4

-1




You learned about


stages of maths





range y y 02



range all real y

Cubic function


EXAMPLE



Solution



y minus25 minus6 1 2 3 10 29

y

x

1

1

-2 2 3 4

-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4



If you already

know the shape of


3






Domain and range


EXAMPLE


Solution




function are

f

f

1 1 2

1 21

4 4 2

64 2

66

3

3

- = - +

= - +

=

= +

= +

=

] ]

] ]

g g

g g


these points

y

x

(-1 1)

(4 66)

Range 1 66 y y










positive number

EXAMPLE


Solution


x 4So $

$


(or zero) So y 0$


59 Exercises


(a) 4 3 y x= +

(b) f x 4= -] g

(c) 3x =



(f) f x xx 12 2= - -] g

(g) 64x y 2 2+ =

(h) f t t 4

3=

-] g

(i) ( ) g 2

5zz

= +

(j) | |f x x=] g


(a) y x=

(b) 2 y x= -

(c) | |f x x2 3= -] g

(d) | | 2 y x= -

(e) f x x2 5= - +] g

(f) | | y x5= -

(g) 2 y x=

(h) y 5x= -

(i) f x xx 1

= +] g

(j)2

4 3 y x

x= -


(a) y x x 5 2= -] g


= +-

(d) g x x x164 2= -] g

(e) 49x y 2 2+ =

You may like to


by dividing by x




4 (a) Solve x1 02$-


f x x1 2= -] g


(a) 2 y x x2= - -

(b) g t t t 62= +] g



in each case


x3 3 -


x2 3 -


x2 1 - (d)


x1 5


0 4x


x3 3 -


x1 1 -


x2 3 -


x4 4 -

(j) y x x7 62= - + - in the

domain 0 7x


function y x 1

3=

+





x=] g


function

find its range(b)


plane

(a) f x x4=] g

(b) y x3= -

(c) y x 34= -

(d) 2 p x x3=] g

(e) 1 g x x3= +] g

(f) 100x y 2 2+ =

(g) 2 1 y x= +


y x 1= -









(b) f x x 22= -] g

(c)1

y x=

(d) f x x3=] g

(e) f x 3x=] g

13 (a) Solve x4 02$-


(i) 4 y x2= -

(ii) y x4

2= - -




DID YOU KNOW




a light



Limits






a a1

1

03

Z

=

=

3

3

-


lim a 0x=

x 3-



asymptote

EXAMPLES

1 Find lim x

x x5x 0

2+

Solution


which is undefined


( )

lim lim

lim

xx x

x

x x

x

5 5

5

5

x x

x

0

2

0 1

1

0

+=

+

= +

=

] g




2 Find limx

x

4

22

-

-

x 2

Solution


lim lim

lim

x

x

x x

x

x

4

2

2 2

2

21

41

2 1

1

-

-=

+ -

-

=+

=

x x2 2

x 2

^ _h i

3 Find limh

h x hx h2 72 2+ -

h 0

Solution

lim lim

lim

h

h x hx h

h

h hx x

hx x

x

2 7 2 7

2 7

7

2 2 2

2

2

+ -=

+ -

= + -

= -

h 0

h h0 0

^ h

Continuity





EXAMPLES

1 Find lim

x

x

1

12

-

-

x 1


11

y x

x2

=-

- Sketch the curve

Solution


xx2

-

- gives

00

CONTINUED




( )

lim lim

limx

x

x

x x

x11

1

1 1

1

2

x x

x

1

2

1

1

-

-=

-

+ -

= +

=

-

] ]g g

11

y xx2

=-





y xx

x

x x

x

11

1

1 1

1

2

=-

-

=

+

= +

-

-^ ^h h


2 Find limx

x x2

2x 2

2

+

+ -

-


x x2

22

= + -

+

Solution


x x2

22

+

+ - gives

00

lim lim

lim

xx x

x

x x

x

22

2

1 2

1

3

x x

x

2

2

2

2

+

+ -=

+

- +

=

= -

-

- -

-

^^ ^

^

hh h

h2 y

x

x xx

y x

x

x

x

22

2

2

1

1


=+

+ -= -

=+

= -

+ -^ ^h h


Remember that x 1




1 Find(a) lim x 52

+x 4

(b) lim t 7-t 3 -

(c) lim x x2 43+ -

x 2

(d) lim xx x32

+

x 0

(e) limh

h h

2

22

-

- -

h 2

(f) lim y

y

5

1253

-

-

y 5

(g) limx

x x

12 12

+

+ +

x 1-

(h) limx

x x4

2 82

+

+ -

x 4 -

(i) limc

c

4

22

-

-

c 2

(j) limx x

x 12

-

-

x 1

(k) lim h

h h h2 73 2+ -

h 0

(l) limh

hx hx h32 2- +

h 0

(m) limh

hx h x hx h2 3 53 2 2- + -

h 0

(n) lim x c x c 3 3

-

-

x c



discontinuous

(a) 3 y x2= -

(b)1

1 y

x=

+

(c) f x x 1= -] g

(d)4

1 y

x2=

+

(e)4

1

y x2=

-



(a)3

y xx x2

= +

(b)33

y x

x x2

=+

+

(c)1

5 4 y

xx x2

=+

+ +

510 Exercises




Regions

Class Investigation


them all







EXAMPLE


Solution



y

x =3

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4







region

EXAMPLE


Solution



included


shown

y

x

1

1

-2 2 3 4

-3

-4

-5

2

3

4

5

-1

-2

-3-4

-1 y = -1



y -intercept 1-






EXAMPLES


1 y x 2$ +

Solution






y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0



2 x y 2 3 61-

Solution


region


side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g








3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g







variable) say x 3=

When x

y

3

3 1

4

=

= +

=







x and the dependent





Investigation










A

B

C

D

E

1

2

3

4


and (E 2)


NOT a function





1

x

y

2

x

y




x

y





graph is a function

y

x



x

y





place





EXAMPLES


Solution



Solution


not a function







2 3 1 4 0 5 1 3 2 4- -^ ^ ^ ^ ^h h h h h Solution


function


y

x

3

Solution

y

x

3





a function




1

2

3

4

5

6

7

8

9 1 3 2 1 3 3 4 0-^ ^ ^ ^h h h h 10 1 3 2 1 2 7 4 0-^ ^ ^ ^h h h h 11

1

2

3

4

5

1

2

3

4

5

12 1

2

3

4

5

1

2

3

4

5

131

2

3

4

5

1

2

3

4

5

51 Exercises







15 A 3

B 4

C 7

D 3

E 5

F 7

G 4

Function notation




EXAMPLES


Solution

When x

y x

3

1

3 1

4

=

= +

= +

=


Solution

f x x

f

1

3 3 1

4

= +

= +

=

]] gg





EXAMPLES

1 If f x x x3 12= + +] g find f 2-] g

Solution

( ) ( )f 2 2 3 2 1

4 6 1

1

2- = - + - +

= - +

= -

] g


Solution

( )

( )

f x x x

f 1 1 1

1 12

3 2

3

= -

- = - - -

= - -

= -

2] ]g g


Solution

( )

ie

( ) ( )

f x

x x

x x

x x

x x

0

3 10 0

5 2 0

5 0 2 0

5 2

2

=

+ - =

+ - =

+ = - =

= - =


when

when f x

x x

x x

3 4 2

2 21

$=

+

-

] g )

Solution

since 4 21-

( ) ( ) since

( ) ( ) since

( ) ( ) since

( ) ( )

f

f

f

f

3 3 3 4 3 2

13

2 3 2 4 2 2

10

0 2 0 0 2

0

4 2 4

8

1

$

$

= +

=

= +

=

= -

=

- = - -

=


when

when

x

x

x

2

1 2

1

2

1

-

-

g x

x

x2 1

5

2

= -] g


when 2x -=







than 2




Solution

( ) ( )

( )

( )

g

g

g

1 2 1 1 1 1 2

1

2 5 2 1

3 3 3 2

9

since

since

since2

1

2

= - -

=

- = - -

=

=

( ) ( ) ( ) g g g 1 2 3 1 5 9

3

So + - - = + -

= -

DID YOU KNOW






f 3-] g

2 If h x x 22= -] g find h h0 2] ]g g

and h 4-] g


and f 2-] g


= - +] g

5 Find f 3-] g if f x x x2 5 43= - +] g


f x 13=] g




f x271

=] g




f x h+] g


g x x x2 32= + +] g


product of factors






15 f x x x

x x

1

1

for

for

32

=] g )


16 f x

x x

x x

x x

2 4 1

3 1 1

1

if

if

if 2

1 1

$

=

-

+ -

-

] gZ

[

]]

]]

Find the values of

f f f 2 2 1- - + -] ] ]g g g

52 Exercises








EXAMPLE


Solution


x x

x x

x x

x x

0 7 8

8 1

8 0 1 0

8 1

2= + -

= + -

+ = - =

= - =

] ]g g


y 0 7 0 8

8

2= + -

= -

] ]g g





Domain and range




domain


image) of f

EXAMPLE


Solution


x

y

CONTINUED






Domain all real x

Range y y 0$















EXAMPLES


x x 3 x 2 x 1

y

Solution


increasing


x 1 x 2

2

x x 3

x 2 x 1

y

Solution



increases again




1 1








=







x

y









x

y




EXAMPLES


Solution

f x x

x

f x

f x x

3

3


2

2

2`

- = - +

= +

=

= +

] ]]

]

g gg

g


Solution

f x x x

x x

x x

f x


3

3

3

3`

- = - - -

= - +

= - -

= -

= -

] ] ]^]

]

g g gh

gg

Investigation






1 f x kxn=] g

2 f x x kn= +] g

3 f x x k n= +

] ]g g






shape of graphs


each function

(a) y x3 2= -

(b) x y 2 5 20 0- + =

(c) x y 3 12 0+ - =

(d) f x x x32= +] g

(e) f x x 42= -] g

(f) p x x x5 62= + +] g

(g) y x x8 152= - +

(h) p x x 53

= +

] g

53 Exercises




(i) y xx

x3

0= + ] g

(j) g x x9 2= -] g


f x x 22= -

] g What type of

function is it


(a) f x2^ h (b) ( )f x 26



an even function


f x x=] g


function


odd function



f x f x- -] ]g g


neither

(a) y x x

x4 2

3

=

-

(b) y x 1

13

=

-

(c) f xx 4

32

=

-

] g

(d) y x

x

33

=+

-

(e) f x x x

x5 2

3

=-] g



f x xn=] g

even(a)

odd(b)


ever be

even(a)

odd(b)



graph is increasing




x

y(a)

x

4

y(b)

2-2

x

y(c)




Investigation


graph







y kx

y kx

y kx

y kx

y xk

(a)

(b)

(c)

(d)

(e)

2

3

4

=

=

=

=

=



y x k

y x k

y x k

y x k

y x k1

(a)

(b)

(c)

(d)

(e)

2

2

3

4

= +

= +

= +

= +

= +

] g



-2

1 2

-4

-1-2

2

4

y

x

(d) y

x

(e)

CONTINUED




Gradient form



Investigation










y x k

y x k

y x k

y x k y

x k

1

(a)

(b)

(c)

(d)(e)

2

3

4

= +

= +

= +

= +

=+

]]

]

gg

g





the right








Linear Function





EXAMPLE



Solution


Find the intercepts


0 3 5

5 3

1

x

x

x32

=

=

=

-


3 5

5

y 0=

= -

-] g

-1

-2

y

5

4

3

2

1 1 23

6

-3

-4

-5

1 4-1-2 32-3-4

x



Range all real y








Special lines






1 y kx=

2 y x k= +






EXAMPLES


Solution




-1

-3

y

4

3

2

1

5

-2

-4

-5

1 4-1-2

x

32-3-4


2 Sketch x 1= -


Solution





-

-

4

3

2

5

-2

-4

-5

1 4-1-2-4

y

x







Domain all real x

Range y y b=

54 Exercises


each function

(a) y x 2= -

(b) f x x2 3= +] g (c) x y 2 1 0+ =-

(d) x y 3 0+ =-

(e) x y 3 6 2 0=- -


line

(a) x 4=

(b) x 3 0=-

(c) y 5=

(d) y 1 0+ =

(e) f x x2 1= -] g (f) y x 4= +

(g) f x x3 2= +] g

(h) x y 3+ =

(i) x y 1 0=- -

(j) x y 2 3 0+ =-


(a) x y 3 2 7 0+ =-

(b) y 2=

(c) x 4= - (d) x 2 0=-

(e) y 3 0=-


are even or odd

(a) y x2=

(b) y 3=

(c) x 4=

(d) y x= -

(e) y x=




meet




Applications






The pronumeral

a is called the


Quadratic Function










EXAMPLES



Solution



x

x

x

0 1

1

1

2

2

= -

=

=


0 1

1

y 2= -

= -



Domain xall real

Range y y 1$ -


Solution


parabola will lie




left



0

1 0

1

x

x

x

1 2= +

+ =

= -

] g


1

y 0 1 2= +

=

] g

-1

-

4

3

2

1

5

-2

-4

-5

-

1 41-2 5-4

y

x









Solution



x x

x xx x

x x

0 6

3 23 0 2 0

3 2

2= + -

= + -

+ = - =

= - =

] ]g g


f 0 0 0 66

2= + -

= -

] ] ]g g g



x -intercepts


23 2

21- +

= -


-c m

f 21

21

21

6

41

21

6

641

2

- = - + - -

= - -

= -

c c cm m m


-

CONTINUED

You will learn more


-1

-3

4

3

2

1

5

-2

-4

-5

1 4-1-2 32-3-4

y

x










41



2

11 -





Solution


So f x 0=] g

0 4 54 5 0

0

x xx x

x x5 1

2

2

= - + +

=

+ =

- -

-] ]g g

x x

x x

5 0 1 0

5 1

- = + =

= = -


f 0 0 4 0 5

5

2= - + +

=

] ] ]g g g

-1

-3

4

2

5

-2

-4

-5

-

y

1 41-2 5-4

x

-2 4






21 5

2- +

=

f 2 2 4 2 59

= - + +

=

2] ] ]g g g










each function

(a) 2 y x x2= +

(b) 3 y x x2= - +

(c) f x x 12= -] g

(d) y x x 22= - -

(e) y x x9 82= +-

2 Sketch

(a) 2 y x2= +

(b) y x 12= - +

(c) f x x 42= -] g

(d) 2 y x x2= +

(e) y x x2= - -

(f) f x x 3= -2] ]g g

55 Exercises

-1

9

8

7

5

4

3

2

6

1

-2

-3

-4

-5

y

2 51 643-1-2-3-4

x




EXAMPLES


Solution









(g) f x x 1 2= +] ]g g

(h) y x x3 42= + -

(i) y x x2 5 32= - +

(j) f x x x3 22= - + -] g



(a) ndash y x x7 122= +

(b) f x x x42= +] g

(c) y x x2 82= - -

(d) y x x6 92= +-

(e) f t t 4 2= -] g


(a) y x 52= -

(b) f x x x6

2= -

] g (c) f x x x 22= - -] g

(d) y x2= -

(e) f x x 7 2= -] ]g g



(a) y x2= for x0 3

(b) y x 42= - + for x1 2 -

(c) f x x 12= -] g for x2 5 -

(d) y x x2 32= + - for x2 4 -

(e) y x x

22= - +- for

x0 4


function is

increasing(i)

decreasing(ii)

(a) y x2=

(b) y x2

= - (c) f x x 92

= -] g

(d) y x x42= - +

(e) f x x 5 2= +] ]g g


function



(a) y x 12= +

(b) f x x 32= -

] g

(c) y x2 2= -

(d) f x x x32= -] g

(e) f x x x2= +] g

(f) y x 42= -

(g) y x x2 32= - -

(h) y x x5 42= +-

(i) p x x 1 2= +] ]g g

(j) y x 2= -2] g




CONTINUED

eg When x 3= -

| | y 3 13 12

= - -

= -

=

x -3 -2 -1 0 1 2 3

y 2 1 0 -1 0 1 2


y

-2

4

3

2

1

5

-1

-3

-4

-5

1 4-1-2 32-3-4

x


| | y x

x x

x x1

1 0

1 0

when

when 1

$= - =

-


y x x1 0$= - ] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x - 1




y x 1= - - x 01] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y =- x - 1



-3

4

3

2

1

5

-2

-1

-4

-5

y y

3-1-2 421-3-4

x

y = - x - 1

y = x - 1


intercepts


So f x 0=

] g

| |

| |

x

x

x

0 1

1

1`

= -

=

=


( ) | |f 0 0 11

= -

= -





-3

4

3

2

1

5

-2

-1

-4

-5

y

4-2 5321-1-3-4

x



Domain all real x

Range y y 1$ -

2 Sketch | | y x 2= +

Solution


| | ( ) y x x xx x2 2 2 0

2 2 0whenwhen 1

$= + = + +

- + +


2 y x= + when x 2 0$+

x 2$ -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x + 2






y-intercept of 1-

CONTINUED




2 y x= - +] g when x 2 01+

ie y x 2= - - when x 21 -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2


-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2

y = x + 2



So 0f x =] g

0 | 2 |

0 2

2

x

x

x

= +

= +

- =


(0) | 0 2 |

2

f = +

=

There is only one

solution for the


Can you see why





-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

If you know how to

sketch the graph of




Investigation




are their equations





1 | |f x k x=] g

2 | |f x x k= +] g

3 | |f x x k= +] g







EXAMPLES

Solve

1 |2 1 | 3x - =

Solution



is x 1 2= -

2 |2 1 | 3 2x x= -+

Solution



3 | 1 | 2x 1+

Solution






and then check them







each function

(a) | | y x=

(b) | |f x x 7= +] g

(c) | |f x x 2= -] g

(d) 5 | | y x=

(e) | |f x x 3= - +] g

(f) | 6 | y x= +

(g) | |f x x3 2= -] g

(h) | 5 4 | y x= +

(i) | 7 1 | y x= -

(j) | |f x x2 9= +] g


plane

(a) | | y x=

(b) | |f x x 1= +] g

(c) | |f x x 3= -] g

(d) 2 | | y x=

(e) | |f x x= -] g

(f) | 1 | y x= +

(g) | |f x x 1= - -] g

(h) | 2 3 | y x= -

(i) | 4 2 | y x= +

(j) | |f x x3 1= +] g


each function

(a) | 1 | y x= -

(b) | |f x x 8= -] g

(c) | |f x x2 5= +] g

(d) 2 | | 3 y x= -

(e) | |f x x 3= - -] g


function is

increasing(i)

decreasing(ii)

(a) | 2 | y x= -

(b) | |f x x 2= +

] g

(c) | |f x x2 3= -] g

(d) 4 | | 1 y x= -

(e) | |f x x= -] g


of each function

(a) | | y x= for x2 2 -

(b) | |f x x 4= - -] g for

x4 3 -

(c) | |f x x 4= +] g for x7 2 -

(d) | 2 5 | y x= - for x3 3 -

(e) | |f x x= -] g for x1 1 -


function increasing

(a) | 3 | y x= +

(b) | |f x x 4= - +] g

(c) | |f x x 9= -] g

(d) | | y x 2 1= - -

(e) | |f x x 2= - +] g

56 Exercises




7 Solve graphically

(a) | | 3x =

(b) | |x 12

(c) | |x 2

(d) | 2 | 1x + =

(e) | 3 | 0x- =

(f) |2 3 | 1x - =

(g) | |x 1 41-

(h) | |x 1 3+

(i) | |x 2 22-

(j) | |x 3 1$-

(k) | |x2 3 5+

(l) | |x2 1 1$-

(m) |3 1 | 3x x- = +

(n) |3 2 | 4x x- = -

(o) | 1 | 1x x- = +

(p) | 3 | 2 2x x+ = + (q) |2 1 | 1x x+ = -

(r) |2 5 | 3x x- = -

(s) | 1 | 2x x- =

(t) |2 3 | 3x x- = +

The Hyperbola


or= =

EXAMPLE

Sketch1

y x=

Solution



x -3 -2 -121-

41- 0

41

21 1 2 3

y 3

1-

2

1- -1 -2 -4 mdash 4 2 1

2

1

3

1

Class Discussion



y is never 0 Why






but never touches)



Investigation




equations




1 y xk

=

21

y x k= +

31

y x k

=+


EXAMPLES


3=

-

] g


Solution



So x 3






be zero

CONTINUED







1

y 0 3

3=

-

= -

-3

4

3

2

1

5

-2

-1

-4

-5

y

-1-2 4 521-3-4

x

x = 3

y = 0

Asymptotes

3

2 Sketch y x2 4

1= -

+

Solution




x

x

2 4

2

-

-






( ) y

2 0 41

41

= -+

= -


a translation of3

y x

=





y

-2

x

-

1

4


a=

+


domain x xb



1 For each graph



exists


(a)2

y x=

(b)1

y x= -

(c) f xx 1

1=

+] g

(d) f xx 2

3=

-

] g

(e)3 6

1 y

x=

+

(f) f xx 3

2= -

-

] g

(g) f xx 1

4=

-

] g

(h)1

2 y

x= -

+

(i) f xx6 3

2=

-

] g

(j)2

6 y

x= -

+

2 Show that f x x2

=] g is an odd

function



(a) f xx2 5

1=

+

] g for x2 2 -

(b)3

1 y

x=

+ for x2 0 -

(c) f xx2 4

5=

-

] g for x3 1 -

57 Exercises




(d) f xx 4

3= -

-

] g for x3 3 -

(e)3 1

2 y

x= -

+ for x0 5



(a)3

y x= for y 1 3

(b)2

y x= - for y 221

- -

(c) f xx 1

1=

-

] g for y 171

- -

(d) f xx2 1

3= -

+

] g for

y 131

- -

(e)3 2

6 y

x=

- for y 1

21

6



Circle gate





Proof

r y

x

( x y)

y

x








c a b

r x y

2 2 2

2 2 2

`

= +

= +

EXAMPLE



Solution


y

x

-2

-2 2

2


one place

y

x

2

2

2

-2

The radius is 4

CONTINUED






Domain 2 2 x x -

Range 2 2 y y -


range y r y r -



Proof


shown

y

x

(a b)

x

y

r

( x y)

a

b x - a

y - b




ndash ndash

c a b

r x a y b

2 2 2

2 2 2

= +

= +] ^g h




EXAMPLES



Solution



y

x 9

9

-9

-9



Domain 9 9 x x -

Range 9 9 y y -



Solution


ndash

ndash ndash

x y

x y

1 2 4

1 2 2

2

2 2

+ + =

+ - =

2

2

] ^

] ]_

g h

g gi

So 1 2a b= = - and 2r =

CONTINUED







y

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

(1 -2)



Domain 1 3 x x -

Range 4 0 y y -


Solution


a b2 1= - = and r 3=

Substituting

ndash ndash

ndash

ndash

x a y b r

x y

x y

2 1 3

2 1 9

2 2 2

2 2 2

2 2

+ =

- - + =

+ + =

] ^]] ^

] ^

g hg g hg h


ndash

ndash

a b a ab b

x x x

x x

a b a ab b

y y y

y y

2

2 2 2 2

4 4

2

1 2 1 1

2 1

So

So

2 2 2

2 2 2

2

2 2 2

2 2 2

2

+ = + +

+ = + +

= + +

= - +

= - +

= - +

]] ] ]]^ ^ ]

gg g ggh h g


ndash

x x y y

x x y y

x x y y

x x y y

4 4 2 1 9

4 2 5 9

4 2 5 9

4 2 4 0

9 9

2

2

2

2

+ + + - + =

+ + - + =

+ + + =

+ + - - =

- -


in Chapter 2




Investigation


two functions





circle






for all values

y

x r

r

-r


Proof

ndash

x y r

y r x y r x

2 2 2

2 2 2

2 2

+ =

=

= -



semi-circles





y 0 for all values

y

x r

-

r

-r


EXAMPLES


1 f x x92

= -] g

Solution



y

x 3

3

-3

Domain 3 3 x x -

Range 0 3 y y




2 y x4 2= - -

Solution



y

x 2

-2

-2

Domain 2 2 x x -

Range 2 0 y y -




range

(a) 9x y 2 2+ =

(b) x y 16 02 2+ =-


(d) 1 9x y 2 2+ + =

] g

(e) ndashx y 2 1 12 2+ + =] ^g h



below the x -axis



range

(a) 25 y x2= - -

(b) 1 y x2= -

(c) 36 y x2= -

(d) 64 y x2= - -

(e) 7 y x2= - -



each circle

(a) 100x y 2 2+ =

(b) 5x y 2 2+ =


(d) ndashx y 5 6 492 2+ + =] ^g h

(e) ndashx y 3 812 2+ =^ h

58 Exercises






grouping symbols)



Centre(c) 1 5-







Centre(j) 4 7- -

^ h and radius 3

Other Graphs




EXAMPLES


Solution



eg When 0x =

y 3

1

c=

=

x

y

1

3

3

1

31

When1

1

= -

=

=

=

-

x 3- 2- 1- 0 1 2 3

y 271

91

31

1 3 9 27



You will meet these

graphs again in the

HSC Course





other point

When

x

y

1

3

3

1

=

=

=

y

x

1

2

1

3







Solution



x minus2 minus1 0 05 1 2 3 4

y minus03 0 03 05 06

y

x

1

2

1 2 3 4

-1




You learned about


stages of maths





range y y 02



range all real y

Cubic function


EXAMPLE



Solution



y minus25 minus6 1 2 3 10 29

y

x

1

1

-2 2 3 4

-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4



If you already

know the shape of


3






Domain and range


EXAMPLE


Solution




function are

f

f

1 1 2

1 21

4 4 2

64 2

66

3

3

- = - +

= - +

=

= +

= +

=

] ]

] ]

g g

g g


these points

y

x

(-1 1)

(4 66)

Range 1 66 y y










positive number

EXAMPLE


Solution


x 4So $

$


(or zero) So y 0$


59 Exercises


(a) 4 3 y x= +

(b) f x 4= -] g

(c) 3x =



(f) f x xx 12 2= - -] g

(g) 64x y 2 2+ =

(h) f t t 4

3=

-] g

(i) ( ) g 2

5zz

= +

(j) | |f x x=] g


(a) y x=

(b) 2 y x= -

(c) | |f x x2 3= -] g

(d) | | 2 y x= -

(e) f x x2 5= - +] g

(f) | | y x5= -

(g) 2 y x=

(h) y 5x= -

(i) f x xx 1

= +] g

(j)2

4 3 y x

x= -


(a) y x x 5 2= -] g


= +-

(d) g x x x164 2= -] g

(e) 49x y 2 2+ =

You may like to


by dividing by x




4 (a) Solve x1 02$-


f x x1 2= -] g


(a) 2 y x x2= - -

(b) g t t t 62= +] g



in each case


x3 3 -


x2 3 -


x2 1 - (d)


x1 5


0 4x


x3 3 -


x1 1 -


x2 3 -


x4 4 -

(j) y x x7 62= - + - in the

domain 0 7x


function y x 1

3=

+





x=] g


function

find its range(b)


plane

(a) f x x4=] g

(b) y x3= -

(c) y x 34= -

(d) 2 p x x3=] g

(e) 1 g x x3= +] g

(f) 100x y 2 2+ =

(g) 2 1 y x= +


y x 1= -









(b) f x x 22= -] g

(c)1

y x=

(d) f x x3=] g

(e) f x 3x=] g

13 (a) Solve x4 02$-


(i) 4 y x2= -

(ii) y x4

2= - -




DID YOU KNOW




a light



Limits






a a1

1

03

Z

=

=

3

3

-


lim a 0x=

x 3-



asymptote

EXAMPLES

1 Find lim x

x x5x 0

2+

Solution


which is undefined


( )

lim lim

lim

xx x

x

x x

x

5 5

5

5

x x

x

0

2

0 1

1

0

+=

+

= +

=

] g




2 Find limx

x

4

22

-

-

x 2

Solution


lim lim

lim

x

x

x x

x

x

4

2

2 2

2

21

41

2 1

1

-

-=

+ -

-

=+

=

x x2 2

x 2

^ _h i

3 Find limh

h x hx h2 72 2+ -

h 0

Solution

lim lim

lim

h

h x hx h

h

h hx x

hx x

x

2 7 2 7

2 7

7

2 2 2

2

2

+ -=

+ -

= + -

= -

h 0

h h0 0

^ h

Continuity





EXAMPLES

1 Find lim

x

x

1

12

-

-

x 1


11

y x

x2

=-

- Sketch the curve

Solution


xx2

-

- gives

00

CONTINUED




( )

lim lim

limx

x

x

x x

x11

1

1 1

1

2

x x

x

1

2

1

1

-

-=

-

+ -

= +

=

-

] ]g g

11

y xx2

=-





y xx

x

x x

x

11

1

1 1

1

2

=-

-

=

+

= +

-

-^ ^h h


2 Find limx

x x2

2x 2

2

+

+ -

-


x x2

22

= + -

+

Solution


x x2

22

+

+ - gives

00

lim lim

lim

xx x

x

x x

x

22

2

1 2

1

3

x x

x

2

2

2

2

+

+ -=

+

- +

=

= -

-

- -

-

^^ ^

^

hh h

h2 y

x

x xx

y x

x

x

x

22

2

2

1

1


=+

+ -= -

=+

= -

+ -^ ^h h


Remember that x 1




1 Find(a) lim x 52

+x 4

(b) lim t 7-t 3 -

(c) lim x x2 43+ -

x 2

(d) lim xx x32

+

x 0

(e) limh

h h

2

22

-

- -

h 2

(f) lim y

y

5

1253

-

-

y 5

(g) limx

x x

12 12

+

+ +

x 1-

(h) limx

x x4

2 82

+

+ -

x 4 -

(i) limc

c

4

22

-

-

c 2

(j) limx x

x 12

-

-

x 1

(k) lim h

h h h2 73 2+ -

h 0

(l) limh

hx hx h32 2- +

h 0

(m) limh

hx h x hx h2 3 53 2 2- + -

h 0

(n) lim x c x c 3 3

-

-

x c



discontinuous

(a) 3 y x2= -

(b)1

1 y

x=

+

(c) f x x 1= -] g

(d)4

1 y

x2=

+

(e)4

1

y x2=

-



(a)3

y xx x2

= +

(b)33

y x

x x2

=+

+

(c)1

5 4 y

xx x2

=+

+ +

510 Exercises




Regions

Class Investigation


them all







EXAMPLE


Solution



y

x =3

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4







region

EXAMPLE


Solution



included


shown

y

x

1

1

-2 2 3 4

-3

-4

-5

2

3

4

5

-1

-2

-3-4

-1 y = -1



y -intercept 1-






EXAMPLES


1 y x 2$ +

Solution






y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0



2 x y 2 3 61-

Solution


region


side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g








3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g







1

x

y

2

x

y




x

y





graph is a function

y

x



x

y





place





EXAMPLES


Solution



Solution


not a function







2 3 1 4 0 5 1 3 2 4- -^ ^ ^ ^ ^h h h h h Solution


function


y

x

3

Solution

y

x

3





a function




1

2

3

4

5

6

7

8

9 1 3 2 1 3 3 4 0-^ ^ ^ ^h h h h 10 1 3 2 1 2 7 4 0-^ ^ ^ ^h h h h 11

1

2

3

4

5

1

2

3

4

5

12 1

2

3

4

5

1

2

3

4

5

131

2

3

4

5

1

2

3

4

5

51 Exercises







15 A 3

B 4

C 7

D 3

E 5

F 7

G 4

Function notation




EXAMPLES


Solution

When x

y x

3

1

3 1

4

=

= +

= +

=


Solution

f x x

f

1

3 3 1

4

= +

= +

=

]] gg





EXAMPLES

1 If f x x x3 12= + +] g find f 2-] g

Solution

( ) ( )f 2 2 3 2 1

4 6 1

1

2- = - + - +

= - +

= -

] g


Solution

( )

( )

f x x x

f 1 1 1

1 12

3 2

3

= -

- = - - -

= - -

= -

2] ]g g


Solution

( )

ie

( ) ( )

f x

x x

x x

x x

x x

0

3 10 0

5 2 0

5 0 2 0

5 2

2

=

+ - =

+ - =

+ = - =

= - =


when

when f x

x x

x x

3 4 2

2 21

$=

+

-

] g )

Solution

since 4 21-

( ) ( ) since

( ) ( ) since

( ) ( ) since

( ) ( )

f

f

f

f

3 3 3 4 3 2

13

2 3 2 4 2 2

10

0 2 0 0 2

0

4 2 4

8

1

$

$

= +

=

= +

=

= -

=

- = - -

=


when

when

x

x

x

2

1 2

1

2

1

-

-

g x

x

x2 1

5

2

= -] g


when 2x -=







than 2




Solution

( ) ( )

( )

( )

g

g

g

1 2 1 1 1 1 2

1

2 5 2 1

3 3 3 2

9

since

since

since2

1

2

= - -

=

- = - -

=

=

( ) ( ) ( ) g g g 1 2 3 1 5 9

3

So + - - = + -

= -

DID YOU KNOW






f 3-] g

2 If h x x 22= -] g find h h0 2] ]g g

and h 4-] g


and f 2-] g


= - +] g

5 Find f 3-] g if f x x x2 5 43= - +] g


f x 13=] g




f x271

=] g




f x h+] g


g x x x2 32= + +] g


product of factors






15 f x x x

x x

1

1

for

for

32

=] g )


16 f x

x x

x x

x x

2 4 1

3 1 1

1

if

if

if 2

1 1

$

=

-

+ -

-

] gZ

[

]]

]]

Find the values of

f f f 2 2 1- - + -] ] ]g g g

52 Exercises








EXAMPLE


Solution


x x

x x

x x

x x

0 7 8

8 1

8 0 1 0

8 1

2= + -

= + -

+ = - =

= - =

] ]g g


y 0 7 0 8

8

2= + -

= -

] ]g g





Domain and range




domain


image) of f

EXAMPLE


Solution


x

y

CONTINUED






Domain all real x

Range y y 0$















EXAMPLES


x x 3 x 2 x 1

y

Solution


increasing


x 1 x 2

2

x x 3

x 2 x 1

y

Solution



increases again




1 1








=







x

y









x

y




EXAMPLES


Solution

f x x

x

f x

f x x

3

3


2

2

2`

- = - +

= +

=

= +

] ]]

]

g gg

g


Solution

f x x x

x x

x x

f x


3

3

3

3`

- = - - -

= - +

= - -

= -

= -

] ] ]^]

]

g g gh

gg

Investigation






1 f x kxn=] g

2 f x x kn= +] g

3 f x x k n= +

] ]g g






shape of graphs


each function

(a) y x3 2= -

(b) x y 2 5 20 0- + =

(c) x y 3 12 0+ - =

(d) f x x x32= +] g

(e) f x x 42= -] g

(f) p x x x5 62= + +] g

(g) y x x8 152= - +

(h) p x x 53

= +

] g

53 Exercises




(i) y xx

x3

0= + ] g

(j) g x x9 2= -] g


f x x 22= -

] g What type of

function is it


(a) f x2^ h (b) ( )f x 26



an even function


f x x=] g


function


odd function



f x f x- -] ]g g


neither

(a) y x x

x4 2

3

=

-

(b) y x 1

13

=

-

(c) f xx 4

32

=

-

] g

(d) y x

x

33

=+

-

(e) f x x x

x5 2

3

=-] g



f x xn=] g

even(a)

odd(b)


ever be

even(a)

odd(b)



graph is increasing




x

y(a)

x

4

y(b)

2-2

x

y(c)




Investigation


graph







y kx

y kx

y kx

y kx

y xk

(a)

(b)

(c)

(d)

(e)

2

3

4

=

=

=

=

=



y x k

y x k

y x k

y x k

y x k1

(a)

(b)

(c)

(d)

(e)

2

2

3

4

= +

= +

= +

= +

= +

] g



-2

1 2

-4

-1-2

2

4

y

x

(d) y

x

(e)

CONTINUED




Gradient form



Investigation










y x k

y x k

y x k

y x k y

x k

1

(a)

(b)

(c)

(d)(e)

2

3

4

= +

= +

= +

= +

=+

]]

]

gg

g





the right








Linear Function





EXAMPLE



Solution


Find the intercepts


0 3 5

5 3

1

x

x

x32

=

=

=

-


3 5

5

y 0=

= -

-] g

-1

-2

y

5

4

3

2

1 1 23

6

-3

-4

-5

1 4-1-2 32-3-4

x



Range all real y








Special lines






1 y kx=

2 y x k= +






EXAMPLES


Solution




-1

-3

y

4

3

2

1

5

-2

-4

-5

1 4-1-2

x

32-3-4


2 Sketch x 1= -


Solution





-

-

4

3

2

5

-2

-4

-5

1 4-1-2-4

y

x







Domain all real x

Range y y b=

54 Exercises


each function

(a) y x 2= -

(b) f x x2 3= +] g (c) x y 2 1 0+ =-

(d) x y 3 0+ =-

(e) x y 3 6 2 0=- -


line

(a) x 4=

(b) x 3 0=-

(c) y 5=

(d) y 1 0+ =

(e) f x x2 1= -] g (f) y x 4= +

(g) f x x3 2= +] g

(h) x y 3+ =

(i) x y 1 0=- -

(j) x y 2 3 0+ =-


(a) x y 3 2 7 0+ =-

(b) y 2=

(c) x 4= - (d) x 2 0=-

(e) y 3 0=-


are even or odd

(a) y x2=

(b) y 3=

(c) x 4=

(d) y x= -

(e) y x=




meet




Applications






The pronumeral

a is called the


Quadratic Function










EXAMPLES



Solution



x

x

x

0 1

1

1

2

2

= -

=

=


0 1

1

y 2= -

= -



Domain xall real

Range y y 1$ -


Solution


parabola will lie




left



0

1 0

1

x

x

x

1 2= +

+ =

= -

] g


1

y 0 1 2= +

=

] g

-1

-

4

3

2

1

5

-2

-4

-5

-

1 41-2 5-4

y

x









Solution



x x

x xx x

x x

0 6

3 23 0 2 0

3 2

2= + -

= + -

+ = - =

= - =

] ]g g


f 0 0 0 66

2= + -

= -

] ] ]g g g



x -intercepts


23 2

21- +

= -


-c m

f 21

21

21

6

41

21

6

641

2

- = - + - -

= - -

= -

c c cm m m


-

CONTINUED

You will learn more


-1

-3

4

3

2

1

5

-2

-4

-5

1 4-1-2 32-3-4

y

x










41



2

11 -





Solution


So f x 0=] g

0 4 54 5 0

0

x xx x

x x5 1

2

2

= - + +

=

+ =

- -

-] ]g g

x x

x x

5 0 1 0

5 1

- = + =

= = -


f 0 0 4 0 5

5

2= - + +

=

] ] ]g g g

-1

-3

4

2

5

-2

-4

-5

-

y

1 41-2 5-4

x

-2 4






21 5

2- +

=

f 2 2 4 2 59

= - + +

=

2] ] ]g g g










each function

(a) 2 y x x2= +

(b) 3 y x x2= - +

(c) f x x 12= -] g

(d) y x x 22= - -

(e) y x x9 82= +-

2 Sketch

(a) 2 y x2= +

(b) y x 12= - +

(c) f x x 42= -] g

(d) 2 y x x2= +

(e) y x x2= - -

(f) f x x 3= -2] ]g g

55 Exercises

-1

9

8

7

5

4

3

2

6

1

-2

-3

-4

-5

y

2 51 643-1-2-3-4

x




EXAMPLES


Solution









(g) f x x 1 2= +] ]g g

(h) y x x3 42= + -

(i) y x x2 5 32= - +

(j) f x x x3 22= - + -] g



(a) ndash y x x7 122= +

(b) f x x x42= +] g

(c) y x x2 82= - -

(d) y x x6 92= +-

(e) f t t 4 2= -] g


(a) y x 52= -

(b) f x x x6

2= -

] g (c) f x x x 22= - -] g

(d) y x2= -

(e) f x x 7 2= -] ]g g



(a) y x2= for x0 3

(b) y x 42= - + for x1 2 -

(c) f x x 12= -] g for x2 5 -

(d) y x x2 32= + - for x2 4 -

(e) y x x

22= - +- for

x0 4


function is

increasing(i)

decreasing(ii)

(a) y x2=

(b) y x2

= - (c) f x x 92

= -] g

(d) y x x42= - +

(e) f x x 5 2= +] ]g g


function



(a) y x 12= +

(b) f x x 32= -

] g

(c) y x2 2= -

(d) f x x x32= -] g

(e) f x x x2= +] g

(f) y x 42= -

(g) y x x2 32= - -

(h) y x x5 42= +-

(i) p x x 1 2= +] ]g g

(j) y x 2= -2] g




CONTINUED

eg When x 3= -

| | y 3 13 12

= - -

= -

=

x -3 -2 -1 0 1 2 3

y 2 1 0 -1 0 1 2


y

-2

4

3

2

1

5

-1

-3

-4

-5

1 4-1-2 32-3-4

x


| | y x

x x

x x1

1 0

1 0

when

when 1

$= - =

-


y x x1 0$= - ] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x - 1




y x 1= - - x 01] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y =- x - 1



-3

4

3

2

1

5

-2

-1

-4

-5

y y

3-1-2 421-3-4

x

y = - x - 1

y = x - 1


intercepts


So f x 0=

] g

| |

| |

x

x

x

0 1

1

1`

= -

=

=


( ) | |f 0 0 11

= -

= -





-3

4

3

2

1

5

-2

-1

-4

-5

y

4-2 5321-1-3-4

x



Domain all real x

Range y y 1$ -

2 Sketch | | y x 2= +

Solution


| | ( ) y x x xx x2 2 2 0

2 2 0whenwhen 1

$= + = + +

- + +


2 y x= + when x 2 0$+

x 2$ -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x + 2






y-intercept of 1-

CONTINUED




2 y x= - +] g when x 2 01+

ie y x 2= - - when x 21 -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2


-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2

y = x + 2



So 0f x =] g

0 | 2 |

0 2

2

x

x

x

= +

= +

- =


(0) | 0 2 |

2

f = +

=

There is only one

solution for the


Can you see why





-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

If you know how to

sketch the graph of




Investigation




are their equations





1 | |f x k x=] g

2 | |f x x k= +] g

3 | |f x x k= +] g







EXAMPLES

Solve

1 |2 1 | 3x - =

Solution



is x 1 2= -

2 |2 1 | 3 2x x= -+

Solution



3 | 1 | 2x 1+

Solution






and then check them







each function

(a) | | y x=

(b) | |f x x 7= +] g

(c) | |f x x 2= -] g

(d) 5 | | y x=

(e) | |f x x 3= - +] g

(f) | 6 | y x= +

(g) | |f x x3 2= -] g

(h) | 5 4 | y x= +

(i) | 7 1 | y x= -

(j) | |f x x2 9= +] g


plane

(a) | | y x=

(b) | |f x x 1= +] g

(c) | |f x x 3= -] g

(d) 2 | | y x=

(e) | |f x x= -] g

(f) | 1 | y x= +

(g) | |f x x 1= - -] g

(h) | 2 3 | y x= -

(i) | 4 2 | y x= +

(j) | |f x x3 1= +] g


each function

(a) | 1 | y x= -

(b) | |f x x 8= -] g

(c) | |f x x2 5= +] g

(d) 2 | | 3 y x= -

(e) | |f x x 3= - -] g


function is

increasing(i)

decreasing(ii)

(a) | 2 | y x= -

(b) | |f x x 2= +

] g

(c) | |f x x2 3= -] g

(d) 4 | | 1 y x= -

(e) | |f x x= -] g


of each function

(a) | | y x= for x2 2 -

(b) | |f x x 4= - -] g for

x4 3 -

(c) | |f x x 4= +] g for x7 2 -

(d) | 2 5 | y x= - for x3 3 -

(e) | |f x x= -] g for x1 1 -


function increasing

(a) | 3 | y x= +

(b) | |f x x 4= - +] g

(c) | |f x x 9= -] g

(d) | | y x 2 1= - -

(e) | |f x x 2= - +] g

56 Exercises




7 Solve graphically

(a) | | 3x =

(b) | |x 12

(c) | |x 2

(d) | 2 | 1x + =

(e) | 3 | 0x- =

(f) |2 3 | 1x - =

(g) | |x 1 41-

(h) | |x 1 3+

(i) | |x 2 22-

(j) | |x 3 1$-

(k) | |x2 3 5+

(l) | |x2 1 1$-

(m) |3 1 | 3x x- = +

(n) |3 2 | 4x x- = -

(o) | 1 | 1x x- = +

(p) | 3 | 2 2x x+ = + (q) |2 1 | 1x x+ = -

(r) |2 5 | 3x x- = -

(s) | 1 | 2x x- =

(t) |2 3 | 3x x- = +

The Hyperbola


or= =

EXAMPLE

Sketch1

y x=

Solution



x -3 -2 -121-

41- 0

41

21 1 2 3

y 3

1-

2

1- -1 -2 -4 mdash 4 2 1

2

1

3

1

Class Discussion



y is never 0 Why






but never touches)



Investigation




equations




1 y xk

=

21

y x k= +

31

y x k

=+


EXAMPLES


3=

-

] g


Solution



So x 3






be zero

CONTINUED







1

y 0 3

3=

-

= -

-3

4

3

2

1

5

-2

-1

-4

-5

y

-1-2 4 521-3-4

x

x = 3

y = 0

Asymptotes

3

2 Sketch y x2 4

1= -

+

Solution




x

x

2 4

2

-

-






( ) y

2 0 41

41

= -+

= -


a translation of3

y x

=





y

-2

x

-

1

4


a=

+


domain x xb



1 For each graph



exists


(a)2

y x=

(b)1

y x= -

(c) f xx 1

1=

+] g

(d) f xx 2

3=

-

] g

(e)3 6

1 y

x=

+

(f) f xx 3

2= -

-

] g

(g) f xx 1

4=

-

] g

(h)1

2 y

x= -

+

(i) f xx6 3

2=

-

] g

(j)2

6 y

x= -

+

2 Show that f x x2

=] g is an odd

function



(a) f xx2 5

1=

+

] g for x2 2 -

(b)3

1 y

x=

+ for x2 0 -

(c) f xx2 4

5=

-

] g for x3 1 -

57 Exercises




(d) f xx 4

3= -

-

] g for x3 3 -

(e)3 1

2 y

x= -

+ for x0 5



(a)3

y x= for y 1 3

(b)2

y x= - for y 221

- -

(c) f xx 1

1=

-

] g for y 171

- -

(d) f xx2 1

3= -

+

] g for

y 131

- -

(e)3 2

6 y

x=

- for y 1

21

6



Circle gate





Proof

r y

x

( x y)

y

x








c a b

r x y

2 2 2

2 2 2

`

= +

= +

EXAMPLE



Solution


y

x

-2

-2 2

2


one place

y

x

2

2

2

-2

The radius is 4

CONTINUED






Domain 2 2 x x -

Range 2 2 y y -


range y r y r -



Proof


shown

y

x

(a b)

x

y

r

( x y)

a

b x - a

y - b




ndash ndash

c a b

r x a y b

2 2 2

2 2 2

= +

= +] ^g h




EXAMPLES



Solution



y

x 9

9

-9

-9



Domain 9 9 x x -

Range 9 9 y y -



Solution


ndash

ndash ndash

x y

x y

1 2 4

1 2 2

2

2 2

+ + =

+ - =

2

2

] ^

] ]_

g h

g gi

So 1 2a b= = - and 2r =

CONTINUED







y

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

(1 -2)



Domain 1 3 x x -

Range 4 0 y y -


Solution


a b2 1= - = and r 3=

Substituting

ndash ndash

ndash

ndash

x a y b r

x y

x y

2 1 3

2 1 9

2 2 2

2 2 2

2 2

+ =

- - + =

+ + =

] ^]] ^

] ^

g hg g hg h


ndash

ndash

a b a ab b

x x x

x x

a b a ab b

y y y

y y

2

2 2 2 2

4 4

2

1 2 1 1

2 1

So

So

2 2 2

2 2 2

2

2 2 2

2 2 2

2

+ = + +

+ = + +

= + +

= - +

= - +

= - +

]] ] ]]^ ^ ]

gg g ggh h g


ndash

x x y y

x x y y

x x y y

x x y y

4 4 2 1 9

4 2 5 9

4 2 5 9

4 2 4 0

9 9

2

2

2

2

+ + + - + =

+ + - + =

+ + + =

+ + - - =

- -


in Chapter 2




Investigation


two functions





circle






for all values

y

x r

r

-r


Proof

ndash

x y r

y r x y r x

2 2 2

2 2 2

2 2

+ =

=

= -



semi-circles





y 0 for all values

y

x r

-

r

-r


EXAMPLES


1 f x x92

= -] g

Solution



y

x 3

3

-3

Domain 3 3 x x -

Range 0 3 y y




2 y x4 2= - -

Solution



y

x 2

-2

-2

Domain 2 2 x x -

Range 2 0 y y -




range

(a) 9x y 2 2+ =

(b) x y 16 02 2+ =-


(d) 1 9x y 2 2+ + =

] g

(e) ndashx y 2 1 12 2+ + =] ^g h



below the x -axis



range

(a) 25 y x2= - -

(b) 1 y x2= -

(c) 36 y x2= -

(d) 64 y x2= - -

(e) 7 y x2= - -



each circle

(a) 100x y 2 2+ =

(b) 5x y 2 2+ =


(d) ndashx y 5 6 492 2+ + =] ^g h

(e) ndashx y 3 812 2+ =^ h

58 Exercises






grouping symbols)



Centre(c) 1 5-







Centre(j) 4 7- -

^ h and radius 3

Other Graphs




EXAMPLES


Solution



eg When 0x =

y 3

1

c=

=

x

y

1

3

3

1

31

When1

1

= -

=

=

=

-

x 3- 2- 1- 0 1 2 3

y 271

91

31

1 3 9 27



You will meet these

graphs again in the

HSC Course





other point

When

x

y

1

3

3

1

=

=

=

y

x

1

2

1

3







Solution



x minus2 minus1 0 05 1 2 3 4

y minus03 0 03 05 06

y

x

1

2

1 2 3 4

-1




You learned about


stages of maths





range y y 02



range all real y

Cubic function


EXAMPLE



Solution



y minus25 minus6 1 2 3 10 29

y

x

1

1

-2 2 3 4

-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4



If you already

know the shape of


3






Domain and range


EXAMPLE


Solution




function are

f

f

1 1 2

1 21

4 4 2

64 2

66

3

3

- = - +

= - +

=

= +

= +

=

] ]

] ]

g g

g g


these points

y

x

(-1 1)

(4 66)

Range 1 66 y y










positive number

EXAMPLE


Solution


x 4So $

$


(or zero) So y 0$


59 Exercises


(a) 4 3 y x= +

(b) f x 4= -] g

(c) 3x =



(f) f x xx 12 2= - -] g

(g) 64x y 2 2+ =

(h) f t t 4

3=

-] g

(i) ( ) g 2

5zz

= +

(j) | |f x x=] g


(a) y x=

(b) 2 y x= -

(c) | |f x x2 3= -] g

(d) | | 2 y x= -

(e) f x x2 5= - +] g

(f) | | y x5= -

(g) 2 y x=

(h) y 5x= -

(i) f x xx 1

= +] g

(j)2

4 3 y x

x= -


(a) y x x 5 2= -] g


= +-

(d) g x x x164 2= -] g

(e) 49x y 2 2+ =

You may like to


by dividing by x




4 (a) Solve x1 02$-


f x x1 2= -] g


(a) 2 y x x2= - -

(b) g t t t 62= +] g



in each case


x3 3 -


x2 3 -


x2 1 - (d)


x1 5


0 4x


x3 3 -


x1 1 -


x2 3 -


x4 4 -

(j) y x x7 62= - + - in the

domain 0 7x


function y x 1

3=

+





x=] g


function

find its range(b)


plane

(a) f x x4=] g

(b) y x3= -

(c) y x 34= -

(d) 2 p x x3=] g

(e) 1 g x x3= +] g

(f) 100x y 2 2+ =

(g) 2 1 y x= +


y x 1= -









(b) f x x 22= -] g

(c)1

y x=

(d) f x x3=] g

(e) f x 3x=] g

13 (a) Solve x4 02$-


(i) 4 y x2= -

(ii) y x4

2= - -




DID YOU KNOW




a light



Limits






a a1

1

03

Z

=

=

3

3

-


lim a 0x=

x 3-



asymptote

EXAMPLES

1 Find lim x

x x5x 0

2+

Solution


which is undefined


( )

lim lim

lim

xx x

x

x x

x

5 5

5

5

x x

x

0

2

0 1

1

0

+=

+

= +

=

] g




2 Find limx

x

4

22

-

-

x 2

Solution


lim lim

lim

x

x

x x

x

x

4

2

2 2

2

21

41

2 1

1

-

-=

+ -

-

=+

=

x x2 2

x 2

^ _h i

3 Find limh

h x hx h2 72 2+ -

h 0

Solution

lim lim

lim

h

h x hx h

h

h hx x

hx x

x

2 7 2 7

2 7

7

2 2 2

2

2

+ -=

+ -

= + -

= -

h 0

h h0 0

^ h

Continuity





EXAMPLES

1 Find lim

x

x

1

12

-

-

x 1


11

y x

x2

=-

- Sketch the curve

Solution


xx2

-

- gives

00

CONTINUED




( )

lim lim

limx

x

x

x x

x11

1

1 1

1

2

x x

x

1

2

1

1

-

-=

-

+ -

= +

=

-

] ]g g

11

y xx2

=-





y xx

x

x x

x

11

1

1 1

1

2

=-

-

=

+

= +

-

-^ ^h h


2 Find limx

x x2

2x 2

2

+

+ -

-


x x2

22

= + -

+

Solution


x x2

22

+

+ - gives

00

lim lim

lim

xx x

x

x x

x

22

2

1 2

1

3

x x

x

2

2

2

2

+

+ -=

+

- +

=

= -

-

- -

-

^^ ^

^

hh h

h2 y

x

x xx

y x

x

x

x

22

2

2

1

1


=+

+ -= -

=+

= -

+ -^ ^h h


Remember that x 1




1 Find(a) lim x 52

+x 4

(b) lim t 7-t 3 -

(c) lim x x2 43+ -

x 2

(d) lim xx x32

+

x 0

(e) limh

h h

2

22

-

- -

h 2

(f) lim y

y

5

1253

-

-

y 5

(g) limx

x x

12 12

+

+ +

x 1-

(h) limx

x x4

2 82

+

+ -

x 4 -

(i) limc

c

4

22

-

-

c 2

(j) limx x

x 12

-

-

x 1

(k) lim h

h h h2 73 2+ -

h 0

(l) limh

hx hx h32 2- +

h 0

(m) limh

hx h x hx h2 3 53 2 2- + -

h 0

(n) lim x c x c 3 3

-

-

x c



discontinuous

(a) 3 y x2= -

(b)1

1 y

x=

+

(c) f x x 1= -] g

(d)4

1 y

x2=

+

(e)4

1

y x2=

-



(a)3

y xx x2

= +

(b)33

y x

x x2

=+

+

(c)1

5 4 y

xx x2

=+

+ +

510 Exercises




Regions

Class Investigation


them all







EXAMPLE


Solution



y

x =3

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4







region

EXAMPLE


Solution



included


shown

y

x

1

1

-2 2 3 4

-3

-4

-5

2

3

4

5

-1

-2

-3-4

-1 y = -1



y -intercept 1-






EXAMPLES


1 y x 2$ +

Solution






y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0



2 x y 2 3 61-

Solution


region


side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g








3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g







graph is a function

y

x



x

y





place





EXAMPLES


Solution



Solution


not a function







2 3 1 4 0 5 1 3 2 4- -^ ^ ^ ^ ^h h h h h Solution


function


y

x

3

Solution

y

x

3





a function




1

2

3

4

5

6

7

8

9 1 3 2 1 3 3 4 0-^ ^ ^ ^h h h h 10 1 3 2 1 2 7 4 0-^ ^ ^ ^h h h h 11

1

2

3

4

5

1

2

3

4

5

12 1

2

3

4

5

1

2

3

4

5

131

2

3

4

5

1

2

3

4

5

51 Exercises







15 A 3

B 4

C 7

D 3

E 5

F 7

G 4

Function notation




EXAMPLES


Solution

When x

y x

3

1

3 1

4

=

= +

= +

=


Solution

f x x

f

1

3 3 1

4

= +

= +

=

]] gg





EXAMPLES

1 If f x x x3 12= + +] g find f 2-] g

Solution

( ) ( )f 2 2 3 2 1

4 6 1

1

2- = - + - +

= - +

= -

] g


Solution

( )

( )

f x x x

f 1 1 1

1 12

3 2

3

= -

- = - - -

= - -

= -

2] ]g g


Solution

( )

ie

( ) ( )

f x

x x

x x

x x

x x

0

3 10 0

5 2 0

5 0 2 0

5 2

2

=

+ - =

+ - =

+ = - =

= - =


when

when f x

x x

x x

3 4 2

2 21

$=

+

-

] g )

Solution

since 4 21-

( ) ( ) since

( ) ( ) since

( ) ( ) since

( ) ( )

f

f

f

f

3 3 3 4 3 2

13

2 3 2 4 2 2

10

0 2 0 0 2

0

4 2 4

8

1

$

$

= +

=

= +

=

= -

=

- = - -

=


when

when

x

x

x

2

1 2

1

2

1

-

-

g x

x

x2 1

5

2

= -] g


when 2x -=







than 2




Solution

( ) ( )

( )

( )

g

g

g

1 2 1 1 1 1 2

1

2 5 2 1

3 3 3 2

9

since

since

since2

1

2

= - -

=

- = - -

=

=

( ) ( ) ( ) g g g 1 2 3 1 5 9

3

So + - - = + -

= -

DID YOU KNOW






f 3-] g

2 If h x x 22= -] g find h h0 2] ]g g

and h 4-] g


and f 2-] g


= - +] g

5 Find f 3-] g if f x x x2 5 43= - +] g


f x 13=] g




f x271

=] g




f x h+] g


g x x x2 32= + +] g


product of factors






15 f x x x

x x

1

1

for

for

32

=] g )


16 f x

x x

x x

x x

2 4 1

3 1 1

1

if

if

if 2

1 1

$

=

-

+ -

-

] gZ

[

]]

]]

Find the values of

f f f 2 2 1- - + -] ] ]g g g

52 Exercises








EXAMPLE


Solution


x x

x x

x x

x x

0 7 8

8 1

8 0 1 0

8 1

2= + -

= + -

+ = - =

= - =

] ]g g


y 0 7 0 8

8

2= + -

= -

] ]g g





Domain and range




domain


image) of f

EXAMPLE


Solution


x

y

CONTINUED






Domain all real x

Range y y 0$















EXAMPLES


x x 3 x 2 x 1

y

Solution


increasing


x 1 x 2

2

x x 3

x 2 x 1

y

Solution



increases again




1 1








=







x

y









x

y




EXAMPLES


Solution

f x x

x

f x

f x x

3

3


2

2

2`

- = - +

= +

=

= +

] ]]

]

g gg

g


Solution

f x x x

x x

x x

f x


3

3

3

3`

- = - - -

= - +

= - -

= -

= -

] ] ]^]

]

g g gh

gg

Investigation






1 f x kxn=] g

2 f x x kn= +] g

3 f x x k n= +

] ]g g






shape of graphs


each function

(a) y x3 2= -

(b) x y 2 5 20 0- + =

(c) x y 3 12 0+ - =

(d) f x x x32= +] g

(e) f x x 42= -] g

(f) p x x x5 62= + +] g

(g) y x x8 152= - +

(h) p x x 53

= +

] g

53 Exercises




(i) y xx

x3

0= + ] g

(j) g x x9 2= -] g


f x x 22= -

] g What type of

function is it


(a) f x2^ h (b) ( )f x 26



an even function


f x x=] g


function


odd function



f x f x- -] ]g g


neither

(a) y x x

x4 2

3

=

-

(b) y x 1

13

=

-

(c) f xx 4

32

=

-

] g

(d) y x

x

33

=+

-

(e) f x x x

x5 2

3

=-] g



f x xn=] g

even(a)

odd(b)


ever be

even(a)

odd(b)



graph is increasing




x

y(a)

x

4

y(b)

2-2

x

y(c)




Investigation


graph







y kx

y kx

y kx

y kx

y xk

(a)

(b)

(c)

(d)

(e)

2

3

4

=

=

=

=

=



y x k

y x k

y x k

y x k

y x k1

(a)

(b)

(c)

(d)

(e)

2

2

3

4

= +

= +

= +

= +

= +

] g



-2

1 2

-4

-1-2

2

4

y

x

(d) y

x

(e)

CONTINUED




Gradient form



Investigation










y x k

y x k

y x k

y x k y

x k

1

(a)

(b)

(c)

(d)(e)

2

3

4

= +

= +

= +

= +

=+

]]

]

gg

g





the right








Linear Function





EXAMPLE



Solution


Find the intercepts


0 3 5

5 3

1

x

x

x32

=

=

=

-


3 5

5

y 0=

= -

-] g

-1

-2

y

5

4

3

2

1 1 23

6

-3

-4

-5

1 4-1-2 32-3-4

x



Range all real y








Special lines






1 y kx=

2 y x k= +






EXAMPLES


Solution




-1

-3

y

4

3

2

1

5

-2

-4

-5

1 4-1-2

x

32-3-4


2 Sketch x 1= -


Solution





-

-

4

3

2

5

-2

-4

-5

1 4-1-2-4

y

x







Domain all real x

Range y y b=

54 Exercises


each function

(a) y x 2= -

(b) f x x2 3= +] g (c) x y 2 1 0+ =-

(d) x y 3 0+ =-

(e) x y 3 6 2 0=- -


line

(a) x 4=

(b) x 3 0=-

(c) y 5=

(d) y 1 0+ =

(e) f x x2 1= -] g (f) y x 4= +

(g) f x x3 2= +] g

(h) x y 3+ =

(i) x y 1 0=- -

(j) x y 2 3 0+ =-


(a) x y 3 2 7 0+ =-

(b) y 2=

(c) x 4= - (d) x 2 0=-

(e) y 3 0=-


are even or odd

(a) y x2=

(b) y 3=

(c) x 4=

(d) y x= -

(e) y x=




meet




Applications






The pronumeral

a is called the


Quadratic Function










EXAMPLES



Solution



x

x

x

0 1

1

1

2

2

= -

=

=


0 1

1

y 2= -

= -



Domain xall real

Range y y 1$ -


Solution


parabola will lie




left



0

1 0

1

x

x

x

1 2= +

+ =

= -

] g


1

y 0 1 2= +

=

] g

-1

-

4

3

2

1

5

-2

-4

-5

-

1 41-2 5-4

y

x









Solution



x x

x xx x

x x

0 6

3 23 0 2 0

3 2

2= + -

= + -

+ = - =

= - =

] ]g g


f 0 0 0 66

2= + -

= -

] ] ]g g g



x -intercepts


23 2

21- +

= -


-c m

f 21

21

21

6

41

21

6

641

2

- = - + - -

= - -

= -

c c cm m m


-

CONTINUED

You will learn more


-1

-3

4

3

2

1

5

-2

-4

-5

1 4-1-2 32-3-4

y

x










41



2

11 -





Solution


So f x 0=] g

0 4 54 5 0

0

x xx x

x x5 1

2

2

= - + +

=

+ =

- -

-] ]g g

x x

x x

5 0 1 0

5 1

- = + =

= = -


f 0 0 4 0 5

5

2= - + +

=

] ] ]g g g

-1

-3

4

2

5

-2

-4

-5

-

y

1 41-2 5-4

x

-2 4






21 5

2- +

=

f 2 2 4 2 59

= - + +

=

2] ] ]g g g










each function

(a) 2 y x x2= +

(b) 3 y x x2= - +

(c) f x x 12= -] g

(d) y x x 22= - -

(e) y x x9 82= +-

2 Sketch

(a) 2 y x2= +

(b) y x 12= - +

(c) f x x 42= -] g

(d) 2 y x x2= +

(e) y x x2= - -

(f) f x x 3= -2] ]g g

55 Exercises

-1

9

8

7

5

4

3

2

6

1

-2

-3

-4

-5

y

2 51 643-1-2-3-4

x




EXAMPLES


Solution









(g) f x x 1 2= +] ]g g

(h) y x x3 42= + -

(i) y x x2 5 32= - +

(j) f x x x3 22= - + -] g



(a) ndash y x x7 122= +

(b) f x x x42= +] g

(c) y x x2 82= - -

(d) y x x6 92= +-

(e) f t t 4 2= -] g


(a) y x 52= -

(b) f x x x6

2= -

] g (c) f x x x 22= - -] g

(d) y x2= -

(e) f x x 7 2= -] ]g g



(a) y x2= for x0 3

(b) y x 42= - + for x1 2 -

(c) f x x 12= -] g for x2 5 -

(d) y x x2 32= + - for x2 4 -

(e) y x x

22= - +- for

x0 4


function is

increasing(i)

decreasing(ii)

(a) y x2=

(b) y x2

= - (c) f x x 92

= -] g

(d) y x x42= - +

(e) f x x 5 2= +] ]g g


function



(a) y x 12= +

(b) f x x 32= -

] g

(c) y x2 2= -

(d) f x x x32= -] g

(e) f x x x2= +] g

(f) y x 42= -

(g) y x x2 32= - -

(h) y x x5 42= +-

(i) p x x 1 2= +] ]g g

(j) y x 2= -2] g




CONTINUED

eg When x 3= -

| | y 3 13 12

= - -

= -

=

x -3 -2 -1 0 1 2 3

y 2 1 0 -1 0 1 2


y

-2

4

3

2

1

5

-1

-3

-4

-5

1 4-1-2 32-3-4

x


| | y x

x x

x x1

1 0

1 0

when

when 1

$= - =

-


y x x1 0$= - ] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x - 1




y x 1= - - x 01] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y =- x - 1



-3

4

3

2

1

5

-2

-1

-4

-5

y y

3-1-2 421-3-4

x

y = - x - 1

y = x - 1


intercepts


So f x 0=

] g

| |

| |

x

x

x

0 1

1

1`

= -

=

=


( ) | |f 0 0 11

= -

= -





-3

4

3

2

1

5

-2

-1

-4

-5

y

4-2 5321-1-3-4

x



Domain all real x

Range y y 1$ -

2 Sketch | | y x 2= +

Solution


| | ( ) y x x xx x2 2 2 0

2 2 0whenwhen 1

$= + = + +

- + +


2 y x= + when x 2 0$+

x 2$ -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x + 2






y-intercept of 1-

CONTINUED




2 y x= - +] g when x 2 01+

ie y x 2= - - when x 21 -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2


-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2

y = x + 2



So 0f x =] g

0 | 2 |

0 2

2

x

x

x

= +

= +

- =


(0) | 0 2 |

2

f = +

=

There is only one

solution for the


Can you see why





-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

If you know how to

sketch the graph of




Investigation




are their equations





1 | |f x k x=] g

2 | |f x x k= +] g

3 | |f x x k= +] g







EXAMPLES

Solve

1 |2 1 | 3x - =

Solution



is x 1 2= -

2 |2 1 | 3 2x x= -+

Solution



3 | 1 | 2x 1+

Solution






and then check them







each function

(a) | | y x=

(b) | |f x x 7= +] g

(c) | |f x x 2= -] g

(d) 5 | | y x=

(e) | |f x x 3= - +] g

(f) | 6 | y x= +

(g) | |f x x3 2= -] g

(h) | 5 4 | y x= +

(i) | 7 1 | y x= -

(j) | |f x x2 9= +] g


plane

(a) | | y x=

(b) | |f x x 1= +] g

(c) | |f x x 3= -] g

(d) 2 | | y x=

(e) | |f x x= -] g

(f) | 1 | y x= +

(g) | |f x x 1= - -] g

(h) | 2 3 | y x= -

(i) | 4 2 | y x= +

(j) | |f x x3 1= +] g


each function

(a) | 1 | y x= -

(b) | |f x x 8= -] g

(c) | |f x x2 5= +] g

(d) 2 | | 3 y x= -

(e) | |f x x 3= - -] g


function is

increasing(i)

decreasing(ii)

(a) | 2 | y x= -

(b) | |f x x 2= +

] g

(c) | |f x x2 3= -] g

(d) 4 | | 1 y x= -

(e) | |f x x= -] g


of each function

(a) | | y x= for x2 2 -

(b) | |f x x 4= - -] g for

x4 3 -

(c) | |f x x 4= +] g for x7 2 -

(d) | 2 5 | y x= - for x3 3 -

(e) | |f x x= -] g for x1 1 -


function increasing

(a) | 3 | y x= +

(b) | |f x x 4= - +] g

(c) | |f x x 9= -] g

(d) | | y x 2 1= - -

(e) | |f x x 2= - +] g

56 Exercises




7 Solve graphically

(a) | | 3x =

(b) | |x 12

(c) | |x 2

(d) | 2 | 1x + =

(e) | 3 | 0x- =

(f) |2 3 | 1x - =

(g) | |x 1 41-

(h) | |x 1 3+

(i) | |x 2 22-

(j) | |x 3 1$-

(k) | |x2 3 5+

(l) | |x2 1 1$-

(m) |3 1 | 3x x- = +

(n) |3 2 | 4x x- = -

(o) | 1 | 1x x- = +

(p) | 3 | 2 2x x+ = + (q) |2 1 | 1x x+ = -

(r) |2 5 | 3x x- = -

(s) | 1 | 2x x- =

(t) |2 3 | 3x x- = +

The Hyperbola


or= =

EXAMPLE

Sketch1

y x=

Solution



x -3 -2 -121-

41- 0

41

21 1 2 3

y 3

1-

2

1- -1 -2 -4 mdash 4 2 1

2

1

3

1

Class Discussion



y is never 0 Why






but never touches)



Investigation




equations




1 y xk

=

21

y x k= +

31

y x k

=+


EXAMPLES


3=

-

] g


Solution



So x 3






be zero

CONTINUED







1

y 0 3

3=

-

= -

-3

4

3

2

1

5

-2

-1

-4

-5

y

-1-2 4 521-3-4

x

x = 3

y = 0

Asymptotes

3

2 Sketch y x2 4

1= -

+

Solution




x

x

2 4

2

-

-






( ) y

2 0 41

41

= -+

= -


a translation of3

y x

=





y

-2

x

-

1

4


a=

+


domain x xb



1 For each graph



exists


(a)2

y x=

(b)1

y x= -

(c) f xx 1

1=

+] g

(d) f xx 2

3=

-

] g

(e)3 6

1 y

x=

+

(f) f xx 3

2= -

-

] g

(g) f xx 1

4=

-

] g

(h)1

2 y

x= -

+

(i) f xx6 3

2=

-

] g

(j)2

6 y

x= -

+

2 Show that f x x2

=] g is an odd

function



(a) f xx2 5

1=

+

] g for x2 2 -

(b)3

1 y

x=

+ for x2 0 -

(c) f xx2 4

5=

-

] g for x3 1 -

57 Exercises




(d) f xx 4

3= -

-

] g for x3 3 -

(e)3 1

2 y

x= -

+ for x0 5



(a)3

y x= for y 1 3

(b)2

y x= - for y 221

- -

(c) f xx 1

1=

-

] g for y 171

- -

(d) f xx2 1

3= -

+

] g for

y 131

- -

(e)3 2

6 y

x=

- for y 1

21

6



Circle gate





Proof

r y

x

( x y)

y

x








c a b

r x y

2 2 2

2 2 2

`

= +

= +

EXAMPLE



Solution


y

x

-2

-2 2

2


one place

y

x

2

2

2

-2

The radius is 4

CONTINUED






Domain 2 2 x x -

Range 2 2 y y -


range y r y r -



Proof


shown

y

x

(a b)

x

y

r

( x y)

a

b x - a

y - b




ndash ndash

c a b

r x a y b

2 2 2

2 2 2

= +

= +] ^g h




EXAMPLES



Solution



y

x 9

9

-9

-9



Domain 9 9 x x -

Range 9 9 y y -



Solution


ndash

ndash ndash

x y

x y

1 2 4

1 2 2

2

2 2

+ + =

+ - =

2

2

] ^

] ]_

g h

g gi

So 1 2a b= = - and 2r =

CONTINUED







y

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

(1 -2)



Domain 1 3 x x -

Range 4 0 y y -


Solution


a b2 1= - = and r 3=

Substituting

ndash ndash

ndash

ndash

x a y b r

x y

x y

2 1 3

2 1 9

2 2 2

2 2 2

2 2

+ =

- - + =

+ + =

] ^]] ^

] ^

g hg g hg h


ndash

ndash

a b a ab b

x x x

x x

a b a ab b

y y y

y y

2

2 2 2 2

4 4

2

1 2 1 1

2 1

So

So

2 2 2

2 2 2

2

2 2 2

2 2 2

2

+ = + +

+ = + +

= + +

= - +

= - +

= - +

]] ] ]]^ ^ ]

gg g ggh h g


ndash

x x y y

x x y y

x x y y

x x y y

4 4 2 1 9

4 2 5 9

4 2 5 9

4 2 4 0

9 9

2

2

2

2

+ + + - + =

+ + - + =

+ + + =

+ + - - =

- -


in Chapter 2




Investigation


two functions





circle






for all values

y

x r

r

-r


Proof

ndash

x y r

y r x y r x

2 2 2

2 2 2

2 2

+ =

=

= -



semi-circles





y 0 for all values

y

x r

-

r

-r


EXAMPLES


1 f x x92

= -] g

Solution



y

x 3

3

-3

Domain 3 3 x x -

Range 0 3 y y




2 y x4 2= - -

Solution



y

x 2

-2

-2

Domain 2 2 x x -

Range 2 0 y y -




range

(a) 9x y 2 2+ =

(b) x y 16 02 2+ =-


(d) 1 9x y 2 2+ + =

] g

(e) ndashx y 2 1 12 2+ + =] ^g h



below the x -axis



range

(a) 25 y x2= - -

(b) 1 y x2= -

(c) 36 y x2= -

(d) 64 y x2= - -

(e) 7 y x2= - -



each circle

(a) 100x y 2 2+ =

(b) 5x y 2 2+ =


(d) ndashx y 5 6 492 2+ + =] ^g h

(e) ndashx y 3 812 2+ =^ h

58 Exercises






grouping symbols)



Centre(c) 1 5-







Centre(j) 4 7- -

^ h and radius 3

Other Graphs




EXAMPLES


Solution



eg When 0x =

y 3

1

c=

=

x

y

1

3

3

1

31

When1

1

= -

=

=

=

-

x 3- 2- 1- 0 1 2 3

y 271

91

31

1 3 9 27



You will meet these

graphs again in the

HSC Course





other point

When

x

y

1

3

3

1

=

=

=

y

x

1

2

1

3







Solution



x minus2 minus1 0 05 1 2 3 4

y minus03 0 03 05 06

y

x

1

2

1 2 3 4

-1




You learned about


stages of maths





range y y 02



range all real y

Cubic function


EXAMPLE



Solution



y minus25 minus6 1 2 3 10 29

y

x

1

1

-2 2 3 4

-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4



If you already

know the shape of


3






Domain and range


EXAMPLE


Solution




function are

f

f

1 1 2

1 21

4 4 2

64 2

66

3

3

- = - +

= - +

=

= +

= +

=

] ]

] ]

g g

g g


these points

y

x

(-1 1)

(4 66)

Range 1 66 y y










positive number

EXAMPLE


Solution


x 4So $

$


(or zero) So y 0$


59 Exercises


(a) 4 3 y x= +

(b) f x 4= -] g

(c) 3x =



(f) f x xx 12 2= - -] g

(g) 64x y 2 2+ =

(h) f t t 4

3=

-] g

(i) ( ) g 2

5zz

= +

(j) | |f x x=] g


(a) y x=

(b) 2 y x= -

(c) | |f x x2 3= -] g

(d) | | 2 y x= -

(e) f x x2 5= - +] g

(f) | | y x5= -

(g) 2 y x=

(h) y 5x= -

(i) f x xx 1

= +] g

(j)2

4 3 y x

x= -


(a) y x x 5 2= -] g


= +-

(d) g x x x164 2= -] g

(e) 49x y 2 2+ =

You may like to


by dividing by x




4 (a) Solve x1 02$-


f x x1 2= -] g


(a) 2 y x x2= - -

(b) g t t t 62= +] g



in each case


x3 3 -


x2 3 -


x2 1 - (d)


x1 5


0 4x


x3 3 -


x1 1 -


x2 3 -


x4 4 -

(j) y x x7 62= - + - in the

domain 0 7x


function y x 1

3=

+





x=] g


function

find its range(b)


plane

(a) f x x4=] g

(b) y x3= -

(c) y x 34= -

(d) 2 p x x3=] g

(e) 1 g x x3= +] g

(f) 100x y 2 2+ =

(g) 2 1 y x= +


y x 1= -









(b) f x x 22= -] g

(c)1

y x=

(d) f x x3=] g

(e) f x 3x=] g

13 (a) Solve x4 02$-


(i) 4 y x2= -

(ii) y x4

2= - -




DID YOU KNOW




a light



Limits






a a1

1

03

Z

=

=

3

3

-


lim a 0x=

x 3-



asymptote

EXAMPLES

1 Find lim x

x x5x 0

2+

Solution


which is undefined


( )

lim lim

lim

xx x

x

x x

x

5 5

5

5

x x

x

0

2

0 1

1

0

+=

+

= +

=

] g




2 Find limx

x

4

22

-

-

x 2

Solution


lim lim

lim

x

x

x x

x

x

4

2

2 2

2

21

41

2 1

1

-

-=

+ -

-

=+

=

x x2 2

x 2

^ _h i

3 Find limh

h x hx h2 72 2+ -

h 0

Solution

lim lim

lim

h

h x hx h

h

h hx x

hx x

x

2 7 2 7

2 7

7

2 2 2

2

2

+ -=

+ -

= + -

= -

h 0

h h0 0

^ h

Continuity





EXAMPLES

1 Find lim

x

x

1

12

-

-

x 1


11

y x

x2

=-

- Sketch the curve

Solution


xx2

-

- gives

00

CONTINUED




( )

lim lim

limx

x

x

x x

x11

1

1 1

1

2

x x

x

1

2

1

1

-

-=

-

+ -

= +

=

-

] ]g g

11

y xx2

=-





y xx

x

x x

x

11

1

1 1

1

2

=-

-

=

+

= +

-

-^ ^h h


2 Find limx

x x2

2x 2

2

+

+ -

-


x x2

22

= + -

+

Solution


x x2

22

+

+ - gives

00

lim lim

lim

xx x

x

x x

x

22

2

1 2

1

3

x x

x

2

2

2

2

+

+ -=

+

- +

=

= -

-

- -

-

^^ ^

^

hh h

h2 y

x

x xx

y x

x

x

x

22

2

2

1

1


=+

+ -= -

=+

= -

+ -^ ^h h


Remember that x 1




1 Find(a) lim x 52

+x 4

(b) lim t 7-t 3 -

(c) lim x x2 43+ -

x 2

(d) lim xx x32

+

x 0

(e) limh

h h

2

22

-

- -

h 2

(f) lim y

y

5

1253

-

-

y 5

(g) limx

x x

12 12

+

+ +

x 1-

(h) limx

x x4

2 82

+

+ -

x 4 -

(i) limc

c

4

22

-

-

c 2

(j) limx x

x 12

-

-

x 1

(k) lim h

h h h2 73 2+ -

h 0

(l) limh

hx hx h32 2- +

h 0

(m) limh

hx h x hx h2 3 53 2 2- + -

h 0

(n) lim x c x c 3 3

-

-

x c



discontinuous

(a) 3 y x2= -

(b)1

1 y

x=

+

(c) f x x 1= -] g

(d)4

1 y

x2=

+

(e)4

1

y x2=

-



(a)3

y xx x2

= +

(b)33

y x

x x2

=+

+

(c)1

5 4 y

xx x2

=+

+ +

510 Exercises




Regions

Class Investigation


them all







EXAMPLE


Solution



y

x =3

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4







region

EXAMPLE


Solution



included


shown

y

x

1

1

-2 2 3 4

-3

-4

-5

2

3

4

5

-1

-2

-3-4

-1 y = -1



y -intercept 1-






EXAMPLES


1 y x 2$ +

Solution






y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0



2 x y 2 3 61-

Solution


region


side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g








3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g






EXAMPLES


Solution



Solution


not a function







2 3 1 4 0 5 1 3 2 4- -^ ^ ^ ^ ^h h h h h Solution


function


y

x

3

Solution

y

x

3





a function




1

2

3

4

5

6

7

8

9 1 3 2 1 3 3 4 0-^ ^ ^ ^h h h h 10 1 3 2 1 2 7 4 0-^ ^ ^ ^h h h h 11

1

2

3

4

5

1

2

3

4

5

12 1

2

3

4

5

1

2

3

4

5

131

2

3

4

5

1

2

3

4

5

51 Exercises







15 A 3

B 4

C 7

D 3

E 5

F 7

G 4

Function notation




EXAMPLES


Solution

When x

y x

3

1

3 1

4

=

= +

= +

=


Solution

f x x

f

1

3 3 1

4

= +

= +

=

]] gg





EXAMPLES

1 If f x x x3 12= + +] g find f 2-] g

Solution

( ) ( )f 2 2 3 2 1

4 6 1

1

2- = - + - +

= - +

= -

] g


Solution

( )

( )

f x x x

f 1 1 1

1 12

3 2

3

= -

- = - - -

= - -

= -

2] ]g g


Solution

( )

ie

( ) ( )

f x

x x

x x

x x

x x

0

3 10 0

5 2 0

5 0 2 0

5 2

2

=

+ - =

+ - =

+ = - =

= - =


when

when f x

x x

x x

3 4 2

2 21

$=

+

-

] g )

Solution

since 4 21-

( ) ( ) since

( ) ( ) since

( ) ( ) since

( ) ( )

f

f

f

f

3 3 3 4 3 2

13

2 3 2 4 2 2

10

0 2 0 0 2

0

4 2 4

8

1

$

$

= +

=

= +

=

= -

=

- = - -

=


when

when

x

x

x

2

1 2

1

2

1

-

-

g x

x

x2 1

5

2

= -] g


when 2x -=







than 2




Solution

( ) ( )

( )

( )

g

g

g

1 2 1 1 1 1 2

1

2 5 2 1

3 3 3 2

9

since

since

since2

1

2

= - -

=

- = - -

=

=

( ) ( ) ( ) g g g 1 2 3 1 5 9

3

So + - - = + -

= -

DID YOU KNOW






f 3-] g

2 If h x x 22= -] g find h h0 2] ]g g

and h 4-] g


and f 2-] g


= - +] g

5 Find f 3-] g if f x x x2 5 43= - +] g


f x 13=] g




f x271

=] g




f x h+] g


g x x x2 32= + +] g


product of factors






15 f x x x

x x

1

1

for

for

32

=] g )


16 f x

x x

x x

x x

2 4 1

3 1 1

1

if

if

if 2

1 1

$

=

-

+ -

-

] gZ

[

]]

]]

Find the values of

f f f 2 2 1- - + -] ] ]g g g

52 Exercises








EXAMPLE


Solution


x x

x x

x x

x x

0 7 8

8 1

8 0 1 0

8 1

2= + -

= + -

+ = - =

= - =

] ]g g


y 0 7 0 8

8

2= + -

= -

] ]g g





Domain and range




domain


image) of f

EXAMPLE


Solution


x

y

CONTINUED






Domain all real x

Range y y 0$















EXAMPLES


x x 3 x 2 x 1

y

Solution


increasing


x 1 x 2

2

x x 3

x 2 x 1

y

Solution



increases again




1 1








=







x

y









x

y




EXAMPLES


Solution

f x x

x

f x

f x x

3

3


2

2

2`

- = - +

= +

=

= +

] ]]

]

g gg

g


Solution

f x x x

x x

x x

f x


3

3

3

3`

- = - - -

= - +

= - -

= -

= -

] ] ]^]

]

g g gh

gg

Investigation






1 f x kxn=] g

2 f x x kn= +] g

3 f x x k n= +

] ]g g






shape of graphs


each function

(a) y x3 2= -

(b) x y 2 5 20 0- + =

(c) x y 3 12 0+ - =

(d) f x x x32= +] g

(e) f x x 42= -] g

(f) p x x x5 62= + +] g

(g) y x x8 152= - +

(h) p x x 53

= +

] g

53 Exercises




(i) y xx

x3

0= + ] g

(j) g x x9 2= -] g


f x x 22= -

] g What type of

function is it


(a) f x2^ h (b) ( )f x 26



an even function


f x x=] g


function


odd function



f x f x- -] ]g g


neither

(a) y x x

x4 2

3

=

-

(b) y x 1

13

=

-

(c) f xx 4

32

=

-

] g

(d) y x

x

33

=+

-

(e) f x x x

x5 2

3

=-] g



f x xn=] g

even(a)

odd(b)


ever be

even(a)

odd(b)



graph is increasing




x

y(a)

x

4

y(b)

2-2

x

y(c)




Investigation


graph







y kx

y kx

y kx

y kx

y xk

(a)

(b)

(c)

(d)

(e)

2

3

4

=

=

=

=

=



y x k

y x k

y x k

y x k

y x k1

(a)

(b)

(c)

(d)

(e)

2

2

3

4

= +

= +

= +

= +

= +

] g



-2

1 2

-4

-1-2

2

4

y

x

(d) y

x

(e)

CONTINUED




Gradient form



Investigation










y x k

y x k

y x k

y x k y

x k

1

(a)

(b)

(c)

(d)(e)

2

3

4

= +

= +

= +

= +

=+

]]

]

gg

g





the right








Linear Function





EXAMPLE



Solution


Find the intercepts


0 3 5

5 3

1

x

x

x32

=

=

=

-


3 5

5

y 0=

= -

-] g

-1

-2

y

5

4

3

2

1 1 23

6

-3

-4

-5

1 4-1-2 32-3-4

x



Range all real y








Special lines






1 y kx=

2 y x k= +






EXAMPLES


Solution




-1

-3

y

4

3

2

1

5

-2

-4

-5

1 4-1-2

x

32-3-4


2 Sketch x 1= -


Solution





-

-

4

3

2

5

-2

-4

-5

1 4-1-2-4

y

x







Domain all real x

Range y y b=

54 Exercises


each function

(a) y x 2= -

(b) f x x2 3= +] g (c) x y 2 1 0+ =-

(d) x y 3 0+ =-

(e) x y 3 6 2 0=- -


line

(a) x 4=

(b) x 3 0=-

(c) y 5=

(d) y 1 0+ =

(e) f x x2 1= -] g (f) y x 4= +

(g) f x x3 2= +] g

(h) x y 3+ =

(i) x y 1 0=- -

(j) x y 2 3 0+ =-


(a) x y 3 2 7 0+ =-

(b) y 2=

(c) x 4= - (d) x 2 0=-

(e) y 3 0=-


are even or odd

(a) y x2=

(b) y 3=

(c) x 4=

(d) y x= -

(e) y x=




meet




Applications






The pronumeral

a is called the


Quadratic Function










EXAMPLES



Solution



x

x

x

0 1

1

1

2

2

= -

=

=


0 1

1

y 2= -

= -



Domain xall real

Range y y 1$ -


Solution


parabola will lie




left



0

1 0

1

x

x

x

1 2= +

+ =

= -

] g


1

y 0 1 2= +

=

] g

-1

-

4

3

2

1

5

-2

-4

-5

-

1 41-2 5-4

y

x









Solution



x x

x xx x

x x

0 6

3 23 0 2 0

3 2

2= + -

= + -

+ = - =

= - =

] ]g g


f 0 0 0 66

2= + -

= -

] ] ]g g g



x -intercepts


23 2

21- +

= -


-c m

f 21

21

21

6

41

21

6

641

2

- = - + - -

= - -

= -

c c cm m m


-

CONTINUED

You will learn more


-1

-3

4

3

2

1

5

-2

-4

-5

1 4-1-2 32-3-4

y

x










41



2

11 -





Solution


So f x 0=] g

0 4 54 5 0

0

x xx x

x x5 1

2

2

= - + +

=

+ =

- -

-] ]g g

x x

x x

5 0 1 0

5 1

- = + =

= = -


f 0 0 4 0 5

5

2= - + +

=

] ] ]g g g

-1

-3

4

2

5

-2

-4

-5

-

y

1 41-2 5-4

x

-2 4






21 5

2- +

=

f 2 2 4 2 59

= - + +

=

2] ] ]g g g










each function

(a) 2 y x x2= +

(b) 3 y x x2= - +

(c) f x x 12= -] g

(d) y x x 22= - -

(e) y x x9 82= +-

2 Sketch

(a) 2 y x2= +

(b) y x 12= - +

(c) f x x 42= -] g

(d) 2 y x x2= +

(e) y x x2= - -

(f) f x x 3= -2] ]g g

55 Exercises

-1

9

8

7

5

4

3

2

6

1

-2

-3

-4

-5

y

2 51 643-1-2-3-4

x




EXAMPLES


Solution









(g) f x x 1 2= +] ]g g

(h) y x x3 42= + -

(i) y x x2 5 32= - +

(j) f x x x3 22= - + -] g



(a) ndash y x x7 122= +

(b) f x x x42= +] g

(c) y x x2 82= - -

(d) y x x6 92= +-

(e) f t t 4 2= -] g


(a) y x 52= -

(b) f x x x6

2= -

] g (c) f x x x 22= - -] g

(d) y x2= -

(e) f x x 7 2= -] ]g g



(a) y x2= for x0 3

(b) y x 42= - + for x1 2 -

(c) f x x 12= -] g for x2 5 -

(d) y x x2 32= + - for x2 4 -

(e) y x x

22= - +- for

x0 4


function is

increasing(i)

decreasing(ii)

(a) y x2=

(b) y x2

= - (c) f x x 92

= -] g

(d) y x x42= - +

(e) f x x 5 2= +] ]g g


function



(a) y x 12= +

(b) f x x 32= -

] g

(c) y x2 2= -

(d) f x x x32= -] g

(e) f x x x2= +] g

(f) y x 42= -

(g) y x x2 32= - -

(h) y x x5 42= +-

(i) p x x 1 2= +] ]g g

(j) y x 2= -2] g




CONTINUED

eg When x 3= -

| | y 3 13 12

= - -

= -

=

x -3 -2 -1 0 1 2 3

y 2 1 0 -1 0 1 2


y

-2

4

3

2

1

5

-1

-3

-4

-5

1 4-1-2 32-3-4

x


| | y x

x x

x x1

1 0

1 0

when

when 1

$= - =

-


y x x1 0$= - ] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x - 1




y x 1= - - x 01] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y =- x - 1



-3

4

3

2

1

5

-2

-1

-4

-5

y y

3-1-2 421-3-4

x

y = - x - 1

y = x - 1


intercepts


So f x 0=

] g

| |

| |

x

x

x

0 1

1

1`

= -

=

=


( ) | |f 0 0 11

= -

= -





-3

4

3

2

1

5

-2

-1

-4

-5

y

4-2 5321-1-3-4

x



Domain all real x

Range y y 1$ -

2 Sketch | | y x 2= +

Solution


| | ( ) y x x xx x2 2 2 0

2 2 0whenwhen 1

$= + = + +

- + +


2 y x= + when x 2 0$+

x 2$ -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x + 2






y-intercept of 1-

CONTINUED




2 y x= - +] g when x 2 01+

ie y x 2= - - when x 21 -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2


-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2

y = x + 2



So 0f x =] g

0 | 2 |

0 2

2

x

x

x

= +

= +

- =


(0) | 0 2 |

2

f = +

=

There is only one

solution for the


Can you see why





-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

If you know how to

sketch the graph of




Investigation




are their equations





1 | |f x k x=] g

2 | |f x x k= +] g

3 | |f x x k= +] g







EXAMPLES

Solve

1 |2 1 | 3x - =

Solution



is x 1 2= -

2 |2 1 | 3 2x x= -+

Solution



3 | 1 | 2x 1+

Solution






and then check them







each function

(a) | | y x=

(b) | |f x x 7= +] g

(c) | |f x x 2= -] g

(d) 5 | | y x=

(e) | |f x x 3= - +] g

(f) | 6 | y x= +

(g) | |f x x3 2= -] g

(h) | 5 4 | y x= +

(i) | 7 1 | y x= -

(j) | |f x x2 9= +] g


plane

(a) | | y x=

(b) | |f x x 1= +] g

(c) | |f x x 3= -] g

(d) 2 | | y x=

(e) | |f x x= -] g

(f) | 1 | y x= +

(g) | |f x x 1= - -] g

(h) | 2 3 | y x= -

(i) | 4 2 | y x= +

(j) | |f x x3 1= +] g


each function

(a) | 1 | y x= -

(b) | |f x x 8= -] g

(c) | |f x x2 5= +] g

(d) 2 | | 3 y x= -

(e) | |f x x 3= - -] g


function is

increasing(i)

decreasing(ii)

(a) | 2 | y x= -

(b) | |f x x 2= +

] g

(c) | |f x x2 3= -] g

(d) 4 | | 1 y x= -

(e) | |f x x= -] g


of each function

(a) | | y x= for x2 2 -

(b) | |f x x 4= - -] g for

x4 3 -

(c) | |f x x 4= +] g for x7 2 -

(d) | 2 5 | y x= - for x3 3 -

(e) | |f x x= -] g for x1 1 -


function increasing

(a) | 3 | y x= +

(b) | |f x x 4= - +] g

(c) | |f x x 9= -] g

(d) | | y x 2 1= - -

(e) | |f x x 2= - +] g

56 Exercises




7 Solve graphically

(a) | | 3x =

(b) | |x 12

(c) | |x 2

(d) | 2 | 1x + =

(e) | 3 | 0x- =

(f) |2 3 | 1x - =

(g) | |x 1 41-

(h) | |x 1 3+

(i) | |x 2 22-

(j) | |x 3 1$-

(k) | |x2 3 5+

(l) | |x2 1 1$-

(m) |3 1 | 3x x- = +

(n) |3 2 | 4x x- = -

(o) | 1 | 1x x- = +

(p) | 3 | 2 2x x+ = + (q) |2 1 | 1x x+ = -

(r) |2 5 | 3x x- = -

(s) | 1 | 2x x- =

(t) |2 3 | 3x x- = +

The Hyperbola


or= =

EXAMPLE

Sketch1

y x=

Solution



x -3 -2 -121-

41- 0

41

21 1 2 3

y 3

1-

2

1- -1 -2 -4 mdash 4 2 1

2

1

3

1

Class Discussion



y is never 0 Why






but never touches)



Investigation




equations




1 y xk

=

21

y x k= +

31

y x k

=+


EXAMPLES


3=

-

] g


Solution



So x 3






be zero

CONTINUED







1

y 0 3

3=

-

= -

-3

4

3

2

1

5

-2

-1

-4

-5

y

-1-2 4 521-3-4

x

x = 3

y = 0

Asymptotes

3

2 Sketch y x2 4

1= -

+

Solution




x

x

2 4

2

-

-






( ) y

2 0 41

41

= -+

= -


a translation of3

y x

=





y

-2

x

-

1

4


a=

+


domain x xb



1 For each graph



exists


(a)2

y x=

(b)1

y x= -

(c) f xx 1

1=

+] g

(d) f xx 2

3=

-

] g

(e)3 6

1 y

x=

+

(f) f xx 3

2= -

-

] g

(g) f xx 1

4=

-

] g

(h)1

2 y

x= -

+

(i) f xx6 3

2=

-

] g

(j)2

6 y

x= -

+

2 Show that f x x2

=] g is an odd

function



(a) f xx2 5

1=

+

] g for x2 2 -

(b)3

1 y

x=

+ for x2 0 -

(c) f xx2 4

5=

-

] g for x3 1 -

57 Exercises




(d) f xx 4

3= -

-

] g for x3 3 -

(e)3 1

2 y

x= -

+ for x0 5



(a)3

y x= for y 1 3

(b)2

y x= - for y 221

- -

(c) f xx 1

1=

-

] g for y 171

- -

(d) f xx2 1

3= -

+

] g for

y 131

- -

(e)3 2

6 y

x=

- for y 1

21

6



Circle gate





Proof

r y

x

( x y)

y

x








c a b

r x y

2 2 2

2 2 2

`

= +

= +

EXAMPLE



Solution


y

x

-2

-2 2

2


one place

y

x

2

2

2

-2

The radius is 4

CONTINUED






Domain 2 2 x x -

Range 2 2 y y -


range y r y r -



Proof


shown

y

x

(a b)

x

y

r

( x y)

a

b x - a

y - b




ndash ndash

c a b

r x a y b

2 2 2

2 2 2

= +

= +] ^g h




EXAMPLES



Solution



y

x 9

9

-9

-9



Domain 9 9 x x -

Range 9 9 y y -



Solution


ndash

ndash ndash

x y

x y

1 2 4

1 2 2

2

2 2

+ + =

+ - =

2

2

] ^

] ]_

g h

g gi

So 1 2a b= = - and 2r =

CONTINUED







y

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

(1 -2)



Domain 1 3 x x -

Range 4 0 y y -


Solution


a b2 1= - = and r 3=

Substituting

ndash ndash

ndash

ndash

x a y b r

x y

x y

2 1 3

2 1 9

2 2 2

2 2 2

2 2

+ =

- - + =

+ + =

] ^]] ^

] ^

g hg g hg h


ndash

ndash

a b a ab b

x x x

x x

a b a ab b

y y y

y y

2

2 2 2 2

4 4

2

1 2 1 1

2 1

So

So

2 2 2

2 2 2

2

2 2 2

2 2 2

2

+ = + +

+ = + +

= + +

= - +

= - +

= - +

]] ] ]]^ ^ ]

gg g ggh h g


ndash

x x y y

x x y y

x x y y

x x y y

4 4 2 1 9

4 2 5 9

4 2 5 9

4 2 4 0

9 9

2

2

2

2

+ + + - + =

+ + - + =

+ + + =

+ + - - =

- -


in Chapter 2




Investigation


two functions





circle






for all values

y

x r

r

-r


Proof

ndash

x y r

y r x y r x

2 2 2

2 2 2

2 2

+ =

=

= -



semi-circles





y 0 for all values

y

x r

-

r

-r


EXAMPLES


1 f x x92

= -] g

Solution



y

x 3

3

-3

Domain 3 3 x x -

Range 0 3 y y




2 y x4 2= - -

Solution



y

x 2

-2

-2

Domain 2 2 x x -

Range 2 0 y y -




range

(a) 9x y 2 2+ =

(b) x y 16 02 2+ =-


(d) 1 9x y 2 2+ + =

] g

(e) ndashx y 2 1 12 2+ + =] ^g h



below the x -axis



range

(a) 25 y x2= - -

(b) 1 y x2= -

(c) 36 y x2= -

(d) 64 y x2= - -

(e) 7 y x2= - -



each circle

(a) 100x y 2 2+ =

(b) 5x y 2 2+ =


(d) ndashx y 5 6 492 2+ + =] ^g h

(e) ndashx y 3 812 2+ =^ h

58 Exercises






grouping symbols)



Centre(c) 1 5-







Centre(j) 4 7- -

^ h and radius 3

Other Graphs




EXAMPLES


Solution



eg When 0x =

y 3

1

c=

=

x

y

1

3

3

1

31

When1

1

= -

=

=

=

-

x 3- 2- 1- 0 1 2 3

y 271

91

31

1 3 9 27



You will meet these

graphs again in the

HSC Course





other point

When

x

y

1

3

3

1

=

=

=

y

x

1

2

1

3







Solution



x minus2 minus1 0 05 1 2 3 4

y minus03 0 03 05 06

y

x

1

2

1 2 3 4

-1




You learned about


stages of maths





range y y 02



range all real y

Cubic function


EXAMPLE



Solution



y minus25 minus6 1 2 3 10 29

y

x

1

1

-2 2 3 4

-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4



If you already

know the shape of


3






Domain and range


EXAMPLE


Solution




function are

f

f

1 1 2

1 21

4 4 2

64 2

66

3

3

- = - +

= - +

=

= +

= +

=

] ]

] ]

g g

g g


these points

y

x

(-1 1)

(4 66)

Range 1 66 y y










positive number

EXAMPLE


Solution


x 4So $

$


(or zero) So y 0$


59 Exercises


(a) 4 3 y x= +

(b) f x 4= -] g

(c) 3x =



(f) f x xx 12 2= - -] g

(g) 64x y 2 2+ =

(h) f t t 4

3=

-] g

(i) ( ) g 2

5zz

= +

(j) | |f x x=] g


(a) y x=

(b) 2 y x= -

(c) | |f x x2 3= -] g

(d) | | 2 y x= -

(e) f x x2 5= - +] g

(f) | | y x5= -

(g) 2 y x=

(h) y 5x= -

(i) f x xx 1

= +] g

(j)2

4 3 y x

x= -


(a) y x x 5 2= -] g


= +-

(d) g x x x164 2= -] g

(e) 49x y 2 2+ =

You may like to


by dividing by x




4 (a) Solve x1 02$-


f x x1 2= -] g


(a) 2 y x x2= - -

(b) g t t t 62= +] g



in each case


x3 3 -


x2 3 -


x2 1 - (d)


x1 5


0 4x


x3 3 -


x1 1 -


x2 3 -


x4 4 -

(j) y x x7 62= - + - in the

domain 0 7x


function y x 1

3=

+





x=] g


function

find its range(b)


plane

(a) f x x4=] g

(b) y x3= -

(c) y x 34= -

(d) 2 p x x3=] g

(e) 1 g x x3= +] g

(f) 100x y 2 2+ =

(g) 2 1 y x= +


y x 1= -









(b) f x x 22= -] g

(c)1

y x=

(d) f x x3=] g

(e) f x 3x=] g

13 (a) Solve x4 02$-


(i) 4 y x2= -

(ii) y x4

2= - -




DID YOU KNOW




a light



Limits






a a1

1

03

Z

=

=

3

3

-


lim a 0x=

x 3-



asymptote

EXAMPLES

1 Find lim x

x x5x 0

2+

Solution


which is undefined


( )

lim lim

lim

xx x

x

x x

x

5 5

5

5

x x

x

0

2

0 1

1

0

+=

+

= +

=

] g




2 Find limx

x

4

22

-

-

x 2

Solution


lim lim

lim

x

x

x x

x

x

4

2

2 2

2

21

41

2 1

1

-

-=

+ -

-

=+

=

x x2 2

x 2

^ _h i

3 Find limh

h x hx h2 72 2+ -

h 0

Solution

lim lim

lim

h

h x hx h

h

h hx x

hx x

x

2 7 2 7

2 7

7

2 2 2

2

2

+ -=

+ -

= + -

= -

h 0

h h0 0

^ h

Continuity





EXAMPLES

1 Find lim

x

x

1

12

-

-

x 1


11

y x

x2

=-

- Sketch the curve

Solution


xx2

-

- gives

00

CONTINUED




( )

lim lim

limx

x

x

x x

x11

1

1 1

1

2

x x

x

1

2

1

1

-

-=

-

+ -

= +

=

-

] ]g g

11

y xx2

=-





y xx

x

x x

x

11

1

1 1

1

2

=-

-

=

+

= +

-

-^ ^h h


2 Find limx

x x2

2x 2

2

+

+ -

-


x x2

22

= + -

+

Solution


x x2

22

+

+ - gives

00

lim lim

lim

xx x

x

x x

x

22

2

1 2

1

3

x x

x

2

2

2

2

+

+ -=

+

- +

=

= -

-

- -

-

^^ ^

^

hh h

h2 y

x

x xx

y x

x

x

x

22

2

2

1

1


=+

+ -= -

=+

= -

+ -^ ^h h


Remember that x 1




1 Find(a) lim x 52

+x 4

(b) lim t 7-t 3 -

(c) lim x x2 43+ -

x 2

(d) lim xx x32

+

x 0

(e) limh

h h

2

22

-

- -

h 2

(f) lim y

y

5

1253

-

-

y 5

(g) limx

x x

12 12

+

+ +

x 1-

(h) limx

x x4

2 82

+

+ -

x 4 -

(i) limc

c

4

22

-

-

c 2

(j) limx x

x 12

-

-

x 1

(k) lim h

h h h2 73 2+ -

h 0

(l) limh

hx hx h32 2- +

h 0

(m) limh

hx h x hx h2 3 53 2 2- + -

h 0

(n) lim x c x c 3 3

-

-

x c



discontinuous

(a) 3 y x2= -

(b)1

1 y

x=

+

(c) f x x 1= -] g

(d)4

1 y

x2=

+

(e)4

1

y x2=

-



(a)3

y xx x2

= +

(b)33

y x

x x2

=+

+

(c)1

5 4 y

xx x2

=+

+ +

510 Exercises




Regions

Class Investigation


them all







EXAMPLE


Solution



y

x =3

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4







region

EXAMPLE


Solution



included


shown

y

x

1

1

-2 2 3 4

-3

-4

-5

2

3

4

5

-1

-2

-3-4

-1 y = -1



y -intercept 1-






EXAMPLES


1 y x 2$ +

Solution






y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0



2 x y 2 3 61-

Solution


region


side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g








3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g







2 3 1 4 0 5 1 3 2 4- -^ ^ ^ ^ ^h h h h h Solution


function


y

x

3

Solution

y

x

3





a function




1

2

3

4

5

6

7

8

9 1 3 2 1 3 3 4 0-^ ^ ^ ^h h h h 10 1 3 2 1 2 7 4 0-^ ^ ^ ^h h h h 11

1

2

3

4

5

1

2

3

4

5

12 1

2

3

4

5

1

2

3

4

5

131

2

3

4

5

1

2

3

4

5

51 Exercises







15 A 3

B 4

C 7

D 3

E 5

F 7

G 4

Function notation




EXAMPLES


Solution

When x

y x

3

1

3 1

4

=

= +

= +

=


Solution

f x x

f

1

3 3 1

4

= +

= +

=

]] gg





EXAMPLES

1 If f x x x3 12= + +] g find f 2-] g

Solution

( ) ( )f 2 2 3 2 1

4 6 1

1

2- = - + - +

= - +

= -

] g


Solution

( )

( )

f x x x

f 1 1 1

1 12

3 2

3

= -

- = - - -

= - -

= -

2] ]g g


Solution

( )

ie

( ) ( )

f x

x x

x x

x x

x x

0

3 10 0

5 2 0

5 0 2 0

5 2

2

=

+ - =

+ - =

+ = - =

= - =


when

when f x

x x

x x

3 4 2

2 21

$=

+

-

] g )

Solution

since 4 21-

( ) ( ) since

( ) ( ) since

( ) ( ) since

( ) ( )

f

f

f

f

3 3 3 4 3 2

13

2 3 2 4 2 2

10

0 2 0 0 2

0

4 2 4

8

1

$

$

= +

=

= +

=

= -

=

- = - -

=


when

when

x

x

x

2

1 2

1

2

1

-

-

g x

x

x2 1

5

2

= -] g


when 2x -=







than 2




Solution

( ) ( )

( )

( )

g

g

g

1 2 1 1 1 1 2

1

2 5 2 1

3 3 3 2

9

since

since

since2

1

2

= - -

=

- = - -

=

=

( ) ( ) ( ) g g g 1 2 3 1 5 9

3

So + - - = + -

= -

DID YOU KNOW






f 3-] g

2 If h x x 22= -] g find h h0 2] ]g g

and h 4-] g


and f 2-] g


= - +] g

5 Find f 3-] g if f x x x2 5 43= - +] g


f x 13=] g




f x271

=] g




f x h+] g


g x x x2 32= + +] g


product of factors






15 f x x x

x x

1

1

for

for

32

=] g )


16 f x

x x

x x

x x

2 4 1

3 1 1

1

if

if

if 2

1 1

$

=

-

+ -

-

] gZ

[

]]

]]

Find the values of

f f f 2 2 1- - + -] ] ]g g g

52 Exercises








EXAMPLE


Solution


x x

x x

x x

x x

0 7 8

8 1

8 0 1 0

8 1

2= + -

= + -

+ = - =

= - =

] ]g g


y 0 7 0 8

8

2= + -

= -

] ]g g





Domain and range




domain


image) of f

EXAMPLE


Solution


x

y

CONTINUED






Domain all real x

Range y y 0$















EXAMPLES


x x 3 x 2 x 1

y

Solution


increasing


x 1 x 2

2

x x 3

x 2 x 1

y

Solution



increases again




1 1








=







x

y









x

y




EXAMPLES


Solution

f x x

x

f x

f x x

3

3


2

2

2`

- = - +

= +

=

= +

] ]]

]

g gg

g


Solution

f x x x

x x

x x

f x


3

3

3

3`

- = - - -

= - +

= - -

= -

= -

] ] ]^]

]

g g gh

gg

Investigation






1 f x kxn=] g

2 f x x kn= +] g

3 f x x k n= +

] ]g g






shape of graphs


each function

(a) y x3 2= -

(b) x y 2 5 20 0- + =

(c) x y 3 12 0+ - =

(d) f x x x32= +] g

(e) f x x 42= -] g

(f) p x x x5 62= + +] g

(g) y x x8 152= - +

(h) p x x 53

= +

] g

53 Exercises




(i) y xx

x3

0= + ] g

(j) g x x9 2= -] g


f x x 22= -

] g What type of

function is it


(a) f x2^ h (b) ( )f x 26



an even function


f x x=] g


function


odd function



f x f x- -] ]g g


neither

(a) y x x

x4 2

3

=

-

(b) y x 1

13

=

-

(c) f xx 4

32

=

-

] g

(d) y x

x

33

=+

-

(e) f x x x

x5 2

3

=-] g



f x xn=] g

even(a)

odd(b)


ever be

even(a)

odd(b)



graph is increasing




x

y(a)

x

4

y(b)

2-2

x

y(c)




Investigation


graph







y kx

y kx

y kx

y kx

y xk

(a)

(b)

(c)

(d)

(e)

2

3

4

=

=

=

=

=



y x k

y x k

y x k

y x k

y x k1

(a)

(b)

(c)

(d)

(e)

2

2

3

4

= +

= +

= +

= +

= +

] g



-2

1 2

-4

-1-2

2

4

y

x

(d) y

x

(e)

CONTINUED




Gradient form



Investigation










y x k

y x k

y x k

y x k y

x k

1

(a)

(b)

(c)

(d)(e)

2

3

4

= +

= +

= +

= +

=+

]]

]

gg

g





the right








Linear Function





EXAMPLE



Solution


Find the intercepts


0 3 5

5 3

1

x

x

x32

=

=

=

-


3 5

5

y 0=

= -

-] g

-1

-2

y

5

4

3

2

1 1 23

6

-3

-4

-5

1 4-1-2 32-3-4

x



Range all real y








Special lines






1 y kx=

2 y x k= +






EXAMPLES


Solution




-1

-3

y

4

3

2

1

5

-2

-4

-5

1 4-1-2

x

32-3-4


2 Sketch x 1= -


Solution





-

-

4

3

2

5

-2

-4

-5

1 4-1-2-4

y

x







Domain all real x

Range y y b=

54 Exercises


each function

(a) y x 2= -

(b) f x x2 3= +] g (c) x y 2 1 0+ =-

(d) x y 3 0+ =-

(e) x y 3 6 2 0=- -


line

(a) x 4=

(b) x 3 0=-

(c) y 5=

(d) y 1 0+ =

(e) f x x2 1= -] g (f) y x 4= +

(g) f x x3 2= +] g

(h) x y 3+ =

(i) x y 1 0=- -

(j) x y 2 3 0+ =-


(a) x y 3 2 7 0+ =-

(b) y 2=

(c) x 4= - (d) x 2 0=-

(e) y 3 0=-


are even or odd

(a) y x2=

(b) y 3=

(c) x 4=

(d) y x= -

(e) y x=




meet




Applications






The pronumeral

a is called the


Quadratic Function










EXAMPLES



Solution



x

x

x

0 1

1

1

2

2

= -

=

=


0 1

1

y 2= -

= -



Domain xall real

Range y y 1$ -


Solution


parabola will lie




left



0

1 0

1

x

x

x

1 2= +

+ =

= -

] g


1

y 0 1 2= +

=

] g

-1

-

4

3

2

1

5

-2

-4

-5

-

1 41-2 5-4

y

x









Solution



x x

x xx x

x x

0 6

3 23 0 2 0

3 2

2= + -

= + -

+ = - =

= - =

] ]g g


f 0 0 0 66

2= + -

= -

] ] ]g g g



x -intercepts


23 2

21- +

= -


-c m

f 21

21

21

6

41

21

6

641

2

- = - + - -

= - -

= -

c c cm m m


-

CONTINUED

You will learn more


-1

-3

4

3

2

1

5

-2

-4

-5

1 4-1-2 32-3-4

y

x










41



2

11 -





Solution


So f x 0=] g

0 4 54 5 0

0

x xx x

x x5 1

2

2

= - + +

=

+ =

- -

-] ]g g

x x

x x

5 0 1 0

5 1

- = + =

= = -


f 0 0 4 0 5

5

2= - + +

=

] ] ]g g g

-1

-3

4

2

5

-2

-4

-5

-

y

1 41-2 5-4

x

-2 4






21 5

2- +

=

f 2 2 4 2 59

= - + +

=

2] ] ]g g g










each function

(a) 2 y x x2= +

(b) 3 y x x2= - +

(c) f x x 12= -] g

(d) y x x 22= - -

(e) y x x9 82= +-

2 Sketch

(a) 2 y x2= +

(b) y x 12= - +

(c) f x x 42= -] g

(d) 2 y x x2= +

(e) y x x2= - -

(f) f x x 3= -2] ]g g

55 Exercises

-1

9

8

7

5

4

3

2

6

1

-2

-3

-4

-5

y

2 51 643-1-2-3-4

x




EXAMPLES


Solution









(g) f x x 1 2= +] ]g g

(h) y x x3 42= + -

(i) y x x2 5 32= - +

(j) f x x x3 22= - + -] g



(a) ndash y x x7 122= +

(b) f x x x42= +] g

(c) y x x2 82= - -

(d) y x x6 92= +-

(e) f t t 4 2= -] g


(a) y x 52= -

(b) f x x x6

2= -

] g (c) f x x x 22= - -] g

(d) y x2= -

(e) f x x 7 2= -] ]g g



(a) y x2= for x0 3

(b) y x 42= - + for x1 2 -

(c) f x x 12= -] g for x2 5 -

(d) y x x2 32= + - for x2 4 -

(e) y x x

22= - +- for

x0 4


function is

increasing(i)

decreasing(ii)

(a) y x2=

(b) y x2

= - (c) f x x 92

= -] g

(d) y x x42= - +

(e) f x x 5 2= +] ]g g


function



(a) y x 12= +

(b) f x x 32= -

] g

(c) y x2 2= -

(d) f x x x32= -] g

(e) f x x x2= +] g

(f) y x 42= -

(g) y x x2 32= - -

(h) y x x5 42= +-

(i) p x x 1 2= +] ]g g

(j) y x 2= -2] g




CONTINUED

eg When x 3= -

| | y 3 13 12

= - -

= -

=

x -3 -2 -1 0 1 2 3

y 2 1 0 -1 0 1 2


y

-2

4

3

2

1

5

-1

-3

-4

-5

1 4-1-2 32-3-4

x


| | y x

x x

x x1

1 0

1 0

when

when 1

$= - =

-


y x x1 0$= - ] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x - 1




y x 1= - - x 01] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y =- x - 1



-3

4

3

2

1

5

-2

-1

-4

-5

y y

3-1-2 421-3-4

x

y = - x - 1

y = x - 1


intercepts


So f x 0=

] g

| |

| |

x

x

x

0 1

1

1`

= -

=

=


( ) | |f 0 0 11

= -

= -





-3

4

3

2

1

5

-2

-1

-4

-5

y

4-2 5321-1-3-4

x



Domain all real x

Range y y 1$ -

2 Sketch | | y x 2= +

Solution


| | ( ) y x x xx x2 2 2 0

2 2 0whenwhen 1

$= + = + +

- + +


2 y x= + when x 2 0$+

x 2$ -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x + 2






y-intercept of 1-

CONTINUED




2 y x= - +] g when x 2 01+

ie y x 2= - - when x 21 -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2


-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2

y = x + 2



So 0f x =] g

0 | 2 |

0 2

2

x

x

x

= +

= +

- =


(0) | 0 2 |

2

f = +

=

There is only one

solution for the


Can you see why





-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

If you know how to

sketch the graph of




Investigation




are their equations





1 | |f x k x=] g

2 | |f x x k= +] g

3 | |f x x k= +] g







EXAMPLES

Solve

1 |2 1 | 3x - =

Solution



is x 1 2= -

2 |2 1 | 3 2x x= -+

Solution



3 | 1 | 2x 1+

Solution






and then check them







each function

(a) | | y x=

(b) | |f x x 7= +] g

(c) | |f x x 2= -] g

(d) 5 | | y x=

(e) | |f x x 3= - +] g

(f) | 6 | y x= +

(g) | |f x x3 2= -] g

(h) | 5 4 | y x= +

(i) | 7 1 | y x= -

(j) | |f x x2 9= +] g


plane

(a) | | y x=

(b) | |f x x 1= +] g

(c) | |f x x 3= -] g

(d) 2 | | y x=

(e) | |f x x= -] g

(f) | 1 | y x= +

(g) | |f x x 1= - -] g

(h) | 2 3 | y x= -

(i) | 4 2 | y x= +

(j) | |f x x3 1= +] g


each function

(a) | 1 | y x= -

(b) | |f x x 8= -] g

(c) | |f x x2 5= +] g

(d) 2 | | 3 y x= -

(e) | |f x x 3= - -] g


function is

increasing(i)

decreasing(ii)

(a) | 2 | y x= -

(b) | |f x x 2= +

] g

(c) | |f x x2 3= -] g

(d) 4 | | 1 y x= -

(e) | |f x x= -] g


of each function

(a) | | y x= for x2 2 -

(b) | |f x x 4= - -] g for

x4 3 -

(c) | |f x x 4= +] g for x7 2 -

(d) | 2 5 | y x= - for x3 3 -

(e) | |f x x= -] g for x1 1 -


function increasing

(a) | 3 | y x= +

(b) | |f x x 4= - +] g

(c) | |f x x 9= -] g

(d) | | y x 2 1= - -

(e) | |f x x 2= - +] g

56 Exercises




7 Solve graphically

(a) | | 3x =

(b) | |x 12

(c) | |x 2

(d) | 2 | 1x + =

(e) | 3 | 0x- =

(f) |2 3 | 1x - =

(g) | |x 1 41-

(h) | |x 1 3+

(i) | |x 2 22-

(j) | |x 3 1$-

(k) | |x2 3 5+

(l) | |x2 1 1$-

(m) |3 1 | 3x x- = +

(n) |3 2 | 4x x- = -

(o) | 1 | 1x x- = +

(p) | 3 | 2 2x x+ = + (q) |2 1 | 1x x+ = -

(r) |2 5 | 3x x- = -

(s) | 1 | 2x x- =

(t) |2 3 | 3x x- = +

The Hyperbola


or= =

EXAMPLE

Sketch1

y x=

Solution



x -3 -2 -121-

41- 0

41

21 1 2 3

y 3

1-

2

1- -1 -2 -4 mdash 4 2 1

2

1

3

1

Class Discussion



y is never 0 Why






but never touches)



Investigation




equations




1 y xk

=

21

y x k= +

31

y x k

=+


EXAMPLES


3=

-

] g


Solution



So x 3






be zero

CONTINUED







1

y 0 3

3=

-

= -

-3

4

3

2

1

5

-2

-1

-4

-5

y

-1-2 4 521-3-4

x

x = 3

y = 0

Asymptotes

3

2 Sketch y x2 4

1= -

+

Solution




x

x

2 4

2

-

-






( ) y

2 0 41

41

= -+

= -


a translation of3

y x

=





y

-2

x

-

1

4


a=

+


domain x xb



1 For each graph



exists


(a)2

y x=

(b)1

y x= -

(c) f xx 1

1=

+] g

(d) f xx 2

3=

-

] g

(e)3 6

1 y

x=

+

(f) f xx 3

2= -

-

] g

(g) f xx 1

4=

-

] g

(h)1

2 y

x= -

+

(i) f xx6 3

2=

-

] g

(j)2

6 y

x= -

+

2 Show that f x x2

=] g is an odd

function



(a) f xx2 5

1=

+

] g for x2 2 -

(b)3

1 y

x=

+ for x2 0 -

(c) f xx2 4

5=

-

] g for x3 1 -

57 Exercises




(d) f xx 4

3= -

-

] g for x3 3 -

(e)3 1

2 y

x= -

+ for x0 5



(a)3

y x= for y 1 3

(b)2

y x= - for y 221

- -

(c) f xx 1

1=

-

] g for y 171

- -

(d) f xx2 1

3= -

+

] g for

y 131

- -

(e)3 2

6 y

x=

- for y 1

21

6



Circle gate





Proof

r y

x

( x y)

y

x








c a b

r x y

2 2 2

2 2 2

`

= +

= +

EXAMPLE



Solution


y

x

-2

-2 2

2


one place

y

x

2

2

2

-2

The radius is 4

CONTINUED






Domain 2 2 x x -

Range 2 2 y y -


range y r y r -



Proof


shown

y

x

(a b)

x

y

r

( x y)

a

b x - a

y - b




ndash ndash

c a b

r x a y b

2 2 2

2 2 2

= +

= +] ^g h




EXAMPLES



Solution



y

x 9

9

-9

-9



Domain 9 9 x x -

Range 9 9 y y -



Solution


ndash

ndash ndash

x y

x y

1 2 4

1 2 2

2

2 2

+ + =

+ - =

2

2

] ^

] ]_

g h

g gi

So 1 2a b= = - and 2r =

CONTINUED







y

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

(1 -2)



Domain 1 3 x x -

Range 4 0 y y -


Solution


a b2 1= - = and r 3=

Substituting

ndash ndash

ndash

ndash

x a y b r

x y

x y

2 1 3

2 1 9

2 2 2

2 2 2

2 2

+ =

- - + =

+ + =

] ^]] ^

] ^

g hg g hg h


ndash

ndash

a b a ab b

x x x

x x

a b a ab b

y y y

y y

2

2 2 2 2

4 4

2

1 2 1 1

2 1

So

So

2 2 2

2 2 2

2

2 2 2

2 2 2

2

+ = + +

+ = + +

= + +

= - +

= - +

= - +

]] ] ]]^ ^ ]

gg g ggh h g


ndash

x x y y

x x y y

x x y y

x x y y

4 4 2 1 9

4 2 5 9

4 2 5 9

4 2 4 0

9 9

2

2

2

2

+ + + - + =

+ + - + =

+ + + =

+ + - - =

- -


in Chapter 2




Investigation


two functions





circle






for all values

y

x r

r

-r


Proof

ndash

x y r

y r x y r x

2 2 2

2 2 2

2 2

+ =

=

= -



semi-circles





y 0 for all values

y

x r

-

r

-r


EXAMPLES


1 f x x92

= -] g

Solution



y

x 3

3

-3

Domain 3 3 x x -

Range 0 3 y y




2 y x4 2= - -

Solution



y

x 2

-2

-2

Domain 2 2 x x -

Range 2 0 y y -




range

(a) 9x y 2 2+ =

(b) x y 16 02 2+ =-


(d) 1 9x y 2 2+ + =

] g

(e) ndashx y 2 1 12 2+ + =] ^g h



below the x -axis



range

(a) 25 y x2= - -

(b) 1 y x2= -

(c) 36 y x2= -

(d) 64 y x2= - -

(e) 7 y x2= - -



each circle

(a) 100x y 2 2+ =

(b) 5x y 2 2+ =


(d) ndashx y 5 6 492 2+ + =] ^g h

(e) ndashx y 3 812 2+ =^ h

58 Exercises






grouping symbols)



Centre(c) 1 5-







Centre(j) 4 7- -

^ h and radius 3

Other Graphs




EXAMPLES


Solution



eg When 0x =

y 3

1

c=

=

x

y

1

3

3

1

31

When1

1

= -

=

=

=

-

x 3- 2- 1- 0 1 2 3

y 271

91

31

1 3 9 27



You will meet these

graphs again in the

HSC Course





other point

When

x

y

1

3

3

1

=

=

=

y

x

1

2

1

3







Solution



x minus2 minus1 0 05 1 2 3 4

y minus03 0 03 05 06

y

x

1

2

1 2 3 4

-1




You learned about


stages of maths





range y y 02



range all real y

Cubic function


EXAMPLE



Solution



y minus25 minus6 1 2 3 10 29

y

x

1

1

-2 2 3 4

-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4



If you already

know the shape of


3






Domain and range


EXAMPLE


Solution




function are

f

f

1 1 2

1 21

4 4 2

64 2

66

3

3

- = - +

= - +

=

= +

= +

=

] ]

] ]

g g

g g


these points

y

x

(-1 1)

(4 66)

Range 1 66 y y










positive number

EXAMPLE


Solution


x 4So $

$


(or zero) So y 0$


59 Exercises


(a) 4 3 y x= +

(b) f x 4= -] g

(c) 3x =



(f) f x xx 12 2= - -] g

(g) 64x y 2 2+ =

(h) f t t 4

3=

-] g

(i) ( ) g 2

5zz

= +

(j) | |f x x=] g


(a) y x=

(b) 2 y x= -

(c) | |f x x2 3= -] g

(d) | | 2 y x= -

(e) f x x2 5= - +] g

(f) | | y x5= -

(g) 2 y x=

(h) y 5x= -

(i) f x xx 1

= +] g

(j)2

4 3 y x

x= -


(a) y x x 5 2= -] g


= +-

(d) g x x x164 2= -] g

(e) 49x y 2 2+ =

You may like to


by dividing by x




4 (a) Solve x1 02$-


f x x1 2= -] g


(a) 2 y x x2= - -

(b) g t t t 62= +] g



in each case


x3 3 -


x2 3 -


x2 1 - (d)


x1 5


0 4x


x3 3 -


x1 1 -


x2 3 -


x4 4 -

(j) y x x7 62= - + - in the

domain 0 7x


function y x 1

3=

+





x=] g


function

find its range(b)


plane

(a) f x x4=] g

(b) y x3= -

(c) y x 34= -

(d) 2 p x x3=] g

(e) 1 g x x3= +] g

(f) 100x y 2 2+ =

(g) 2 1 y x= +


y x 1= -









(b) f x x 22= -] g

(c)1

y x=

(d) f x x3=] g

(e) f x 3x=] g

13 (a) Solve x4 02$-


(i) 4 y x2= -

(ii) y x4

2= - -




DID YOU KNOW




a light



Limits






a a1

1

03

Z

=

=

3

3

-


lim a 0x=

x 3-



asymptote

EXAMPLES

1 Find lim x

x x5x 0

2+

Solution


which is undefined


( )

lim lim

lim

xx x

x

x x

x

5 5

5

5

x x

x

0

2

0 1

1

0

+=

+

= +

=

] g




2 Find limx

x

4

22

-

-

x 2

Solution


lim lim

lim

x

x

x x

x

x

4

2

2 2

2

21

41

2 1

1

-

-=

+ -

-

=+

=

x x2 2

x 2

^ _h i

3 Find limh

h x hx h2 72 2+ -

h 0

Solution

lim lim

lim

h

h x hx h

h

h hx x

hx x

x

2 7 2 7

2 7

7

2 2 2

2

2

+ -=

+ -

= + -

= -

h 0

h h0 0

^ h

Continuity





EXAMPLES

1 Find lim

x

x

1

12

-

-

x 1


11

y x

x2

=-

- Sketch the curve

Solution


xx2

-

- gives

00

CONTINUED




( )

lim lim

limx

x

x

x x

x11

1

1 1

1

2

x x

x

1

2

1

1

-

-=

-

+ -

= +

=

-

] ]g g

11

y xx2

=-





y xx

x

x x

x

11

1

1 1

1

2

=-

-

=

+

= +

-

-^ ^h h


2 Find limx

x x2

2x 2

2

+

+ -

-


x x2

22

= + -

+

Solution


x x2

22

+

+ - gives

00

lim lim

lim

xx x

x

x x

x

22

2

1 2

1

3

x x

x

2

2

2

2

+

+ -=

+

- +

=

= -

-

- -

-

^^ ^

^

hh h

h2 y

x

x xx

y x

x

x

x

22

2

2

1

1


=+

+ -= -

=+

= -

+ -^ ^h h


Remember that x 1




1 Find(a) lim x 52

+x 4

(b) lim t 7-t 3 -

(c) lim x x2 43+ -

x 2

(d) lim xx x32

+

x 0

(e) limh

h h

2

22

-

- -

h 2

(f) lim y

y

5

1253

-

-

y 5

(g) limx

x x

12 12

+

+ +

x 1-

(h) limx

x x4

2 82

+

+ -

x 4 -

(i) limc

c

4

22

-

-

c 2

(j) limx x

x 12

-

-

x 1

(k) lim h

h h h2 73 2+ -

h 0

(l) limh

hx hx h32 2- +

h 0

(m) limh

hx h x hx h2 3 53 2 2- + -

h 0

(n) lim x c x c 3 3

-

-

x c



discontinuous

(a) 3 y x2= -

(b)1

1 y

x=

+

(c) f x x 1= -] g

(d)4

1 y

x2=

+

(e)4

1

y x2=

-



(a)3

y xx x2

= +

(b)33

y x

x x2

=+

+

(c)1

5 4 y

xx x2

=+

+ +

510 Exercises




Regions

Class Investigation


them all







EXAMPLE


Solution



y

x =3

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4







region

EXAMPLE


Solution



included


shown

y

x

1

1

-2 2 3 4

-3

-4

-5

2

3

4

5

-1

-2

-3-4

-1 y = -1



y -intercept 1-






EXAMPLES


1 y x 2$ +

Solution






y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0



2 x y 2 3 61-

Solution


region


side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g








3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g






1

2

3

4

5

6

7

8

9 1 3 2 1 3 3 4 0-^ ^ ^ ^h h h h 10 1 3 2 1 2 7 4 0-^ ^ ^ ^h h h h 11

1

2

3

4

5

1

2

3

4

5

12 1

2

3

4

5

1

2

3

4

5

131

2

3

4

5

1

2

3

4

5

51 Exercises







15 A 3

B 4

C 7

D 3

E 5

F 7

G 4

Function notation




EXAMPLES


Solution

When x

y x

3

1

3 1

4

=

= +

= +

=


Solution

f x x

f

1

3 3 1

4

= +

= +

=

]] gg





EXAMPLES

1 If f x x x3 12= + +] g find f 2-] g

Solution

( ) ( )f 2 2 3 2 1

4 6 1

1

2- = - + - +

= - +

= -

] g


Solution

( )

( )

f x x x

f 1 1 1

1 12

3 2

3

= -

- = - - -

= - -

= -

2] ]g g


Solution

( )

ie

( ) ( )

f x

x x

x x

x x

x x

0

3 10 0

5 2 0

5 0 2 0

5 2

2

=

+ - =

+ - =

+ = - =

= - =


when

when f x

x x

x x

3 4 2

2 21

$=

+

-

] g )

Solution

since 4 21-

( ) ( ) since

( ) ( ) since

( ) ( ) since

( ) ( )

f

f

f

f

3 3 3 4 3 2

13

2 3 2 4 2 2

10

0 2 0 0 2

0

4 2 4

8

1

$

$

= +

=

= +

=

= -

=

- = - -

=


when

when

x

x

x

2

1 2

1

2

1

-

-

g x

x

x2 1

5

2

= -] g


when 2x -=







than 2




Solution

( ) ( )

( )

( )

g

g

g

1 2 1 1 1 1 2

1

2 5 2 1

3 3 3 2

9

since

since

since2

1

2

= - -

=

- = - -

=

=

( ) ( ) ( ) g g g 1 2 3 1 5 9

3

So + - - = + -

= -

DID YOU KNOW






f 3-] g

2 If h x x 22= -] g find h h0 2] ]g g

and h 4-] g


and f 2-] g


= - +] g

5 Find f 3-] g if f x x x2 5 43= - +] g


f x 13=] g




f x271

=] g




f x h+] g


g x x x2 32= + +] g


product of factors






15 f x x x

x x

1

1

for

for

32

=] g )


16 f x

x x

x x

x x

2 4 1

3 1 1

1

if

if

if 2

1 1

$

=

-

+ -

-

] gZ

[

]]

]]

Find the values of

f f f 2 2 1- - + -] ] ]g g g

52 Exercises








EXAMPLE


Solution


x x

x x

x x

x x

0 7 8

8 1

8 0 1 0

8 1

2= + -

= + -

+ = - =

= - =

] ]g g


y 0 7 0 8

8

2= + -

= -

] ]g g





Domain and range




domain


image) of f

EXAMPLE


Solution


x

y

CONTINUED






Domain all real x

Range y y 0$















EXAMPLES


x x 3 x 2 x 1

y

Solution


increasing


x 1 x 2

2

x x 3

x 2 x 1

y

Solution



increases again




1 1








=







x

y









x

y




EXAMPLES


Solution

f x x

x

f x

f x x

3

3


2

2

2`

- = - +

= +

=

= +

] ]]

]

g gg

g


Solution

f x x x

x x

x x

f x


3

3

3

3`

- = - - -

= - +

= - -

= -

= -

] ] ]^]

]

g g gh

gg

Investigation






1 f x kxn=] g

2 f x x kn= +] g

3 f x x k n= +

] ]g g






shape of graphs


each function

(a) y x3 2= -

(b) x y 2 5 20 0- + =

(c) x y 3 12 0+ - =

(d) f x x x32= +] g

(e) f x x 42= -] g

(f) p x x x5 62= + +] g

(g) y x x8 152= - +

(h) p x x 53

= +

] g

53 Exercises




(i) y xx

x3

0= + ] g

(j) g x x9 2= -] g


f x x 22= -

] g What type of

function is it


(a) f x2^ h (b) ( )f x 26



an even function


f x x=] g


function


odd function



f x f x- -] ]g g


neither

(a) y x x

x4 2

3

=

-

(b) y x 1

13

=

-

(c) f xx 4

32

=

-

] g

(d) y x

x

33

=+

-

(e) f x x x

x5 2

3

=-] g



f x xn=] g

even(a)

odd(b)


ever be

even(a)

odd(b)



graph is increasing




x

y(a)

x

4

y(b)

2-2

x

y(c)




Investigation


graph







y kx

y kx

y kx

y kx

y xk

(a)

(b)

(c)

(d)

(e)

2

3

4

=

=

=

=

=



y x k

y x k

y x k

y x k

y x k1

(a)

(b)

(c)

(d)

(e)

2

2

3

4

= +

= +

= +

= +

= +

] g



-2

1 2

-4

-1-2

2

4

y

x

(d) y

x

(e)

CONTINUED




Gradient form



Investigation










y x k

y x k

y x k

y x k y

x k

1

(a)

(b)

(c)

(d)(e)

2

3

4

= +

= +

= +

= +

=+

]]

]

gg

g





the right








Linear Function





EXAMPLE



Solution


Find the intercepts


0 3 5

5 3

1

x

x

x32

=

=

=

-


3 5

5

y 0=

= -

-] g

-1

-2

y

5

4

3

2

1 1 23

6

-3

-4

-5

1 4-1-2 32-3-4

x



Range all real y








Special lines






1 y kx=

2 y x k= +






EXAMPLES


Solution




-1

-3

y

4

3

2

1

5

-2

-4

-5

1 4-1-2

x

32-3-4


2 Sketch x 1= -


Solution





-

-

4

3

2

5

-2

-4

-5

1 4-1-2-4

y

x







Domain all real x

Range y y b=

54 Exercises


each function

(a) y x 2= -

(b) f x x2 3= +] g (c) x y 2 1 0+ =-

(d) x y 3 0+ =-

(e) x y 3 6 2 0=- -


line

(a) x 4=

(b) x 3 0=-

(c) y 5=

(d) y 1 0+ =

(e) f x x2 1= -] g (f) y x 4= +

(g) f x x3 2= +] g

(h) x y 3+ =

(i) x y 1 0=- -

(j) x y 2 3 0+ =-


(a) x y 3 2 7 0+ =-

(b) y 2=

(c) x 4= - (d) x 2 0=-

(e) y 3 0=-


are even or odd

(a) y x2=

(b) y 3=

(c) x 4=

(d) y x= -

(e) y x=




meet




Applications






The pronumeral

a is called the


Quadratic Function










EXAMPLES



Solution



x

x

x

0 1

1

1

2

2

= -

=

=


0 1

1

y 2= -

= -



Domain xall real

Range y y 1$ -


Solution


parabola will lie




left



0

1 0

1

x

x

x

1 2= +

+ =

= -

] g


1

y 0 1 2= +

=

] g

-1

-

4

3

2

1

5

-2

-4

-5

-

1 41-2 5-4

y

x









Solution



x x

x xx x

x x

0 6

3 23 0 2 0

3 2

2= + -

= + -

+ = - =

= - =

] ]g g


f 0 0 0 66

2= + -

= -

] ] ]g g g



x -intercepts


23 2

21- +

= -


-c m

f 21

21

21

6

41

21

6

641

2

- = - + - -

= - -

= -

c c cm m m


-

CONTINUED

You will learn more


-1

-3

4

3

2

1

5

-2

-4

-5

1 4-1-2 32-3-4

y

x










41



2

11 -





Solution


So f x 0=] g

0 4 54 5 0

0

x xx x

x x5 1

2

2

= - + +

=

+ =

- -

-] ]g g

x x

x x

5 0 1 0

5 1

- = + =

= = -


f 0 0 4 0 5

5

2= - + +

=

] ] ]g g g

-1

-3

4

2

5

-2

-4

-5

-

y

1 41-2 5-4

x

-2 4






21 5

2- +

=

f 2 2 4 2 59

= - + +

=

2] ] ]g g g










each function

(a) 2 y x x2= +

(b) 3 y x x2= - +

(c) f x x 12= -] g

(d) y x x 22= - -

(e) y x x9 82= +-

2 Sketch

(a) 2 y x2= +

(b) y x 12= - +

(c) f x x 42= -] g

(d) 2 y x x2= +

(e) y x x2= - -

(f) f x x 3= -2] ]g g

55 Exercises

-1

9

8

7

5

4

3

2

6

1

-2

-3

-4

-5

y

2 51 643-1-2-3-4

x




EXAMPLES


Solution









(g) f x x 1 2= +] ]g g

(h) y x x3 42= + -

(i) y x x2 5 32= - +

(j) f x x x3 22= - + -] g



(a) ndash y x x7 122= +

(b) f x x x42= +] g

(c) y x x2 82= - -

(d) y x x6 92= +-

(e) f t t 4 2= -] g


(a) y x 52= -

(b) f x x x6

2= -

] g (c) f x x x 22= - -] g

(d) y x2= -

(e) f x x 7 2= -] ]g g



(a) y x2= for x0 3

(b) y x 42= - + for x1 2 -

(c) f x x 12= -] g for x2 5 -

(d) y x x2 32= + - for x2 4 -

(e) y x x

22= - +- for

x0 4


function is

increasing(i)

decreasing(ii)

(a) y x2=

(b) y x2

= - (c) f x x 92

= -] g

(d) y x x42= - +

(e) f x x 5 2= +] ]g g


function



(a) y x 12= +

(b) f x x 32= -

] g

(c) y x2 2= -

(d) f x x x32= -] g

(e) f x x x2= +] g

(f) y x 42= -

(g) y x x2 32= - -

(h) y x x5 42= +-

(i) p x x 1 2= +] ]g g

(j) y x 2= -2] g




CONTINUED

eg When x 3= -

| | y 3 13 12

= - -

= -

=

x -3 -2 -1 0 1 2 3

y 2 1 0 -1 0 1 2


y

-2

4

3

2

1

5

-1

-3

-4

-5

1 4-1-2 32-3-4

x


| | y x

x x

x x1

1 0

1 0

when

when 1

$= - =

-


y x x1 0$= - ] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x - 1




y x 1= - - x 01] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y =- x - 1



-3

4

3

2

1

5

-2

-1

-4

-5

y y

3-1-2 421-3-4

x

y = - x - 1

y = x - 1


intercepts


So f x 0=

] g

| |

| |

x

x

x

0 1

1

1`

= -

=

=


( ) | |f 0 0 11

= -

= -





-3

4

3

2

1

5

-2

-1

-4

-5

y

4-2 5321-1-3-4

x



Domain all real x

Range y y 1$ -

2 Sketch | | y x 2= +

Solution


| | ( ) y x x xx x2 2 2 0

2 2 0whenwhen 1

$= + = + +

- + +


2 y x= + when x 2 0$+

x 2$ -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x + 2






y-intercept of 1-

CONTINUED




2 y x= - +] g when x 2 01+

ie y x 2= - - when x 21 -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2


-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2

y = x + 2



So 0f x =] g

0 | 2 |

0 2

2

x

x

x

= +

= +

- =


(0) | 0 2 |

2

f = +

=

There is only one

solution for the


Can you see why





-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

If you know how to

sketch the graph of




Investigation




are their equations





1 | |f x k x=] g

2 | |f x x k= +] g

3 | |f x x k= +] g







EXAMPLES

Solve

1 |2 1 | 3x - =

Solution



is x 1 2= -

2 |2 1 | 3 2x x= -+

Solution



3 | 1 | 2x 1+

Solution






and then check them







each function

(a) | | y x=

(b) | |f x x 7= +] g

(c) | |f x x 2= -] g

(d) 5 | | y x=

(e) | |f x x 3= - +] g

(f) | 6 | y x= +

(g) | |f x x3 2= -] g

(h) | 5 4 | y x= +

(i) | 7 1 | y x= -

(j) | |f x x2 9= +] g


plane

(a) | | y x=

(b) | |f x x 1= +] g

(c) | |f x x 3= -] g

(d) 2 | | y x=

(e) | |f x x= -] g

(f) | 1 | y x= +

(g) | |f x x 1= - -] g

(h) | 2 3 | y x= -

(i) | 4 2 | y x= +

(j) | |f x x3 1= +] g


each function

(a) | 1 | y x= -

(b) | |f x x 8= -] g

(c) | |f x x2 5= +] g

(d) 2 | | 3 y x= -

(e) | |f x x 3= - -] g


function is

increasing(i)

decreasing(ii)

(a) | 2 | y x= -

(b) | |f x x 2= +

] g

(c) | |f x x2 3= -] g

(d) 4 | | 1 y x= -

(e) | |f x x= -] g


of each function

(a) | | y x= for x2 2 -

(b) | |f x x 4= - -] g for

x4 3 -

(c) | |f x x 4= +] g for x7 2 -

(d) | 2 5 | y x= - for x3 3 -

(e) | |f x x= -] g for x1 1 -


function increasing

(a) | 3 | y x= +

(b) | |f x x 4= - +] g

(c) | |f x x 9= -] g

(d) | | y x 2 1= - -

(e) | |f x x 2= - +] g

56 Exercises




7 Solve graphically

(a) | | 3x =

(b) | |x 12

(c) | |x 2

(d) | 2 | 1x + =

(e) | 3 | 0x- =

(f) |2 3 | 1x - =

(g) | |x 1 41-

(h) | |x 1 3+

(i) | |x 2 22-

(j) | |x 3 1$-

(k) | |x2 3 5+

(l) | |x2 1 1$-

(m) |3 1 | 3x x- = +

(n) |3 2 | 4x x- = -

(o) | 1 | 1x x- = +

(p) | 3 | 2 2x x+ = + (q) |2 1 | 1x x+ = -

(r) |2 5 | 3x x- = -

(s) | 1 | 2x x- =

(t) |2 3 | 3x x- = +

The Hyperbola


or= =

EXAMPLE

Sketch1

y x=

Solution



x -3 -2 -121-

41- 0

41

21 1 2 3

y 3

1-

2

1- -1 -2 -4 mdash 4 2 1

2

1

3

1

Class Discussion



y is never 0 Why






but never touches)



Investigation




equations




1 y xk

=

21

y x k= +

31

y x k

=+


EXAMPLES


3=

-

] g


Solution



So x 3






be zero

CONTINUED







1

y 0 3

3=

-

= -

-3

4

3

2

1

5

-2

-1

-4

-5

y

-1-2 4 521-3-4

x

x = 3

y = 0

Asymptotes

3

2 Sketch y x2 4

1= -

+

Solution




x

x

2 4

2

-

-






( ) y

2 0 41

41

= -+

= -


a translation of3

y x

=





y

-2

x

-

1

4


a=

+


domain x xb



1 For each graph



exists


(a)2

y x=

(b)1

y x= -

(c) f xx 1

1=

+] g

(d) f xx 2

3=

-

] g

(e)3 6

1 y

x=

+

(f) f xx 3

2= -

-

] g

(g) f xx 1

4=

-

] g

(h)1

2 y

x= -

+

(i) f xx6 3

2=

-

] g

(j)2

6 y

x= -

+

2 Show that f x x2

=] g is an odd

function



(a) f xx2 5

1=

+

] g for x2 2 -

(b)3

1 y

x=

+ for x2 0 -

(c) f xx2 4

5=

-

] g for x3 1 -

57 Exercises




(d) f xx 4

3= -

-

] g for x3 3 -

(e)3 1

2 y

x= -

+ for x0 5



(a)3

y x= for y 1 3

(b)2

y x= - for y 221

- -

(c) f xx 1

1=

-

] g for y 171

- -

(d) f xx2 1

3= -

+

] g for

y 131

- -

(e)3 2

6 y

x=

- for y 1

21

6



Circle gate





Proof

r y

x

( x y)

y

x








c a b

r x y

2 2 2

2 2 2

`

= +

= +

EXAMPLE



Solution


y

x

-2

-2 2

2


one place

y

x

2

2

2

-2

The radius is 4

CONTINUED






Domain 2 2 x x -

Range 2 2 y y -


range y r y r -



Proof


shown

y

x

(a b)

x

y

r

( x y)

a

b x - a

y - b




ndash ndash

c a b

r x a y b

2 2 2

2 2 2

= +

= +] ^g h




EXAMPLES



Solution



y

x 9

9

-9

-9



Domain 9 9 x x -

Range 9 9 y y -



Solution


ndash

ndash ndash

x y

x y

1 2 4

1 2 2

2

2 2

+ + =

+ - =

2

2

] ^

] ]_

g h

g gi

So 1 2a b= = - and 2r =

CONTINUED







y

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

(1 -2)



Domain 1 3 x x -

Range 4 0 y y -


Solution


a b2 1= - = and r 3=

Substituting

ndash ndash

ndash

ndash

x a y b r

x y

x y

2 1 3

2 1 9

2 2 2

2 2 2

2 2

+ =

- - + =

+ + =

] ^]] ^

] ^

g hg g hg h


ndash

ndash

a b a ab b

x x x

x x

a b a ab b

y y y

y y

2

2 2 2 2

4 4

2

1 2 1 1

2 1

So

So

2 2 2

2 2 2

2

2 2 2

2 2 2

2

+ = + +

+ = + +

= + +

= - +

= - +

= - +

]] ] ]]^ ^ ]

gg g ggh h g


ndash

x x y y

x x y y

x x y y

x x y y

4 4 2 1 9

4 2 5 9

4 2 5 9

4 2 4 0

9 9

2

2

2

2

+ + + - + =

+ + - + =

+ + + =

+ + - - =

- -


in Chapter 2




Investigation


two functions





circle






for all values

y

x r

r

-r


Proof

ndash

x y r

y r x y r x

2 2 2

2 2 2

2 2

+ =

=

= -



semi-circles





y 0 for all values

y

x r

-

r

-r


EXAMPLES


1 f x x92

= -] g

Solution



y

x 3

3

-3

Domain 3 3 x x -

Range 0 3 y y




2 y x4 2= - -

Solution



y

x 2

-2

-2

Domain 2 2 x x -

Range 2 0 y y -




range

(a) 9x y 2 2+ =

(b) x y 16 02 2+ =-


(d) 1 9x y 2 2+ + =

] g

(e) ndashx y 2 1 12 2+ + =] ^g h



below the x -axis



range

(a) 25 y x2= - -

(b) 1 y x2= -

(c) 36 y x2= -

(d) 64 y x2= - -

(e) 7 y x2= - -



each circle

(a) 100x y 2 2+ =

(b) 5x y 2 2+ =


(d) ndashx y 5 6 492 2+ + =] ^g h

(e) ndashx y 3 812 2+ =^ h

58 Exercises






grouping symbols)



Centre(c) 1 5-







Centre(j) 4 7- -

^ h and radius 3

Other Graphs




EXAMPLES


Solution



eg When 0x =

y 3

1

c=

=

x

y

1

3

3

1

31

When1

1

= -

=

=

=

-

x 3- 2- 1- 0 1 2 3

y 271

91

31

1 3 9 27



You will meet these

graphs again in the

HSC Course





other point

When

x

y

1

3

3

1

=

=

=

y

x

1

2

1

3







Solution



x minus2 minus1 0 05 1 2 3 4

y minus03 0 03 05 06

y

x

1

2

1 2 3 4

-1




You learned about


stages of maths





range y y 02



range all real y

Cubic function


EXAMPLE



Solution



y minus25 minus6 1 2 3 10 29

y

x

1

1

-2 2 3 4

-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4



If you already

know the shape of


3






Domain and range


EXAMPLE


Solution




function are

f

f

1 1 2

1 21

4 4 2

64 2

66

3

3

- = - +

= - +

=

= +

= +

=

] ]

] ]

g g

g g


these points

y

x

(-1 1)

(4 66)

Range 1 66 y y










positive number

EXAMPLE


Solution


x 4So $

$


(or zero) So y 0$


59 Exercises


(a) 4 3 y x= +

(b) f x 4= -] g

(c) 3x =



(f) f x xx 12 2= - -] g

(g) 64x y 2 2+ =

(h) f t t 4

3=

-] g

(i) ( ) g 2

5zz

= +

(j) | |f x x=] g


(a) y x=

(b) 2 y x= -

(c) | |f x x2 3= -] g

(d) | | 2 y x= -

(e) f x x2 5= - +] g

(f) | | y x5= -

(g) 2 y x=

(h) y 5x= -

(i) f x xx 1

= +] g

(j)2

4 3 y x

x= -


(a) y x x 5 2= -] g


= +-

(d) g x x x164 2= -] g

(e) 49x y 2 2+ =

You may like to


by dividing by x




4 (a) Solve x1 02$-


f x x1 2= -] g


(a) 2 y x x2= - -

(b) g t t t 62= +] g



in each case


x3 3 -


x2 3 -


x2 1 - (d)


x1 5


0 4x


x3 3 -


x1 1 -


x2 3 -


x4 4 -

(j) y x x7 62= - + - in the

domain 0 7x


function y x 1

3=

+





x=] g


function

find its range(b)


plane

(a) f x x4=] g

(b) y x3= -

(c) y x 34= -

(d) 2 p x x3=] g

(e) 1 g x x3= +] g

(f) 100x y 2 2+ =

(g) 2 1 y x= +


y x 1= -









(b) f x x 22= -] g

(c)1

y x=

(d) f x x3=] g

(e) f x 3x=] g

13 (a) Solve x4 02$-


(i) 4 y x2= -

(ii) y x4

2= - -




DID YOU KNOW




a light



Limits






a a1

1

03

Z

=

=

3

3

-


lim a 0x=

x 3-



asymptote

EXAMPLES

1 Find lim x

x x5x 0

2+

Solution


which is undefined


( )

lim lim

lim

xx x

x

x x

x

5 5

5

5

x x

x

0

2

0 1

1

0

+=

+

= +

=

] g




2 Find limx

x

4

22

-

-

x 2

Solution


lim lim

lim

x

x

x x

x

x

4

2

2 2

2

21

41

2 1

1

-

-=

+ -

-

=+

=

x x2 2

x 2

^ _h i

3 Find limh

h x hx h2 72 2+ -

h 0

Solution

lim lim

lim

h

h x hx h

h

h hx x

hx x

x

2 7 2 7

2 7

7

2 2 2

2

2

+ -=

+ -

= + -

= -

h 0

h h0 0

^ h

Continuity





EXAMPLES

1 Find lim

x

x

1

12

-

-

x 1


11

y x

x2

=-

- Sketch the curve

Solution


xx2

-

- gives

00

CONTINUED




( )

lim lim

limx

x

x

x x

x11

1

1 1

1

2

x x

x

1

2

1

1

-

-=

-

+ -

= +

=

-

] ]g g

11

y xx2

=-





y xx

x

x x

x

11

1

1 1

1

2

=-

-

=

+

= +

-

-^ ^h h


2 Find limx

x x2

2x 2

2

+

+ -

-


x x2

22

= + -

+

Solution


x x2

22

+

+ - gives

00

lim lim

lim

xx x

x

x x

x

22

2

1 2

1

3

x x

x

2

2

2

2

+

+ -=

+

- +

=

= -

-

- -

-

^^ ^

^

hh h

h2 y

x

x xx

y x

x

x

x

22

2

2

1

1


=+

+ -= -

=+

= -

+ -^ ^h h


Remember that x 1




1 Find(a) lim x 52

+x 4

(b) lim t 7-t 3 -

(c) lim x x2 43+ -

x 2

(d) lim xx x32

+

x 0

(e) limh

h h

2

22

-

- -

h 2

(f) lim y

y

5

1253

-

-

y 5

(g) limx

x x

12 12

+

+ +

x 1-

(h) limx

x x4

2 82

+

+ -

x 4 -

(i) limc

c

4

22

-

-

c 2

(j) limx x

x 12

-

-

x 1

(k) lim h

h h h2 73 2+ -

h 0

(l) limh

hx hx h32 2- +

h 0

(m) limh

hx h x hx h2 3 53 2 2- + -

h 0

(n) lim x c x c 3 3

-

-

x c



discontinuous

(a) 3 y x2= -

(b)1

1 y

x=

+

(c) f x x 1= -] g

(d)4

1 y

x2=

+

(e)4

1

y x2=

-



(a)3

y xx x2

= +

(b)33

y x

x x2

=+

+

(c)1

5 4 y

xx x2

=+

+ +

510 Exercises




Regions

Class Investigation


them all







EXAMPLE


Solution



y

x =3

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4







region

EXAMPLE


Solution



included


shown

y

x

1

1

-2 2 3 4

-3

-4

-5

2

3

4

5

-1

-2

-3-4

-1 y = -1



y -intercept 1-






EXAMPLES


1 y x 2$ +

Solution






y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0



2 x y 2 3 61-

Solution


region


side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g








3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g








15 A 3

B 4

C 7

D 3

E 5

F 7

G 4

Function notation




EXAMPLES


Solution

When x

y x

3

1

3 1

4

=

= +

= +

=


Solution

f x x

f

1

3 3 1

4

= +

= +

=

]] gg





EXAMPLES

1 If f x x x3 12= + +] g find f 2-] g

Solution

( ) ( )f 2 2 3 2 1

4 6 1

1

2- = - + - +

= - +

= -

] g


Solution

( )

( )

f x x x

f 1 1 1

1 12

3 2

3

= -

- = - - -

= - -

= -

2] ]g g


Solution

( )

ie

( ) ( )

f x

x x

x x

x x

x x

0

3 10 0

5 2 0

5 0 2 0

5 2

2

=

+ - =

+ - =

+ = - =

= - =


when

when f x

x x

x x

3 4 2

2 21

$=

+

-

] g )

Solution

since 4 21-

( ) ( ) since

( ) ( ) since

( ) ( ) since

( ) ( )

f

f

f

f

3 3 3 4 3 2

13

2 3 2 4 2 2

10

0 2 0 0 2

0

4 2 4

8

1

$

$

= +

=

= +

=

= -

=

- = - -

=


when

when

x

x

x

2

1 2

1

2

1

-

-

g x

x

x2 1

5

2

= -] g


when 2x -=







than 2




Solution

( ) ( )

( )

( )

g

g

g

1 2 1 1 1 1 2

1

2 5 2 1

3 3 3 2

9

since

since

since2

1

2

= - -

=

- = - -

=

=

( ) ( ) ( ) g g g 1 2 3 1 5 9

3

So + - - = + -

= -

DID YOU KNOW






f 3-] g

2 If h x x 22= -] g find h h0 2] ]g g

and h 4-] g


and f 2-] g


= - +] g

5 Find f 3-] g if f x x x2 5 43= - +] g


f x 13=] g




f x271

=] g




f x h+] g


g x x x2 32= + +] g


product of factors






15 f x x x

x x

1

1

for

for

32

=] g )


16 f x

x x

x x

x x

2 4 1

3 1 1

1

if

if

if 2

1 1

$

=

-

+ -

-

] gZ

[

]]

]]

Find the values of

f f f 2 2 1- - + -] ] ]g g g

52 Exercises








EXAMPLE


Solution


x x

x x

x x

x x

0 7 8

8 1

8 0 1 0

8 1

2= + -

= + -

+ = - =

= - =

] ]g g


y 0 7 0 8

8

2= + -

= -

] ]g g





Domain and range




domain


image) of f

EXAMPLE


Solution


x

y

CONTINUED






Domain all real x

Range y y 0$















EXAMPLES


x x 3 x 2 x 1

y

Solution


increasing


x 1 x 2

2

x x 3

x 2 x 1

y

Solution



increases again




1 1








=







x

y









x

y




EXAMPLES


Solution

f x x

x

f x

f x x

3

3


2

2

2`

- = - +

= +

=

= +

] ]]

]

g gg

g


Solution

f x x x

x x

x x

f x


3

3

3

3`

- = - - -

= - +

= - -

= -

= -

] ] ]^]

]

g g gh

gg

Investigation






1 f x kxn=] g

2 f x x kn= +] g

3 f x x k n= +

] ]g g






shape of graphs


each function

(a) y x3 2= -

(b) x y 2 5 20 0- + =

(c) x y 3 12 0+ - =

(d) f x x x32= +] g

(e) f x x 42= -] g

(f) p x x x5 62= + +] g

(g) y x x8 152= - +

(h) p x x 53

= +

] g

53 Exercises




(i) y xx

x3

0= + ] g

(j) g x x9 2= -] g


f x x 22= -

] g What type of

function is it


(a) f x2^ h (b) ( )f x 26



an even function


f x x=] g


function


odd function



f x f x- -] ]g g


neither

(a) y x x

x4 2

3

=

-

(b) y x 1

13

=

-

(c) f xx 4

32

=

-

] g

(d) y x

x

33

=+

-

(e) f x x x

x5 2

3

=-] g



f x xn=] g

even(a)

odd(b)


ever be

even(a)

odd(b)



graph is increasing




x

y(a)

x

4

y(b)

2-2

x

y(c)




Investigation


graph







y kx

y kx

y kx

y kx

y xk

(a)

(b)

(c)

(d)

(e)

2

3

4

=

=

=

=

=



y x k

y x k

y x k

y x k

y x k1

(a)

(b)

(c)

(d)

(e)

2

2

3

4

= +

= +

= +

= +

= +

] g



-2

1 2

-4

-1-2

2

4

y

x

(d) y

x

(e)

CONTINUED




Gradient form



Investigation










y x k

y x k

y x k

y x k y

x k

1

(a)

(b)

(c)

(d)(e)

2

3

4

= +

= +

= +

= +

=+

]]

]

gg

g





the right








Linear Function





EXAMPLE



Solution


Find the intercepts


0 3 5

5 3

1

x

x

x32

=

=

=

-


3 5

5

y 0=

= -

-] g

-1

-2

y

5

4

3

2

1 1 23

6

-3

-4

-5

1 4-1-2 32-3-4

x



Range all real y








Special lines






1 y kx=

2 y x k= +






EXAMPLES


Solution




-1

-3

y

4

3

2

1

5

-2

-4

-5

1 4-1-2

x

32-3-4


2 Sketch x 1= -


Solution





-

-

4

3

2

5

-2

-4

-5

1 4-1-2-4

y

x







Domain all real x

Range y y b=

54 Exercises


each function

(a) y x 2= -

(b) f x x2 3= +] g (c) x y 2 1 0+ =-

(d) x y 3 0+ =-

(e) x y 3 6 2 0=- -


line

(a) x 4=

(b) x 3 0=-

(c) y 5=

(d) y 1 0+ =

(e) f x x2 1= -] g (f) y x 4= +

(g) f x x3 2= +] g

(h) x y 3+ =

(i) x y 1 0=- -

(j) x y 2 3 0+ =-


(a) x y 3 2 7 0+ =-

(b) y 2=

(c) x 4= - (d) x 2 0=-

(e) y 3 0=-


are even or odd

(a) y x2=

(b) y 3=

(c) x 4=

(d) y x= -

(e) y x=




meet




Applications






The pronumeral

a is called the


Quadratic Function










EXAMPLES



Solution



x

x

x

0 1

1

1

2

2

= -

=

=


0 1

1

y 2= -

= -



Domain xall real

Range y y 1$ -


Solution


parabola will lie




left



0

1 0

1

x

x

x

1 2= +

+ =

= -

] g


1

y 0 1 2= +

=

] g

-1

-

4

3

2

1

5

-2

-4

-5

-

1 41-2 5-4

y

x









Solution



x x

x xx x

x x

0 6

3 23 0 2 0

3 2

2= + -

= + -

+ = - =

= - =

] ]g g


f 0 0 0 66

2= + -

= -

] ] ]g g g



x -intercepts


23 2

21- +

= -


-c m

f 21

21

21

6

41

21

6

641

2

- = - + - -

= - -

= -

c c cm m m


-

CONTINUED

You will learn more


-1

-3

4

3

2

1

5

-2

-4

-5

1 4-1-2 32-3-4

y

x










41



2

11 -





Solution


So f x 0=] g

0 4 54 5 0

0

x xx x

x x5 1

2

2

= - + +

=

+ =

- -

-] ]g g

x x

x x

5 0 1 0

5 1

- = + =

= = -


f 0 0 4 0 5

5

2= - + +

=

] ] ]g g g

-1

-3

4

2

5

-2

-4

-5

-

y

1 41-2 5-4

x

-2 4






21 5

2- +

=

f 2 2 4 2 59

= - + +

=

2] ] ]g g g










each function

(a) 2 y x x2= +

(b) 3 y x x2= - +

(c) f x x 12= -] g

(d) y x x 22= - -

(e) y x x9 82= +-

2 Sketch

(a) 2 y x2= +

(b) y x 12= - +

(c) f x x 42= -] g

(d) 2 y x x2= +

(e) y x x2= - -

(f) f x x 3= -2] ]g g

55 Exercises

-1

9

8

7

5

4

3

2

6

1

-2

-3

-4

-5

y

2 51 643-1-2-3-4

x




EXAMPLES


Solution









(g) f x x 1 2= +] ]g g

(h) y x x3 42= + -

(i) y x x2 5 32= - +

(j) f x x x3 22= - + -] g



(a) ndash y x x7 122= +

(b) f x x x42= +] g

(c) y x x2 82= - -

(d) y x x6 92= +-

(e) f t t 4 2= -] g


(a) y x 52= -

(b) f x x x6

2= -

] g (c) f x x x 22= - -] g

(d) y x2= -

(e) f x x 7 2= -] ]g g



(a) y x2= for x0 3

(b) y x 42= - + for x1 2 -

(c) f x x 12= -] g for x2 5 -

(d) y x x2 32= + - for x2 4 -

(e) y x x

22= - +- for

x0 4


function is

increasing(i)

decreasing(ii)

(a) y x2=

(b) y x2

= - (c) f x x 92

= -] g

(d) y x x42= - +

(e) f x x 5 2= +] ]g g


function



(a) y x 12= +

(b) f x x 32= -

] g

(c) y x2 2= -

(d) f x x x32= -] g

(e) f x x x2= +] g

(f) y x 42= -

(g) y x x2 32= - -

(h) y x x5 42= +-

(i) p x x 1 2= +] ]g g

(j) y x 2= -2] g




CONTINUED

eg When x 3= -

| | y 3 13 12

= - -

= -

=

x -3 -2 -1 0 1 2 3

y 2 1 0 -1 0 1 2


y

-2

4

3

2

1

5

-1

-3

-4

-5

1 4-1-2 32-3-4

x


| | y x

x x

x x1

1 0

1 0

when

when 1

$= - =

-


y x x1 0$= - ] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x - 1




y x 1= - - x 01] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y =- x - 1



-3

4

3

2

1

5

-2

-1

-4

-5

y y

3-1-2 421-3-4

x

y = - x - 1

y = x - 1


intercepts


So f x 0=

] g

| |

| |

x

x

x

0 1

1

1`

= -

=

=


( ) | |f 0 0 11

= -

= -





-3

4

3

2

1

5

-2

-1

-4

-5

y

4-2 5321-1-3-4

x



Domain all real x

Range y y 1$ -

2 Sketch | | y x 2= +

Solution


| | ( ) y x x xx x2 2 2 0

2 2 0whenwhen 1

$= + = + +

- + +


2 y x= + when x 2 0$+

x 2$ -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x + 2






y-intercept of 1-

CONTINUED




2 y x= - +] g when x 2 01+

ie y x 2= - - when x 21 -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2


-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2

y = x + 2



So 0f x =] g

0 | 2 |

0 2

2

x

x

x

= +

= +

- =


(0) | 0 2 |

2

f = +

=

There is only one

solution for the


Can you see why





-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

If you know how to

sketch the graph of




Investigation




are their equations





1 | |f x k x=] g

2 | |f x x k= +] g

3 | |f x x k= +] g







EXAMPLES

Solve

1 |2 1 | 3x - =

Solution



is x 1 2= -

2 |2 1 | 3 2x x= -+

Solution



3 | 1 | 2x 1+

Solution






and then check them







each function

(a) | | y x=

(b) | |f x x 7= +] g

(c) | |f x x 2= -] g

(d) 5 | | y x=

(e) | |f x x 3= - +] g

(f) | 6 | y x= +

(g) | |f x x3 2= -] g

(h) | 5 4 | y x= +

(i) | 7 1 | y x= -

(j) | |f x x2 9= +] g


plane

(a) | | y x=

(b) | |f x x 1= +] g

(c) | |f x x 3= -] g

(d) 2 | | y x=

(e) | |f x x= -] g

(f) | 1 | y x= +

(g) | |f x x 1= - -] g

(h) | 2 3 | y x= -

(i) | 4 2 | y x= +

(j) | |f x x3 1= +] g


each function

(a) | 1 | y x= -

(b) | |f x x 8= -] g

(c) | |f x x2 5= +] g

(d) 2 | | 3 y x= -

(e) | |f x x 3= - -] g


function is

increasing(i)

decreasing(ii)

(a) | 2 | y x= -

(b) | |f x x 2= +

] g

(c) | |f x x2 3= -] g

(d) 4 | | 1 y x= -

(e) | |f x x= -] g


of each function

(a) | | y x= for x2 2 -

(b) | |f x x 4= - -] g for

x4 3 -

(c) | |f x x 4= +] g for x7 2 -

(d) | 2 5 | y x= - for x3 3 -

(e) | |f x x= -] g for x1 1 -


function increasing

(a) | 3 | y x= +

(b) | |f x x 4= - +] g

(c) | |f x x 9= -] g

(d) | | y x 2 1= - -

(e) | |f x x 2= - +] g

56 Exercises




7 Solve graphically

(a) | | 3x =

(b) | |x 12

(c) | |x 2

(d) | 2 | 1x + =

(e) | 3 | 0x- =

(f) |2 3 | 1x - =

(g) | |x 1 41-

(h) | |x 1 3+

(i) | |x 2 22-

(j) | |x 3 1$-

(k) | |x2 3 5+

(l) | |x2 1 1$-

(m) |3 1 | 3x x- = +

(n) |3 2 | 4x x- = -

(o) | 1 | 1x x- = +

(p) | 3 | 2 2x x+ = + (q) |2 1 | 1x x+ = -

(r) |2 5 | 3x x- = -

(s) | 1 | 2x x- =

(t) |2 3 | 3x x- = +

The Hyperbola


or= =

EXAMPLE

Sketch1

y x=

Solution



x -3 -2 -121-

41- 0

41

21 1 2 3

y 3

1-

2

1- -1 -2 -4 mdash 4 2 1

2

1

3

1

Class Discussion



y is never 0 Why






but never touches)



Investigation




equations




1 y xk

=

21

y x k= +

31

y x k

=+


EXAMPLES


3=

-

] g


Solution



So x 3






be zero

CONTINUED







1

y 0 3

3=

-

= -

-3

4

3

2

1

5

-2

-1

-4

-5

y

-1-2 4 521-3-4

x

x = 3

y = 0

Asymptotes

3

2 Sketch y x2 4

1= -

+

Solution




x

x

2 4

2

-

-






( ) y

2 0 41

41

= -+

= -


a translation of3

y x

=





y

-2

x

-

1

4


a=

+


domain x xb



1 For each graph



exists


(a)2

y x=

(b)1

y x= -

(c) f xx 1

1=

+] g

(d) f xx 2

3=

-

] g

(e)3 6

1 y

x=

+

(f) f xx 3

2= -

-

] g

(g) f xx 1

4=

-

] g

(h)1

2 y

x= -

+

(i) f xx6 3

2=

-

] g

(j)2

6 y

x= -

+

2 Show that f x x2

=] g is an odd

function



(a) f xx2 5

1=

+

] g for x2 2 -

(b)3

1 y

x=

+ for x2 0 -

(c) f xx2 4

5=

-

] g for x3 1 -

57 Exercises




(d) f xx 4

3= -

-

] g for x3 3 -

(e)3 1

2 y

x= -

+ for x0 5



(a)3

y x= for y 1 3

(b)2

y x= - for y 221

- -

(c) f xx 1

1=

-

] g for y 171

- -

(d) f xx2 1

3= -

+

] g for

y 131

- -

(e)3 2

6 y

x=

- for y 1

21

6



Circle gate





Proof

r y

x

( x y)

y

x








c a b

r x y

2 2 2

2 2 2

`

= +

= +

EXAMPLE



Solution


y

x

-2

-2 2

2


one place

y

x

2

2

2

-2

The radius is 4

CONTINUED






Domain 2 2 x x -

Range 2 2 y y -


range y r y r -



Proof


shown

y

x

(a b)

x

y

r

( x y)

a

b x - a

y - b




ndash ndash

c a b

r x a y b

2 2 2

2 2 2

= +

= +] ^g h




EXAMPLES



Solution



y

x 9

9

-9

-9



Domain 9 9 x x -

Range 9 9 y y -



Solution


ndash

ndash ndash

x y

x y

1 2 4

1 2 2

2

2 2

+ + =

+ - =

2

2

] ^

] ]_

g h

g gi

So 1 2a b= = - and 2r =

CONTINUED







y

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

(1 -2)



Domain 1 3 x x -

Range 4 0 y y -


Solution


a b2 1= - = and r 3=

Substituting

ndash ndash

ndash

ndash

x a y b r

x y

x y

2 1 3

2 1 9

2 2 2

2 2 2

2 2

+ =

- - + =

+ + =

] ^]] ^

] ^

g hg g hg h


ndash

ndash

a b a ab b

x x x

x x

a b a ab b

y y y

y y

2

2 2 2 2

4 4

2

1 2 1 1

2 1

So

So

2 2 2

2 2 2

2

2 2 2

2 2 2

2

+ = + +

+ = + +

= + +

= - +

= - +

= - +

]] ] ]]^ ^ ]

gg g ggh h g


ndash

x x y y

x x y y

x x y y

x x y y

4 4 2 1 9

4 2 5 9

4 2 5 9

4 2 4 0

9 9

2

2

2

2

+ + + - + =

+ + - + =

+ + + =

+ + - - =

- -


in Chapter 2




Investigation


two functions





circle






for all values

y

x r

r

-r


Proof

ndash

x y r

y r x y r x

2 2 2

2 2 2

2 2

+ =

=

= -



semi-circles





y 0 for all values

y

x r

-

r

-r


EXAMPLES


1 f x x92

= -] g

Solution



y

x 3

3

-3

Domain 3 3 x x -

Range 0 3 y y




2 y x4 2= - -

Solution



y

x 2

-2

-2

Domain 2 2 x x -

Range 2 0 y y -




range

(a) 9x y 2 2+ =

(b) x y 16 02 2+ =-


(d) 1 9x y 2 2+ + =

] g

(e) ndashx y 2 1 12 2+ + =] ^g h



below the x -axis



range

(a) 25 y x2= - -

(b) 1 y x2= -

(c) 36 y x2= -

(d) 64 y x2= - -

(e) 7 y x2= - -



each circle

(a) 100x y 2 2+ =

(b) 5x y 2 2+ =


(d) ndashx y 5 6 492 2+ + =] ^g h

(e) ndashx y 3 812 2+ =^ h

58 Exercises






grouping symbols)



Centre(c) 1 5-







Centre(j) 4 7- -

^ h and radius 3

Other Graphs




EXAMPLES


Solution



eg When 0x =

y 3

1

c=

=

x

y

1

3

3

1

31

When1

1

= -

=

=

=

-

x 3- 2- 1- 0 1 2 3

y 271

91

31

1 3 9 27



You will meet these

graphs again in the

HSC Course





other point

When

x

y

1

3

3

1

=

=

=

y

x

1

2

1

3







Solution



x minus2 minus1 0 05 1 2 3 4

y minus03 0 03 05 06

y

x

1

2

1 2 3 4

-1




You learned about


stages of maths





range y y 02



range all real y

Cubic function


EXAMPLE



Solution



y minus25 minus6 1 2 3 10 29

y

x

1

1

-2 2 3 4

-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4



If you already

know the shape of


3






Domain and range


EXAMPLE


Solution




function are

f

f

1 1 2

1 21

4 4 2

64 2

66

3

3

- = - +

= - +

=

= +

= +

=

] ]

] ]

g g

g g


these points

y

x

(-1 1)

(4 66)

Range 1 66 y y










positive number

EXAMPLE


Solution


x 4So $

$


(or zero) So y 0$


59 Exercises


(a) 4 3 y x= +

(b) f x 4= -] g

(c) 3x =



(f) f x xx 12 2= - -] g

(g) 64x y 2 2+ =

(h) f t t 4

3=

-] g

(i) ( ) g 2

5zz

= +

(j) | |f x x=] g


(a) y x=

(b) 2 y x= -

(c) | |f x x2 3= -] g

(d) | | 2 y x= -

(e) f x x2 5= - +] g

(f) | | y x5= -

(g) 2 y x=

(h) y 5x= -

(i) f x xx 1

= +] g

(j)2

4 3 y x

x= -


(a) y x x 5 2= -] g


= +-

(d) g x x x164 2= -] g

(e) 49x y 2 2+ =

You may like to


by dividing by x




4 (a) Solve x1 02$-


f x x1 2= -] g


(a) 2 y x x2= - -

(b) g t t t 62= +] g



in each case


x3 3 -


x2 3 -


x2 1 - (d)


x1 5


0 4x


x3 3 -


x1 1 -


x2 3 -


x4 4 -

(j) y x x7 62= - + - in the

domain 0 7x


function y x 1

3=

+





x=] g


function

find its range(b)


plane

(a) f x x4=] g

(b) y x3= -

(c) y x 34= -

(d) 2 p x x3=] g

(e) 1 g x x3= +] g

(f) 100x y 2 2+ =

(g) 2 1 y x= +


y x 1= -









(b) f x x 22= -] g

(c)1

y x=

(d) f x x3=] g

(e) f x 3x=] g

13 (a) Solve x4 02$-


(i) 4 y x2= -

(ii) y x4

2= - -




DID YOU KNOW




a light



Limits






a a1

1

03

Z

=

=

3

3

-


lim a 0x=

x 3-



asymptote

EXAMPLES

1 Find lim x

x x5x 0

2+

Solution


which is undefined


( )

lim lim

lim

xx x

x

x x

x

5 5

5

5

x x

x

0

2

0 1

1

0

+=

+

= +

=

] g




2 Find limx

x

4

22

-

-

x 2

Solution


lim lim

lim

x

x

x x

x

x

4

2

2 2

2

21

41

2 1

1

-

-=

+ -

-

=+

=

x x2 2

x 2

^ _h i

3 Find limh

h x hx h2 72 2+ -

h 0

Solution

lim lim

lim

h

h x hx h

h

h hx x

hx x

x

2 7 2 7

2 7

7

2 2 2

2

2

+ -=

+ -

= + -

= -

h 0

h h0 0

^ h

Continuity





EXAMPLES

1 Find lim

x

x

1

12

-

-

x 1


11

y x

x2

=-

- Sketch the curve

Solution


xx2

-

- gives

00

CONTINUED




( )

lim lim

limx

x

x

x x

x11

1

1 1

1

2

x x

x

1

2

1

1

-

-=

-

+ -

= +

=

-

] ]g g

11

y xx2

=-





y xx

x

x x

x

11

1

1 1

1

2

=-

-

=

+

= +

-

-^ ^h h


2 Find limx

x x2

2x 2

2

+

+ -

-


x x2

22

= + -

+

Solution


x x2

22

+

+ - gives

00

lim lim

lim

xx x

x

x x

x

22

2

1 2

1

3

x x

x

2

2

2

2

+

+ -=

+

- +

=

= -

-

- -

-

^^ ^

^

hh h

h2 y

x

x xx

y x

x

x

x

22

2

2

1

1


=+

+ -= -

=+

= -

+ -^ ^h h


Remember that x 1




1 Find(a) lim x 52

+x 4

(b) lim t 7-t 3 -

(c) lim x x2 43+ -

x 2

(d) lim xx x32

+

x 0

(e) limh

h h

2

22

-

- -

h 2

(f) lim y

y

5

1253

-

-

y 5

(g) limx

x x

12 12

+

+ +

x 1-

(h) limx

x x4

2 82

+

+ -

x 4 -

(i) limc

c

4

22

-

-

c 2

(j) limx x

x 12

-

-

x 1

(k) lim h

h h h2 73 2+ -

h 0

(l) limh

hx hx h32 2- +

h 0

(m) limh

hx h x hx h2 3 53 2 2- + -

h 0

(n) lim x c x c 3 3

-

-

x c



discontinuous

(a) 3 y x2= -

(b)1

1 y

x=

+

(c) f x x 1= -] g

(d)4

1 y

x2=

+

(e)4

1

y x2=

-



(a)3

y xx x2

= +

(b)33

y x

x x2

=+

+

(c)1

5 4 y

xx x2

=+

+ +

510 Exercises




Regions

Class Investigation


them all







EXAMPLE


Solution



y

x =3

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4







region

EXAMPLE


Solution



included


shown

y

x

1

1

-2 2 3 4

-3

-4

-5

2

3

4

5

-1

-2

-3-4

-1 y = -1



y -intercept 1-






EXAMPLES


1 y x 2$ +

Solution






y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0



2 x y 2 3 61-

Solution


region


side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g








3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g






EXAMPLES

1 If f x x x3 12= + +] g find f 2-] g

Solution

( ) ( )f 2 2 3 2 1

4 6 1

1

2- = - + - +

= - +

= -

] g


Solution

( )

( )

f x x x

f 1 1 1

1 12

3 2

3

= -

- = - - -

= - -

= -

2] ]g g


Solution

( )

ie

( ) ( )

f x

x x

x x

x x

x x

0

3 10 0

5 2 0

5 0 2 0

5 2

2

=

+ - =

+ - =

+ = - =

= - =


when

when f x

x x

x x

3 4 2

2 21

$=

+

-

] g )

Solution

since 4 21-

( ) ( ) since

( ) ( ) since

( ) ( ) since

( ) ( )

f

f

f

f

3 3 3 4 3 2

13

2 3 2 4 2 2

10

0 2 0 0 2

0

4 2 4

8

1

$

$

= +

=

= +

=

= -

=

- = - -

=


when

when

x

x

x

2

1 2

1

2

1

-

-

g x

x

x2 1

5

2

= -] g


when 2x -=







than 2




Solution

( ) ( )

( )

( )

g

g

g

1 2 1 1 1 1 2

1

2 5 2 1

3 3 3 2

9

since

since

since2

1

2

= - -

=

- = - -

=

=

( ) ( ) ( ) g g g 1 2 3 1 5 9

3

So + - - = + -

= -

DID YOU KNOW






f 3-] g

2 If h x x 22= -] g find h h0 2] ]g g

and h 4-] g


and f 2-] g


= - +] g

5 Find f 3-] g if f x x x2 5 43= - +] g


f x 13=] g




f x271

=] g




f x h+] g


g x x x2 32= + +] g


product of factors






15 f x x x

x x

1

1

for

for

32

=] g )


16 f x

x x

x x

x x

2 4 1

3 1 1

1

if

if

if 2

1 1

$

=

-

+ -

-

] gZ

[

]]

]]

Find the values of

f f f 2 2 1- - + -] ] ]g g g

52 Exercises








EXAMPLE


Solution


x x

x x

x x

x x

0 7 8

8 1

8 0 1 0

8 1

2= + -

= + -

+ = - =

= - =

] ]g g


y 0 7 0 8

8

2= + -

= -

] ]g g





Domain and range




domain


image) of f

EXAMPLE


Solution


x

y

CONTINUED






Domain all real x

Range y y 0$















EXAMPLES


x x 3 x 2 x 1

y

Solution


increasing


x 1 x 2

2

x x 3

x 2 x 1

y

Solution



increases again




1 1








=







x

y









x

y




EXAMPLES


Solution

f x x

x

f x

f x x

3

3


2

2

2`

- = - +

= +

=

= +

] ]]

]

g gg

g


Solution

f x x x

x x

x x

f x


3

3

3

3`

- = - - -

= - +

= - -

= -

= -

] ] ]^]

]

g g gh

gg

Investigation






1 f x kxn=] g

2 f x x kn= +] g

3 f x x k n= +

] ]g g






shape of graphs


each function

(a) y x3 2= -

(b) x y 2 5 20 0- + =

(c) x y 3 12 0+ - =

(d) f x x x32= +] g

(e) f x x 42= -] g

(f) p x x x5 62= + +] g

(g) y x x8 152= - +

(h) p x x 53

= +

] g

53 Exercises




(i) y xx

x3

0= + ] g

(j) g x x9 2= -] g


f x x 22= -

] g What type of

function is it


(a) f x2^ h (b) ( )f x 26



an even function


f x x=] g


function


odd function



f x f x- -] ]g g


neither

(a) y x x

x4 2

3

=

-

(b) y x 1

13

=

-

(c) f xx 4

32

=

-

] g

(d) y x

x

33

=+

-

(e) f x x x

x5 2

3

=-] g



f x xn=] g

even(a)

odd(b)


ever be

even(a)

odd(b)



graph is increasing




x

y(a)

x

4

y(b)

2-2

x

y(c)




Investigation


graph







y kx

y kx

y kx

y kx

y xk

(a)

(b)

(c)

(d)

(e)

2

3

4

=

=

=

=

=



y x k

y x k

y x k

y x k

y x k1

(a)

(b)

(c)

(d)

(e)

2

2

3

4

= +

= +

= +

= +

= +

] g



-2

1 2

-4

-1-2

2

4

y

x

(d) y

x

(e)

CONTINUED




Gradient form



Investigation










y x k

y x k

y x k

y x k y

x k

1

(a)

(b)

(c)

(d)(e)

2

3

4

= +

= +

= +

= +

=+

]]

]

gg

g





the right








Linear Function





EXAMPLE



Solution


Find the intercepts


0 3 5

5 3

1

x

x

x32

=

=

=

-


3 5

5

y 0=

= -

-] g

-1

-2

y

5

4

3

2

1 1 23

6

-3

-4

-5

1 4-1-2 32-3-4

x



Range all real y








Special lines






1 y kx=

2 y x k= +






EXAMPLES


Solution




-1

-3

y

4

3

2

1

5

-2

-4

-5

1 4-1-2

x

32-3-4


2 Sketch x 1= -


Solution





-

-

4

3

2

5

-2

-4

-5

1 4-1-2-4

y

x







Domain all real x

Range y y b=

54 Exercises


each function

(a) y x 2= -

(b) f x x2 3= +] g (c) x y 2 1 0+ =-

(d) x y 3 0+ =-

(e) x y 3 6 2 0=- -


line

(a) x 4=

(b) x 3 0=-

(c) y 5=

(d) y 1 0+ =

(e) f x x2 1= -] g (f) y x 4= +

(g) f x x3 2= +] g

(h) x y 3+ =

(i) x y 1 0=- -

(j) x y 2 3 0+ =-


(a) x y 3 2 7 0+ =-

(b) y 2=

(c) x 4= - (d) x 2 0=-

(e) y 3 0=-


are even or odd

(a) y x2=

(b) y 3=

(c) x 4=

(d) y x= -

(e) y x=




meet




Applications






The pronumeral

a is called the


Quadratic Function










EXAMPLES



Solution



x

x

x

0 1

1

1

2

2

= -

=

=


0 1

1

y 2= -

= -



Domain xall real

Range y y 1$ -


Solution


parabola will lie




left



0

1 0

1

x

x

x

1 2= +

+ =

= -

] g


1

y 0 1 2= +

=

] g

-1

-

4

3

2

1

5

-2

-4

-5

-

1 41-2 5-4

y

x









Solution



x x

x xx x

x x

0 6

3 23 0 2 0

3 2

2= + -

= + -

+ = - =

= - =

] ]g g


f 0 0 0 66

2= + -

= -

] ] ]g g g



x -intercepts


23 2

21- +

= -


-c m

f 21

21

21

6

41

21

6

641

2

- = - + - -

= - -

= -

c c cm m m


-

CONTINUED

You will learn more


-1

-3

4

3

2

1

5

-2

-4

-5

1 4-1-2 32-3-4

y

x










41



2

11 -





Solution


So f x 0=] g

0 4 54 5 0

0

x xx x

x x5 1

2

2

= - + +

=

+ =

- -

-] ]g g

x x

x x

5 0 1 0

5 1

- = + =

= = -


f 0 0 4 0 5

5

2= - + +

=

] ] ]g g g

-1

-3

4

2

5

-2

-4

-5

-

y

1 41-2 5-4

x

-2 4






21 5

2- +

=

f 2 2 4 2 59

= - + +

=

2] ] ]g g g










each function

(a) 2 y x x2= +

(b) 3 y x x2= - +

(c) f x x 12= -] g

(d) y x x 22= - -

(e) y x x9 82= +-

2 Sketch

(a) 2 y x2= +

(b) y x 12= - +

(c) f x x 42= -] g

(d) 2 y x x2= +

(e) y x x2= - -

(f) f x x 3= -2] ]g g

55 Exercises

-1

9

8

7

5

4

3

2

6

1

-2

-3

-4

-5

y

2 51 643-1-2-3-4

x




EXAMPLES


Solution









(g) f x x 1 2= +] ]g g

(h) y x x3 42= + -

(i) y x x2 5 32= - +

(j) f x x x3 22= - + -] g



(a) ndash y x x7 122= +

(b) f x x x42= +] g

(c) y x x2 82= - -

(d) y x x6 92= +-

(e) f t t 4 2= -] g


(a) y x 52= -

(b) f x x x6

2= -

] g (c) f x x x 22= - -] g

(d) y x2= -

(e) f x x 7 2= -] ]g g



(a) y x2= for x0 3

(b) y x 42= - + for x1 2 -

(c) f x x 12= -] g for x2 5 -

(d) y x x2 32= + - for x2 4 -

(e) y x x

22= - +- for

x0 4


function is

increasing(i)

decreasing(ii)

(a) y x2=

(b) y x2

= - (c) f x x 92

= -] g

(d) y x x42= - +

(e) f x x 5 2= +] ]g g


function



(a) y x 12= +

(b) f x x 32= -

] g

(c) y x2 2= -

(d) f x x x32= -] g

(e) f x x x2= +] g

(f) y x 42= -

(g) y x x2 32= - -

(h) y x x5 42= +-

(i) p x x 1 2= +] ]g g

(j) y x 2= -2] g




CONTINUED

eg When x 3= -

| | y 3 13 12

= - -

= -

=

x -3 -2 -1 0 1 2 3

y 2 1 0 -1 0 1 2


y

-2

4

3

2

1

5

-1

-3

-4

-5

1 4-1-2 32-3-4

x


| | y x

x x

x x1

1 0

1 0

when

when 1

$= - =

-


y x x1 0$= - ] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x - 1




y x 1= - - x 01] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y =- x - 1



-3

4

3

2

1

5

-2

-1

-4

-5

y y

3-1-2 421-3-4

x

y = - x - 1

y = x - 1


intercepts


So f x 0=

] g

| |

| |

x

x

x

0 1

1

1`

= -

=

=


( ) | |f 0 0 11

= -

= -





-3

4

3

2

1

5

-2

-1

-4

-5

y

4-2 5321-1-3-4

x



Domain all real x

Range y y 1$ -

2 Sketch | | y x 2= +

Solution


| | ( ) y x x xx x2 2 2 0

2 2 0whenwhen 1

$= + = + +

- + +


2 y x= + when x 2 0$+

x 2$ -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x + 2






y-intercept of 1-

CONTINUED




2 y x= - +] g when x 2 01+

ie y x 2= - - when x 21 -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2


-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2

y = x + 2



So 0f x =] g

0 | 2 |

0 2

2

x

x

x

= +

= +

- =


(0) | 0 2 |

2

f = +

=

There is only one

solution for the


Can you see why





-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

If you know how to

sketch the graph of




Investigation




are their equations





1 | |f x k x=] g

2 | |f x x k= +] g

3 | |f x x k= +] g







EXAMPLES

Solve

1 |2 1 | 3x - =

Solution



is x 1 2= -

2 |2 1 | 3 2x x= -+

Solution



3 | 1 | 2x 1+

Solution






and then check them







each function

(a) | | y x=

(b) | |f x x 7= +] g

(c) | |f x x 2= -] g

(d) 5 | | y x=

(e) | |f x x 3= - +] g

(f) | 6 | y x= +

(g) | |f x x3 2= -] g

(h) | 5 4 | y x= +

(i) | 7 1 | y x= -

(j) | |f x x2 9= +] g


plane

(a) | | y x=

(b) | |f x x 1= +] g

(c) | |f x x 3= -] g

(d) 2 | | y x=

(e) | |f x x= -] g

(f) | 1 | y x= +

(g) | |f x x 1= - -] g

(h) | 2 3 | y x= -

(i) | 4 2 | y x= +

(j) | |f x x3 1= +] g


each function

(a) | 1 | y x= -

(b) | |f x x 8= -] g

(c) | |f x x2 5= +] g

(d) 2 | | 3 y x= -

(e) | |f x x 3= - -] g


function is

increasing(i)

decreasing(ii)

(a) | 2 | y x= -

(b) | |f x x 2= +

] g

(c) | |f x x2 3= -] g

(d) 4 | | 1 y x= -

(e) | |f x x= -] g


of each function

(a) | | y x= for x2 2 -

(b) | |f x x 4= - -] g for

x4 3 -

(c) | |f x x 4= +] g for x7 2 -

(d) | 2 5 | y x= - for x3 3 -

(e) | |f x x= -] g for x1 1 -


function increasing

(a) | 3 | y x= +

(b) | |f x x 4= - +] g

(c) | |f x x 9= -] g

(d) | | y x 2 1= - -

(e) | |f x x 2= - +] g

56 Exercises




7 Solve graphically

(a) | | 3x =

(b) | |x 12

(c) | |x 2

(d) | 2 | 1x + =

(e) | 3 | 0x- =

(f) |2 3 | 1x - =

(g) | |x 1 41-

(h) | |x 1 3+

(i) | |x 2 22-

(j) | |x 3 1$-

(k) | |x2 3 5+

(l) | |x2 1 1$-

(m) |3 1 | 3x x- = +

(n) |3 2 | 4x x- = -

(o) | 1 | 1x x- = +

(p) | 3 | 2 2x x+ = + (q) |2 1 | 1x x+ = -

(r) |2 5 | 3x x- = -

(s) | 1 | 2x x- =

(t) |2 3 | 3x x- = +

The Hyperbola


or= =

EXAMPLE

Sketch1

y x=

Solution



x -3 -2 -121-

41- 0

41

21 1 2 3

y 3

1-

2

1- -1 -2 -4 mdash 4 2 1

2

1

3

1

Class Discussion



y is never 0 Why






but never touches)



Investigation




equations




1 y xk

=

21

y x k= +

31

y x k

=+


EXAMPLES


3=

-

] g


Solution



So x 3






be zero

CONTINUED







1

y 0 3

3=

-

= -

-3

4

3

2

1

5

-2

-1

-4

-5

y

-1-2 4 521-3-4

x

x = 3

y = 0

Asymptotes

3

2 Sketch y x2 4

1= -

+

Solution




x

x

2 4

2

-

-






( ) y

2 0 41

41

= -+

= -


a translation of3

y x

=





y

-2

x

-

1

4


a=

+


domain x xb



1 For each graph



exists


(a)2

y x=

(b)1

y x= -

(c) f xx 1

1=

+] g

(d) f xx 2

3=

-

] g

(e)3 6

1 y

x=

+

(f) f xx 3

2= -

-

] g

(g) f xx 1

4=

-

] g

(h)1

2 y

x= -

+

(i) f xx6 3

2=

-

] g

(j)2

6 y

x= -

+

2 Show that f x x2

=] g is an odd

function



(a) f xx2 5

1=

+

] g for x2 2 -

(b)3

1 y

x=

+ for x2 0 -

(c) f xx2 4

5=

-

] g for x3 1 -

57 Exercises




(d) f xx 4

3= -

-

] g for x3 3 -

(e)3 1

2 y

x= -

+ for x0 5



(a)3

y x= for y 1 3

(b)2

y x= - for y 221

- -

(c) f xx 1

1=

-

] g for y 171

- -

(d) f xx2 1

3= -

+

] g for

y 131

- -

(e)3 2

6 y

x=

- for y 1

21

6



Circle gate





Proof

r y

x

( x y)

y

x








c a b

r x y

2 2 2

2 2 2

`

= +

= +

EXAMPLE



Solution


y

x

-2

-2 2

2


one place

y

x

2

2

2

-2

The radius is 4

CONTINUED






Domain 2 2 x x -

Range 2 2 y y -


range y r y r -



Proof


shown

y

x

(a b)

x

y

r

( x y)

a

b x - a

y - b




ndash ndash

c a b

r x a y b

2 2 2

2 2 2

= +

= +] ^g h




EXAMPLES



Solution



y

x 9

9

-9

-9



Domain 9 9 x x -

Range 9 9 y y -



Solution


ndash

ndash ndash

x y

x y

1 2 4

1 2 2

2

2 2

+ + =

+ - =

2

2

] ^

] ]_

g h

g gi

So 1 2a b= = - and 2r =

CONTINUED







y

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

(1 -2)



Domain 1 3 x x -

Range 4 0 y y -


Solution


a b2 1= - = and r 3=

Substituting

ndash ndash

ndash

ndash

x a y b r

x y

x y

2 1 3

2 1 9

2 2 2

2 2 2

2 2

+ =

- - + =

+ + =

] ^]] ^

] ^

g hg g hg h


ndash

ndash

a b a ab b

x x x

x x

a b a ab b

y y y

y y

2

2 2 2 2

4 4

2

1 2 1 1

2 1

So

So

2 2 2

2 2 2

2

2 2 2

2 2 2

2

+ = + +

+ = + +

= + +

= - +

= - +

= - +

]] ] ]]^ ^ ]

gg g ggh h g


ndash

x x y y

x x y y

x x y y

x x y y

4 4 2 1 9

4 2 5 9

4 2 5 9

4 2 4 0

9 9

2

2

2

2

+ + + - + =

+ + - + =

+ + + =

+ + - - =

- -


in Chapter 2




Investigation


two functions





circle






for all values

y

x r

r

-r


Proof

ndash

x y r

y r x y r x

2 2 2

2 2 2

2 2

+ =

=

= -



semi-circles





y 0 for all values

y

x r

-

r

-r


EXAMPLES


1 f x x92

= -] g

Solution



y

x 3

3

-3

Domain 3 3 x x -

Range 0 3 y y




2 y x4 2= - -

Solution



y

x 2

-2

-2

Domain 2 2 x x -

Range 2 0 y y -




range

(a) 9x y 2 2+ =

(b) x y 16 02 2+ =-


(d) 1 9x y 2 2+ + =

] g

(e) ndashx y 2 1 12 2+ + =] ^g h



below the x -axis



range

(a) 25 y x2= - -

(b) 1 y x2= -

(c) 36 y x2= -

(d) 64 y x2= - -

(e) 7 y x2= - -



each circle

(a) 100x y 2 2+ =

(b) 5x y 2 2+ =


(d) ndashx y 5 6 492 2+ + =] ^g h

(e) ndashx y 3 812 2+ =^ h

58 Exercises






grouping symbols)



Centre(c) 1 5-







Centre(j) 4 7- -

^ h and radius 3

Other Graphs




EXAMPLES


Solution



eg When 0x =

y 3

1

c=

=

x

y

1

3

3

1

31

When1

1

= -

=

=

=

-

x 3- 2- 1- 0 1 2 3

y 271

91

31

1 3 9 27



You will meet these

graphs again in the

HSC Course





other point

When

x

y

1

3

3

1

=

=

=

y

x

1

2

1

3







Solution



x minus2 minus1 0 05 1 2 3 4

y minus03 0 03 05 06

y

x

1

2

1 2 3 4

-1




You learned about


stages of maths





range y y 02



range all real y

Cubic function


EXAMPLE



Solution



y minus25 minus6 1 2 3 10 29

y

x

1

1

-2 2 3 4

-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4



If you already

know the shape of


3






Domain and range


EXAMPLE


Solution




function are

f

f

1 1 2

1 21

4 4 2

64 2

66

3

3

- = - +

= - +

=

= +

= +

=

] ]

] ]

g g

g g


these points

y

x

(-1 1)

(4 66)

Range 1 66 y y










positive number

EXAMPLE


Solution


x 4So $

$


(or zero) So y 0$


59 Exercises


(a) 4 3 y x= +

(b) f x 4= -] g

(c) 3x =



(f) f x xx 12 2= - -] g

(g) 64x y 2 2+ =

(h) f t t 4

3=

-] g

(i) ( ) g 2

5zz

= +

(j) | |f x x=] g


(a) y x=

(b) 2 y x= -

(c) | |f x x2 3= -] g

(d) | | 2 y x= -

(e) f x x2 5= - +] g

(f) | | y x5= -

(g) 2 y x=

(h) y 5x= -

(i) f x xx 1

= +] g

(j)2

4 3 y x

x= -


(a) y x x 5 2= -] g


= +-

(d) g x x x164 2= -] g

(e) 49x y 2 2+ =

You may like to


by dividing by x




4 (a) Solve x1 02$-


f x x1 2= -] g


(a) 2 y x x2= - -

(b) g t t t 62= +] g



in each case


x3 3 -


x2 3 -


x2 1 - (d)


x1 5


0 4x


x3 3 -


x1 1 -


x2 3 -


x4 4 -

(j) y x x7 62= - + - in the

domain 0 7x


function y x 1

3=

+





x=] g


function

find its range(b)


plane

(a) f x x4=] g

(b) y x3= -

(c) y x 34= -

(d) 2 p x x3=] g

(e) 1 g x x3= +] g

(f) 100x y 2 2+ =

(g) 2 1 y x= +


y x 1= -









(b) f x x 22= -] g

(c)1

y x=

(d) f x x3=] g

(e) f x 3x=] g

13 (a) Solve x4 02$-


(i) 4 y x2= -

(ii) y x4

2= - -




DID YOU KNOW




a light



Limits






a a1

1

03

Z

=

=

3

3

-


lim a 0x=

x 3-



asymptote

EXAMPLES

1 Find lim x

x x5x 0

2+

Solution


which is undefined


( )

lim lim

lim

xx x

x

x x

x

5 5

5

5

x x

x

0

2

0 1

1

0

+=

+

= +

=

] g




2 Find limx

x

4

22

-

-

x 2

Solution


lim lim

lim

x

x

x x

x

x

4

2

2 2

2

21

41

2 1

1

-

-=

+ -

-

=+

=

x x2 2

x 2

^ _h i

3 Find limh

h x hx h2 72 2+ -

h 0

Solution

lim lim

lim

h

h x hx h

h

h hx x

hx x

x

2 7 2 7

2 7

7

2 2 2

2

2

+ -=

+ -

= + -

= -

h 0

h h0 0

^ h

Continuity





EXAMPLES

1 Find lim

x

x

1

12

-

-

x 1


11

y x

x2

=-

- Sketch the curve

Solution


xx2

-

- gives

00

CONTINUED




( )

lim lim

limx

x

x

x x

x11

1

1 1

1

2

x x

x

1

2

1

1

-

-=

-

+ -

= +

=

-

] ]g g

11

y xx2

=-





y xx

x

x x

x

11

1

1 1

1

2

=-

-

=

+

= +

-

-^ ^h h


2 Find limx

x x2

2x 2

2

+

+ -

-


x x2

22

= + -

+

Solution


x x2

22

+

+ - gives

00

lim lim

lim

xx x

x

x x

x

22

2

1 2

1

3

x x

x

2

2

2

2

+

+ -=

+

- +

=

= -

-

- -

-

^^ ^

^

hh h

h2 y

x

x xx

y x

x

x

x

22

2

2

1

1


=+

+ -= -

=+

= -

+ -^ ^h h


Remember that x 1




1 Find(a) lim x 52

+x 4

(b) lim t 7-t 3 -

(c) lim x x2 43+ -

x 2

(d) lim xx x32

+

x 0

(e) limh

h h

2

22

-

- -

h 2

(f) lim y

y

5

1253

-

-

y 5

(g) limx

x x

12 12

+

+ +

x 1-

(h) limx

x x4

2 82

+

+ -

x 4 -

(i) limc

c

4

22

-

-

c 2

(j) limx x

x 12

-

-

x 1

(k) lim h

h h h2 73 2+ -

h 0

(l) limh

hx hx h32 2- +

h 0

(m) limh

hx h x hx h2 3 53 2 2- + -

h 0

(n) lim x c x c 3 3

-

-

x c



discontinuous

(a) 3 y x2= -

(b)1

1 y

x=

+

(c) f x x 1= -] g

(d)4

1 y

x2=

+

(e)4

1

y x2=

-



(a)3

y xx x2

= +

(b)33

y x

x x2

=+

+

(c)1

5 4 y

xx x2

=+

+ +

510 Exercises




Regions

Class Investigation


them all







EXAMPLE


Solution



y

x =3

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4







region

EXAMPLE


Solution



included


shown

y

x

1

1

-2 2 3 4

-3

-4

-5

2

3

4

5

-1

-2

-3-4

-1 y = -1



y -intercept 1-






EXAMPLES


1 y x 2$ +

Solution






y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0



2 x y 2 3 61-

Solution


region


side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g








3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g






Solution

( ) ( )

( )

( )

g

g

g

1 2 1 1 1 1 2

1

2 5 2 1

3 3 3 2

9

since

since

since2

1

2

= - -

=

- = - -

=

=

( ) ( ) ( ) g g g 1 2 3 1 5 9

3

So + - - = + -

= -

DID YOU KNOW






f 3-] g

2 If h x x 22= -] g find h h0 2] ]g g

and h 4-] g


and f 2-] g


= - +] g

5 Find f 3-] g if f x x x2 5 43= - +] g


f x 13=] g




f x271

=] g




f x h+] g


g x x x2 32= + +] g


product of factors






15 f x x x

x x

1

1

for

for

32

=] g )


16 f x

x x

x x

x x

2 4 1

3 1 1

1

if

if

if 2

1 1

$

=

-

+ -

-

] gZ

[

]]

]]

Find the values of

f f f 2 2 1- - + -] ] ]g g g

52 Exercises








EXAMPLE


Solution


x x

x x

x x

x x

0 7 8

8 1

8 0 1 0

8 1

2= + -

= + -

+ = - =

= - =

] ]g g


y 0 7 0 8

8

2= + -

= -

] ]g g





Domain and range




domain


image) of f

EXAMPLE


Solution


x

y

CONTINUED






Domain all real x

Range y y 0$















EXAMPLES


x x 3 x 2 x 1

y

Solution


increasing


x 1 x 2

2

x x 3

x 2 x 1

y

Solution



increases again




1 1








=







x

y









x

y




EXAMPLES


Solution

f x x

x

f x

f x x

3

3


2

2

2`

- = - +

= +

=

= +

] ]]

]

g gg

g


Solution

f x x x

x x

x x

f x


3

3

3

3`

- = - - -

= - +

= - -

= -

= -

] ] ]^]

]

g g gh

gg

Investigation






1 f x kxn=] g

2 f x x kn= +] g

3 f x x k n= +

] ]g g






shape of graphs


each function

(a) y x3 2= -

(b) x y 2 5 20 0- + =

(c) x y 3 12 0+ - =

(d) f x x x32= +] g

(e) f x x 42= -] g

(f) p x x x5 62= + +] g

(g) y x x8 152= - +

(h) p x x 53

= +

] g

53 Exercises




(i) y xx

x3

0= + ] g

(j) g x x9 2= -] g


f x x 22= -

] g What type of

function is it


(a) f x2^ h (b) ( )f x 26



an even function


f x x=] g


function


odd function



f x f x- -] ]g g


neither

(a) y x x

x4 2

3

=

-

(b) y x 1

13

=

-

(c) f xx 4

32

=

-

] g

(d) y x

x

33

=+

-

(e) f x x x

x5 2

3

=-] g



f x xn=] g

even(a)

odd(b)


ever be

even(a)

odd(b)



graph is increasing




x

y(a)

x

4

y(b)

2-2

x

y(c)




Investigation


graph







y kx

y kx

y kx

y kx

y xk

(a)

(b)

(c)

(d)

(e)

2

3

4

=

=

=

=

=



y x k

y x k

y x k

y x k

y x k1

(a)

(b)

(c)

(d)

(e)

2

2

3

4

= +

= +

= +

= +

= +

] g



-2

1 2

-4

-1-2

2

4

y

x

(d) y

x

(e)

CONTINUED




Gradient form



Investigation










y x k

y x k

y x k

y x k y

x k

1

(a)

(b)

(c)

(d)(e)

2

3

4

= +

= +

= +

= +

=+

]]

]

gg

g





the right








Linear Function





EXAMPLE



Solution


Find the intercepts


0 3 5

5 3

1

x

x

x32

=

=

=

-


3 5

5

y 0=

= -

-] g

-1

-2

y

5

4

3

2

1 1 23

6

-3

-4

-5

1 4-1-2 32-3-4

x



Range all real y








Special lines






1 y kx=

2 y x k= +






EXAMPLES


Solution




-1

-3

y

4

3

2

1

5

-2

-4

-5

1 4-1-2

x

32-3-4


2 Sketch x 1= -


Solution





-

-

4

3

2

5

-2

-4

-5

1 4-1-2-4

y

x







Domain all real x

Range y y b=

54 Exercises


each function

(a) y x 2= -

(b) f x x2 3= +] g (c) x y 2 1 0+ =-

(d) x y 3 0+ =-

(e) x y 3 6 2 0=- -


line

(a) x 4=

(b) x 3 0=-

(c) y 5=

(d) y 1 0+ =

(e) f x x2 1= -] g (f) y x 4= +

(g) f x x3 2= +] g

(h) x y 3+ =

(i) x y 1 0=- -

(j) x y 2 3 0+ =-


(a) x y 3 2 7 0+ =-

(b) y 2=

(c) x 4= - (d) x 2 0=-

(e) y 3 0=-


are even or odd

(a) y x2=

(b) y 3=

(c) x 4=

(d) y x= -

(e) y x=




meet




Applications






The pronumeral

a is called the


Quadratic Function










EXAMPLES



Solution



x

x

x

0 1

1

1

2

2

= -

=

=


0 1

1

y 2= -

= -



Domain xall real

Range y y 1$ -


Solution


parabola will lie




left



0

1 0

1

x

x

x

1 2= +

+ =

= -

] g


1

y 0 1 2= +

=

] g

-1

-

4

3

2

1

5

-2

-4

-5

-

1 41-2 5-4

y

x









Solution



x x

x xx x

x x

0 6

3 23 0 2 0

3 2

2= + -

= + -

+ = - =

= - =

] ]g g


f 0 0 0 66

2= + -

= -

] ] ]g g g



x -intercepts


23 2

21- +

= -


-c m

f 21

21

21

6

41

21

6

641

2

- = - + - -

= - -

= -

c c cm m m


-

CONTINUED

You will learn more


-1

-3

4

3

2

1

5

-2

-4

-5

1 4-1-2 32-3-4

y

x










41



2

11 -





Solution


So f x 0=] g

0 4 54 5 0

0

x xx x

x x5 1

2

2

= - + +

=

+ =

- -

-] ]g g

x x

x x

5 0 1 0

5 1

- = + =

= = -


f 0 0 4 0 5

5

2= - + +

=

] ] ]g g g

-1

-3

4

2

5

-2

-4

-5

-

y

1 41-2 5-4

x

-2 4






21 5

2- +

=

f 2 2 4 2 59

= - + +

=

2] ] ]g g g










each function

(a) 2 y x x2= +

(b) 3 y x x2= - +

(c) f x x 12= -] g

(d) y x x 22= - -

(e) y x x9 82= +-

2 Sketch

(a) 2 y x2= +

(b) y x 12= - +

(c) f x x 42= -] g

(d) 2 y x x2= +

(e) y x x2= - -

(f) f x x 3= -2] ]g g

55 Exercises

-1

9

8

7

5

4

3

2

6

1

-2

-3

-4

-5

y

2 51 643-1-2-3-4

x




EXAMPLES


Solution









(g) f x x 1 2= +] ]g g

(h) y x x3 42= + -

(i) y x x2 5 32= - +

(j) f x x x3 22= - + -] g



(a) ndash y x x7 122= +

(b) f x x x42= +] g

(c) y x x2 82= - -

(d) y x x6 92= +-

(e) f t t 4 2= -] g


(a) y x 52= -

(b) f x x x6

2= -

] g (c) f x x x 22= - -] g

(d) y x2= -

(e) f x x 7 2= -] ]g g



(a) y x2= for x0 3

(b) y x 42= - + for x1 2 -

(c) f x x 12= -] g for x2 5 -

(d) y x x2 32= + - for x2 4 -

(e) y x x

22= - +- for

x0 4


function is

increasing(i)

decreasing(ii)

(a) y x2=

(b) y x2

= - (c) f x x 92

= -] g

(d) y x x42= - +

(e) f x x 5 2= +] ]g g


function



(a) y x 12= +

(b) f x x 32= -

] g

(c) y x2 2= -

(d) f x x x32= -] g

(e) f x x x2= +] g

(f) y x 42= -

(g) y x x2 32= - -

(h) y x x5 42= +-

(i) p x x 1 2= +] ]g g

(j) y x 2= -2] g




CONTINUED

eg When x 3= -

| | y 3 13 12

= - -

= -

=

x -3 -2 -1 0 1 2 3

y 2 1 0 -1 0 1 2


y

-2

4

3

2

1

5

-1

-3

-4

-5

1 4-1-2 32-3-4

x


| | y x

x x

x x1

1 0

1 0

when

when 1

$= - =

-


y x x1 0$= - ] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x - 1




y x 1= - - x 01] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y =- x - 1



-3

4

3

2

1

5

-2

-1

-4

-5

y y

3-1-2 421-3-4

x

y = - x - 1

y = x - 1


intercepts


So f x 0=

] g

| |

| |

x

x

x

0 1

1

1`

= -

=

=


( ) | |f 0 0 11

= -

= -





-3

4

3

2

1

5

-2

-1

-4

-5

y

4-2 5321-1-3-4

x



Domain all real x

Range y y 1$ -

2 Sketch | | y x 2= +

Solution


| | ( ) y x x xx x2 2 2 0

2 2 0whenwhen 1

$= + = + +

- + +


2 y x= + when x 2 0$+

x 2$ -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x + 2






y-intercept of 1-

CONTINUED




2 y x= - +] g when x 2 01+

ie y x 2= - - when x 21 -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2


-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2

y = x + 2



So 0f x =] g

0 | 2 |

0 2

2

x

x

x

= +

= +

- =


(0) | 0 2 |

2

f = +

=

There is only one

solution for the


Can you see why





-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

If you know how to

sketch the graph of




Investigation




are their equations





1 | |f x k x=] g

2 | |f x x k= +] g

3 | |f x x k= +] g







EXAMPLES

Solve

1 |2 1 | 3x - =

Solution



is x 1 2= -

2 |2 1 | 3 2x x= -+

Solution



3 | 1 | 2x 1+

Solution






and then check them







each function

(a) | | y x=

(b) | |f x x 7= +] g

(c) | |f x x 2= -] g

(d) 5 | | y x=

(e) | |f x x 3= - +] g

(f) | 6 | y x= +

(g) | |f x x3 2= -] g

(h) | 5 4 | y x= +

(i) | 7 1 | y x= -

(j) | |f x x2 9= +] g


plane

(a) | | y x=

(b) | |f x x 1= +] g

(c) | |f x x 3= -] g

(d) 2 | | y x=

(e) | |f x x= -] g

(f) | 1 | y x= +

(g) | |f x x 1= - -] g

(h) | 2 3 | y x= -

(i) | 4 2 | y x= +

(j) | |f x x3 1= +] g


each function

(a) | 1 | y x= -

(b) | |f x x 8= -] g

(c) | |f x x2 5= +] g

(d) 2 | | 3 y x= -

(e) | |f x x 3= - -] g


function is

increasing(i)

decreasing(ii)

(a) | 2 | y x= -

(b) | |f x x 2= +

] g

(c) | |f x x2 3= -] g

(d) 4 | | 1 y x= -

(e) | |f x x= -] g


of each function

(a) | | y x= for x2 2 -

(b) | |f x x 4= - -] g for

x4 3 -

(c) | |f x x 4= +] g for x7 2 -

(d) | 2 5 | y x= - for x3 3 -

(e) | |f x x= -] g for x1 1 -


function increasing

(a) | 3 | y x= +

(b) | |f x x 4= - +] g

(c) | |f x x 9= -] g

(d) | | y x 2 1= - -

(e) | |f x x 2= - +] g

56 Exercises




7 Solve graphically

(a) | | 3x =

(b) | |x 12

(c) | |x 2

(d) | 2 | 1x + =

(e) | 3 | 0x- =

(f) |2 3 | 1x - =

(g) | |x 1 41-

(h) | |x 1 3+

(i) | |x 2 22-

(j) | |x 3 1$-

(k) | |x2 3 5+

(l) | |x2 1 1$-

(m) |3 1 | 3x x- = +

(n) |3 2 | 4x x- = -

(o) | 1 | 1x x- = +

(p) | 3 | 2 2x x+ = + (q) |2 1 | 1x x+ = -

(r) |2 5 | 3x x- = -

(s) | 1 | 2x x- =

(t) |2 3 | 3x x- = +

The Hyperbola


or= =

EXAMPLE

Sketch1

y x=

Solution



x -3 -2 -121-

41- 0

41

21 1 2 3

y 3

1-

2

1- -1 -2 -4 mdash 4 2 1

2

1

3

1

Class Discussion



y is never 0 Why






but never touches)



Investigation




equations




1 y xk

=

21

y x k= +

31

y x k

=+


EXAMPLES


3=

-

] g


Solution



So x 3






be zero

CONTINUED







1

y 0 3

3=

-

= -

-3

4

3

2

1

5

-2

-1

-4

-5

y

-1-2 4 521-3-4

x

x = 3

y = 0

Asymptotes

3

2 Sketch y x2 4

1= -

+

Solution




x

x

2 4

2

-

-






( ) y

2 0 41

41

= -+

= -


a translation of3

y x

=





y

-2

x

-

1

4


a=

+


domain x xb



1 For each graph



exists


(a)2

y x=

(b)1

y x= -

(c) f xx 1

1=

+] g

(d) f xx 2

3=

-

] g

(e)3 6

1 y

x=

+

(f) f xx 3

2= -

-

] g

(g) f xx 1

4=

-

] g

(h)1

2 y

x= -

+

(i) f xx6 3

2=

-

] g

(j)2

6 y

x= -

+

2 Show that f x x2

=] g is an odd

function



(a) f xx2 5

1=

+

] g for x2 2 -

(b)3

1 y

x=

+ for x2 0 -

(c) f xx2 4

5=

-

] g for x3 1 -

57 Exercises




(d) f xx 4

3= -

-

] g for x3 3 -

(e)3 1

2 y

x= -

+ for x0 5



(a)3

y x= for y 1 3

(b)2

y x= - for y 221

- -

(c) f xx 1

1=

-

] g for y 171

- -

(d) f xx2 1

3= -

+

] g for

y 131

- -

(e)3 2

6 y

x=

- for y 1

21

6



Circle gate





Proof

r y

x

( x y)

y

x








c a b

r x y

2 2 2

2 2 2

`

= +

= +

EXAMPLE



Solution


y

x

-2

-2 2

2


one place

y

x

2

2

2

-2

The radius is 4

CONTINUED






Domain 2 2 x x -

Range 2 2 y y -


range y r y r -



Proof


shown

y

x

(a b)

x

y

r

( x y)

a

b x - a

y - b




ndash ndash

c a b

r x a y b

2 2 2

2 2 2

= +

= +] ^g h




EXAMPLES



Solution



y

x 9

9

-9

-9



Domain 9 9 x x -

Range 9 9 y y -



Solution


ndash

ndash ndash

x y

x y

1 2 4

1 2 2

2

2 2

+ + =

+ - =

2

2

] ^

] ]_

g h

g gi

So 1 2a b= = - and 2r =

CONTINUED







y

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

(1 -2)



Domain 1 3 x x -

Range 4 0 y y -


Solution


a b2 1= - = and r 3=

Substituting

ndash ndash

ndash

ndash

x a y b r

x y

x y

2 1 3

2 1 9

2 2 2

2 2 2

2 2

+ =

- - + =

+ + =

] ^]] ^

] ^

g hg g hg h


ndash

ndash

a b a ab b

x x x

x x

a b a ab b

y y y

y y

2

2 2 2 2

4 4

2

1 2 1 1

2 1

So

So

2 2 2

2 2 2

2

2 2 2

2 2 2

2

+ = + +

+ = + +

= + +

= - +

= - +

= - +

]] ] ]]^ ^ ]

gg g ggh h g


ndash

x x y y

x x y y

x x y y

x x y y

4 4 2 1 9

4 2 5 9

4 2 5 9

4 2 4 0

9 9

2

2

2

2

+ + + - + =

+ + - + =

+ + + =

+ + - - =

- -


in Chapter 2




Investigation


two functions





circle






for all values

y

x r

r

-r


Proof

ndash

x y r

y r x y r x

2 2 2

2 2 2

2 2

+ =

=

= -



semi-circles





y 0 for all values

y

x r

-

r

-r


EXAMPLES


1 f x x92

= -] g

Solution



y

x 3

3

-3

Domain 3 3 x x -

Range 0 3 y y




2 y x4 2= - -

Solution



y

x 2

-2

-2

Domain 2 2 x x -

Range 2 0 y y -




range

(a) 9x y 2 2+ =

(b) x y 16 02 2+ =-


(d) 1 9x y 2 2+ + =

] g

(e) ndashx y 2 1 12 2+ + =] ^g h



below the x -axis



range

(a) 25 y x2= - -

(b) 1 y x2= -

(c) 36 y x2= -

(d) 64 y x2= - -

(e) 7 y x2= - -



each circle

(a) 100x y 2 2+ =

(b) 5x y 2 2+ =


(d) ndashx y 5 6 492 2+ + =] ^g h

(e) ndashx y 3 812 2+ =^ h

58 Exercises






grouping symbols)



Centre(c) 1 5-







Centre(j) 4 7- -

^ h and radius 3

Other Graphs




EXAMPLES


Solution



eg When 0x =

y 3

1

c=

=

x

y

1

3

3

1

31

When1

1

= -

=

=

=

-

x 3- 2- 1- 0 1 2 3

y 271

91

31

1 3 9 27



You will meet these

graphs again in the

HSC Course





other point

When

x

y

1

3

3

1

=

=

=

y

x

1

2

1

3







Solution



x minus2 minus1 0 05 1 2 3 4

y minus03 0 03 05 06

y

x

1

2

1 2 3 4

-1




You learned about


stages of maths





range y y 02



range all real y

Cubic function


EXAMPLE



Solution



y minus25 minus6 1 2 3 10 29

y

x

1

1

-2 2 3 4

-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4



If you already

know the shape of


3






Domain and range


EXAMPLE


Solution




function are

f

f

1 1 2

1 21

4 4 2

64 2

66

3

3

- = - +

= - +

=

= +

= +

=

] ]

] ]

g g

g g


these points

y

x

(-1 1)

(4 66)

Range 1 66 y y










positive number

EXAMPLE


Solution


x 4So $

$


(or zero) So y 0$


59 Exercises


(a) 4 3 y x= +

(b) f x 4= -] g

(c) 3x =



(f) f x xx 12 2= - -] g

(g) 64x y 2 2+ =

(h) f t t 4

3=

-] g

(i) ( ) g 2

5zz

= +

(j) | |f x x=] g


(a) y x=

(b) 2 y x= -

(c) | |f x x2 3= -] g

(d) | | 2 y x= -

(e) f x x2 5= - +] g

(f) | | y x5= -

(g) 2 y x=

(h) y 5x= -

(i) f x xx 1

= +] g

(j)2

4 3 y x

x= -


(a) y x x 5 2= -] g


= +-

(d) g x x x164 2= -] g

(e) 49x y 2 2+ =

You may like to


by dividing by x




4 (a) Solve x1 02$-


f x x1 2= -] g


(a) 2 y x x2= - -

(b) g t t t 62= +] g



in each case


x3 3 -


x2 3 -


x2 1 - (d)


x1 5


0 4x


x3 3 -


x1 1 -


x2 3 -


x4 4 -

(j) y x x7 62= - + - in the

domain 0 7x


function y x 1

3=

+





x=] g


function

find its range(b)


plane

(a) f x x4=] g

(b) y x3= -

(c) y x 34= -

(d) 2 p x x3=] g

(e) 1 g x x3= +] g

(f) 100x y 2 2+ =

(g) 2 1 y x= +


y x 1= -









(b) f x x 22= -] g

(c)1

y x=

(d) f x x3=] g

(e) f x 3x=] g

13 (a) Solve x4 02$-


(i) 4 y x2= -

(ii) y x4

2= - -




DID YOU KNOW




a light



Limits






a a1

1

03

Z

=

=

3

3

-


lim a 0x=

x 3-



asymptote

EXAMPLES

1 Find lim x

x x5x 0

2+

Solution


which is undefined


( )

lim lim

lim

xx x

x

x x

x

5 5

5

5

x x

x

0

2

0 1

1

0

+=

+

= +

=

] g




2 Find limx

x

4

22

-

-

x 2

Solution


lim lim

lim

x

x

x x

x

x

4

2

2 2

2

21

41

2 1

1

-

-=

+ -

-

=+

=

x x2 2

x 2

^ _h i

3 Find limh

h x hx h2 72 2+ -

h 0

Solution

lim lim

lim

h

h x hx h

h

h hx x

hx x

x

2 7 2 7

2 7

7

2 2 2

2

2

+ -=

+ -

= + -

= -

h 0

h h0 0

^ h

Continuity





EXAMPLES

1 Find lim

x

x

1

12

-

-

x 1


11

y x

x2

=-

- Sketch the curve

Solution


xx2

-

- gives

00

CONTINUED




( )

lim lim

limx

x

x

x x

x11

1

1 1

1

2

x x

x

1

2

1

1

-

-=

-

+ -

= +

=

-

] ]g g

11

y xx2

=-





y xx

x

x x

x

11

1

1 1

1

2

=-

-

=

+

= +

-

-^ ^h h


2 Find limx

x x2

2x 2

2

+

+ -

-


x x2

22

= + -

+

Solution


x x2

22

+

+ - gives

00

lim lim

lim

xx x

x

x x

x

22

2

1 2

1

3

x x

x

2

2

2

2

+

+ -=

+

- +

=

= -

-

- -

-

^^ ^

^

hh h

h2 y

x

x xx

y x

x

x

x

22

2

2

1

1


=+

+ -= -

=+

= -

+ -^ ^h h


Remember that x 1




1 Find(a) lim x 52

+x 4

(b) lim t 7-t 3 -

(c) lim x x2 43+ -

x 2

(d) lim xx x32

+

x 0

(e) limh

h h

2

22

-

- -

h 2

(f) lim y

y

5

1253

-

-

y 5

(g) limx

x x

12 12

+

+ +

x 1-

(h) limx

x x4

2 82

+

+ -

x 4 -

(i) limc

c

4

22

-

-

c 2

(j) limx x

x 12

-

-

x 1

(k) lim h

h h h2 73 2+ -

h 0

(l) limh

hx hx h32 2- +

h 0

(m) limh

hx h x hx h2 3 53 2 2- + -

h 0

(n) lim x c x c 3 3

-

-

x c



discontinuous

(a) 3 y x2= -

(b)1

1 y

x=

+

(c) f x x 1= -] g

(d)4

1 y

x2=

+

(e)4

1

y x2=

-



(a)3

y xx x2

= +

(b)33

y x

x x2

=+

+

(c)1

5 4 y

xx x2

=+

+ +

510 Exercises




Regions

Class Investigation


them all







EXAMPLE


Solution



y

x =3

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4







region

EXAMPLE


Solution



included


shown

y

x

1

1

-2 2 3 4

-3

-4

-5

2

3

4

5

-1

-2

-3-4

-1 y = -1



y -intercept 1-






EXAMPLES


1 y x 2$ +

Solution






y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0



2 x y 2 3 61-

Solution


region


side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g








3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g








EXAMPLE


Solution


x x

x x

x x

x x

0 7 8

8 1

8 0 1 0

8 1

2= + -

= + -

+ = - =

= - =

] ]g g


y 0 7 0 8

8

2= + -

= -

] ]g g





Domain and range




domain


image) of f

EXAMPLE


Solution


x

y

CONTINUED






Domain all real x

Range y y 0$















EXAMPLES


x x 3 x 2 x 1

y

Solution


increasing


x 1 x 2

2

x x 3

x 2 x 1

y

Solution



increases again




1 1








=







x

y









x

y




EXAMPLES


Solution

f x x

x

f x

f x x

3

3


2

2

2`

- = - +

= +

=

= +

] ]]

]

g gg

g


Solution

f x x x

x x

x x

f x


3

3

3

3`

- = - - -

= - +

= - -

= -

= -

] ] ]^]

]

g g gh

gg

Investigation






1 f x kxn=] g

2 f x x kn= +] g

3 f x x k n= +

] ]g g






shape of graphs


each function

(a) y x3 2= -

(b) x y 2 5 20 0- + =

(c) x y 3 12 0+ - =

(d) f x x x32= +] g

(e) f x x 42= -] g

(f) p x x x5 62= + +] g

(g) y x x8 152= - +

(h) p x x 53

= +

] g

53 Exercises




(i) y xx

x3

0= + ] g

(j) g x x9 2= -] g


f x x 22= -

] g What type of

function is it


(a) f x2^ h (b) ( )f x 26



an even function


f x x=] g


function


odd function



f x f x- -] ]g g


neither

(a) y x x

x4 2

3

=

-

(b) y x 1

13

=

-

(c) f xx 4

32

=

-

] g

(d) y x

x

33

=+

-

(e) f x x x

x5 2

3

=-] g



f x xn=] g

even(a)

odd(b)


ever be

even(a)

odd(b)



graph is increasing




x

y(a)

x

4

y(b)

2-2

x

y(c)




Investigation


graph







y kx

y kx

y kx

y kx

y xk

(a)

(b)

(c)

(d)

(e)

2

3

4

=

=

=

=

=



y x k

y x k

y x k

y x k

y x k1

(a)

(b)

(c)

(d)

(e)

2

2

3

4

= +

= +

= +

= +

= +

] g



-2

1 2

-4

-1-2

2

4

y

x

(d) y

x

(e)

CONTINUED




Gradient form



Investigation










y x k

y x k

y x k

y x k y

x k

1

(a)

(b)

(c)

(d)(e)

2

3

4

= +

= +

= +

= +

=+

]]

]

gg

g





the right








Linear Function





EXAMPLE



Solution


Find the intercepts


0 3 5

5 3

1

x

x

x32

=

=

=

-


3 5

5

y 0=

= -

-] g

-1

-2

y

5

4

3

2

1 1 23

6

-3

-4

-5

1 4-1-2 32-3-4

x



Range all real y








Special lines






1 y kx=

2 y x k= +






EXAMPLES


Solution




-1

-3

y

4

3

2

1

5

-2

-4

-5

1 4-1-2

x

32-3-4


2 Sketch x 1= -


Solution





-

-

4

3

2

5

-2

-4

-5

1 4-1-2-4

y

x







Domain all real x

Range y y b=

54 Exercises


each function

(a) y x 2= -

(b) f x x2 3= +] g (c) x y 2 1 0+ =-

(d) x y 3 0+ =-

(e) x y 3 6 2 0=- -


line

(a) x 4=

(b) x 3 0=-

(c) y 5=

(d) y 1 0+ =

(e) f x x2 1= -] g (f) y x 4= +

(g) f x x3 2= +] g

(h) x y 3+ =

(i) x y 1 0=- -

(j) x y 2 3 0+ =-


(a) x y 3 2 7 0+ =-

(b) y 2=

(c) x 4= - (d) x 2 0=-

(e) y 3 0=-


are even or odd

(a) y x2=

(b) y 3=

(c) x 4=

(d) y x= -

(e) y x=




meet




Applications






The pronumeral

a is called the


Quadratic Function










EXAMPLES



Solution



x

x

x

0 1

1

1

2

2

= -

=

=


0 1

1

y 2= -

= -



Domain xall real

Range y y 1$ -


Solution


parabola will lie




left



0

1 0

1

x

x

x

1 2= +

+ =

= -

] g


1

y 0 1 2= +

=

] g

-1

-

4

3

2

1

5

-2

-4

-5

-

1 41-2 5-4

y

x









Solution



x x

x xx x

x x

0 6

3 23 0 2 0

3 2

2= + -

= + -

+ = - =

= - =

] ]g g


f 0 0 0 66

2= + -

= -

] ] ]g g g



x -intercepts


23 2

21- +

= -


-c m

f 21

21

21

6

41

21

6

641

2

- = - + - -

= - -

= -

c c cm m m


-

CONTINUED

You will learn more


-1

-3

4

3

2

1

5

-2

-4

-5

1 4-1-2 32-3-4

y

x










41



2

11 -





Solution


So f x 0=] g

0 4 54 5 0

0

x xx x

x x5 1

2

2

= - + +

=

+ =

- -

-] ]g g

x x

x x

5 0 1 0

5 1

- = + =

= = -


f 0 0 4 0 5

5

2= - + +

=

] ] ]g g g

-1

-3

4

2

5

-2

-4

-5

-

y

1 41-2 5-4

x

-2 4






21 5

2- +

=

f 2 2 4 2 59

= - + +

=

2] ] ]g g g










each function

(a) 2 y x x2= +

(b) 3 y x x2= - +

(c) f x x 12= -] g

(d) y x x 22= - -

(e) y x x9 82= +-

2 Sketch

(a) 2 y x2= +

(b) y x 12= - +

(c) f x x 42= -] g

(d) 2 y x x2= +

(e) y x x2= - -

(f) f x x 3= -2] ]g g

55 Exercises

-1

9

8

7

5

4

3

2

6

1

-2

-3

-4

-5

y

2 51 643-1-2-3-4

x




EXAMPLES


Solution









(g) f x x 1 2= +] ]g g

(h) y x x3 42= + -

(i) y x x2 5 32= - +

(j) f x x x3 22= - + -] g



(a) ndash y x x7 122= +

(b) f x x x42= +] g

(c) y x x2 82= - -

(d) y x x6 92= +-

(e) f t t 4 2= -] g


(a) y x 52= -

(b) f x x x6

2= -

] g (c) f x x x 22= - -] g

(d) y x2= -

(e) f x x 7 2= -] ]g g



(a) y x2= for x0 3

(b) y x 42= - + for x1 2 -

(c) f x x 12= -] g for x2 5 -

(d) y x x2 32= + - for x2 4 -

(e) y x x

22= - +- for

x0 4


function is

increasing(i)

decreasing(ii)

(a) y x2=

(b) y x2

= - (c) f x x 92

= -] g

(d) y x x42= - +

(e) f x x 5 2= +] ]g g


function



(a) y x 12= +

(b) f x x 32= -

] g

(c) y x2 2= -

(d) f x x x32= -] g

(e) f x x x2= +] g

(f) y x 42= -

(g) y x x2 32= - -

(h) y x x5 42= +-

(i) p x x 1 2= +] ]g g

(j) y x 2= -2] g




CONTINUED

eg When x 3= -

| | y 3 13 12

= - -

= -

=

x -3 -2 -1 0 1 2 3

y 2 1 0 -1 0 1 2


y

-2

4

3

2

1

5

-1

-3

-4

-5

1 4-1-2 32-3-4

x


| | y x

x x

x x1

1 0

1 0

when

when 1

$= - =

-


y x x1 0$= - ] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x - 1




y x 1= - - x 01] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y =- x - 1



-3

4

3

2

1

5

-2

-1

-4

-5

y y

3-1-2 421-3-4

x

y = - x - 1

y = x - 1


intercepts


So f x 0=

] g

| |

| |

x

x

x

0 1

1

1`

= -

=

=


( ) | |f 0 0 11

= -

= -





-3

4

3

2

1

5

-2

-1

-4

-5

y

4-2 5321-1-3-4

x



Domain all real x

Range y y 1$ -

2 Sketch | | y x 2= +

Solution


| | ( ) y x x xx x2 2 2 0

2 2 0whenwhen 1

$= + = + +

- + +


2 y x= + when x 2 0$+

x 2$ -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x + 2






y-intercept of 1-

CONTINUED




2 y x= - +] g when x 2 01+

ie y x 2= - - when x 21 -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2


-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2

y = x + 2



So 0f x =] g

0 | 2 |

0 2

2

x

x

x

= +

= +

- =


(0) | 0 2 |

2

f = +

=

There is only one

solution for the


Can you see why





-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

If you know how to

sketch the graph of




Investigation




are their equations





1 | |f x k x=] g

2 | |f x x k= +] g

3 | |f x x k= +] g







EXAMPLES

Solve

1 |2 1 | 3x - =

Solution



is x 1 2= -

2 |2 1 | 3 2x x= -+

Solution



3 | 1 | 2x 1+

Solution






and then check them







each function

(a) | | y x=

(b) | |f x x 7= +] g

(c) | |f x x 2= -] g

(d) 5 | | y x=

(e) | |f x x 3= - +] g

(f) | 6 | y x= +

(g) | |f x x3 2= -] g

(h) | 5 4 | y x= +

(i) | 7 1 | y x= -

(j) | |f x x2 9= +] g


plane

(a) | | y x=

(b) | |f x x 1= +] g

(c) | |f x x 3= -] g

(d) 2 | | y x=

(e) | |f x x= -] g

(f) | 1 | y x= +

(g) | |f x x 1= - -] g

(h) | 2 3 | y x= -

(i) | 4 2 | y x= +

(j) | |f x x3 1= +] g


each function

(a) | 1 | y x= -

(b) | |f x x 8= -] g

(c) | |f x x2 5= +] g

(d) 2 | | 3 y x= -

(e) | |f x x 3= - -] g


function is

increasing(i)

decreasing(ii)

(a) | 2 | y x= -

(b) | |f x x 2= +

] g

(c) | |f x x2 3= -] g

(d) 4 | | 1 y x= -

(e) | |f x x= -] g


of each function

(a) | | y x= for x2 2 -

(b) | |f x x 4= - -] g for

x4 3 -

(c) | |f x x 4= +] g for x7 2 -

(d) | 2 5 | y x= - for x3 3 -

(e) | |f x x= -] g for x1 1 -


function increasing

(a) | 3 | y x= +

(b) | |f x x 4= - +] g

(c) | |f x x 9= -] g

(d) | | y x 2 1= - -

(e) | |f x x 2= - +] g

56 Exercises




7 Solve graphically

(a) | | 3x =

(b) | |x 12

(c) | |x 2

(d) | 2 | 1x + =

(e) | 3 | 0x- =

(f) |2 3 | 1x - =

(g) | |x 1 41-

(h) | |x 1 3+

(i) | |x 2 22-

(j) | |x 3 1$-

(k) | |x2 3 5+

(l) | |x2 1 1$-

(m) |3 1 | 3x x- = +

(n) |3 2 | 4x x- = -

(o) | 1 | 1x x- = +

(p) | 3 | 2 2x x+ = + (q) |2 1 | 1x x+ = -

(r) |2 5 | 3x x- = -

(s) | 1 | 2x x- =

(t) |2 3 | 3x x- = +

The Hyperbola


or= =

EXAMPLE

Sketch1

y x=

Solution



x -3 -2 -121-

41- 0

41

21 1 2 3

y 3

1-

2

1- -1 -2 -4 mdash 4 2 1

2

1

3

1

Class Discussion



y is never 0 Why






but never touches)



Investigation




equations




1 y xk

=

21

y x k= +

31

y x k

=+


EXAMPLES


3=

-

] g


Solution



So x 3






be zero

CONTINUED







1

y 0 3

3=

-

= -

-3

4

3

2

1

5

-2

-1

-4

-5

y

-1-2 4 521-3-4

x

x = 3

y = 0

Asymptotes

3

2 Sketch y x2 4

1= -

+

Solution




x

x

2 4

2

-

-






( ) y

2 0 41

41

= -+

= -


a translation of3

y x

=





y

-2

x

-

1

4


a=

+


domain x xb



1 For each graph



exists


(a)2

y x=

(b)1

y x= -

(c) f xx 1

1=

+] g

(d) f xx 2

3=

-

] g

(e)3 6

1 y

x=

+

(f) f xx 3

2= -

-

] g

(g) f xx 1

4=

-

] g

(h)1

2 y

x= -

+

(i) f xx6 3

2=

-

] g

(j)2

6 y

x= -

+

2 Show that f x x2

=] g is an odd

function



(a) f xx2 5

1=

+

] g for x2 2 -

(b)3

1 y

x=

+ for x2 0 -

(c) f xx2 4

5=

-

] g for x3 1 -

57 Exercises




(d) f xx 4

3= -

-

] g for x3 3 -

(e)3 1

2 y

x= -

+ for x0 5



(a)3

y x= for y 1 3

(b)2

y x= - for y 221

- -

(c) f xx 1

1=

-

] g for y 171

- -

(d) f xx2 1

3= -

+

] g for

y 131

- -

(e)3 2

6 y

x=

- for y 1

21

6



Circle gate





Proof

r y

x

( x y)

y

x








c a b

r x y

2 2 2

2 2 2

`

= +

= +

EXAMPLE



Solution


y

x

-2

-2 2

2


one place

y

x

2

2

2

-2

The radius is 4

CONTINUED






Domain 2 2 x x -

Range 2 2 y y -


range y r y r -



Proof


shown

y

x

(a b)

x

y

r

( x y)

a

b x - a

y - b




ndash ndash

c a b

r x a y b

2 2 2

2 2 2

= +

= +] ^g h




EXAMPLES



Solution



y

x 9

9

-9

-9



Domain 9 9 x x -

Range 9 9 y y -



Solution


ndash

ndash ndash

x y

x y

1 2 4

1 2 2

2

2 2

+ + =

+ - =

2

2

] ^

] ]_

g h

g gi

So 1 2a b= = - and 2r =

CONTINUED







y

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

(1 -2)



Domain 1 3 x x -

Range 4 0 y y -


Solution


a b2 1= - = and r 3=

Substituting

ndash ndash

ndash

ndash

x a y b r

x y

x y

2 1 3

2 1 9

2 2 2

2 2 2

2 2

+ =

- - + =

+ + =

] ^]] ^

] ^

g hg g hg h


ndash

ndash

a b a ab b

x x x

x x

a b a ab b

y y y

y y

2

2 2 2 2

4 4

2

1 2 1 1

2 1

So

So

2 2 2

2 2 2

2

2 2 2

2 2 2

2

+ = + +

+ = + +

= + +

= - +

= - +

= - +

]] ] ]]^ ^ ]

gg g ggh h g


ndash

x x y y

x x y y

x x y y

x x y y

4 4 2 1 9

4 2 5 9

4 2 5 9

4 2 4 0

9 9

2

2

2

2

+ + + - + =

+ + - + =

+ + + =

+ + - - =

- -


in Chapter 2




Investigation


two functions





circle






for all values

y

x r

r

-r


Proof

ndash

x y r

y r x y r x

2 2 2

2 2 2

2 2

+ =

=

= -



semi-circles





y 0 for all values

y

x r

-

r

-r


EXAMPLES


1 f x x92

= -] g

Solution



y

x 3

3

-3

Domain 3 3 x x -

Range 0 3 y y




2 y x4 2= - -

Solution



y

x 2

-2

-2

Domain 2 2 x x -

Range 2 0 y y -




range

(a) 9x y 2 2+ =

(b) x y 16 02 2+ =-


(d) 1 9x y 2 2+ + =

] g

(e) ndashx y 2 1 12 2+ + =] ^g h



below the x -axis



range

(a) 25 y x2= - -

(b) 1 y x2= -

(c) 36 y x2= -

(d) 64 y x2= - -

(e) 7 y x2= - -



each circle

(a) 100x y 2 2+ =

(b) 5x y 2 2+ =


(d) ndashx y 5 6 492 2+ + =] ^g h

(e) ndashx y 3 812 2+ =^ h

58 Exercises






grouping symbols)



Centre(c) 1 5-







Centre(j) 4 7- -

^ h and radius 3

Other Graphs




EXAMPLES


Solution



eg When 0x =

y 3

1

c=

=

x

y

1

3

3

1

31

When1

1

= -

=

=

=

-

x 3- 2- 1- 0 1 2 3

y 271

91

31

1 3 9 27



You will meet these

graphs again in the

HSC Course





other point

When

x

y

1

3

3

1

=

=

=

y

x

1

2

1

3







Solution



x minus2 minus1 0 05 1 2 3 4

y minus03 0 03 05 06

y

x

1

2

1 2 3 4

-1




You learned about


stages of maths





range y y 02



range all real y

Cubic function


EXAMPLE



Solution



y minus25 minus6 1 2 3 10 29

y

x

1

1

-2 2 3 4

-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4



If you already

know the shape of


3






Domain and range


EXAMPLE


Solution




function are

f

f

1 1 2

1 21

4 4 2

64 2

66

3

3

- = - +

= - +

=

= +

= +

=

] ]

] ]

g g

g g


these points

y

x

(-1 1)

(4 66)

Range 1 66 y y










positive number

EXAMPLE


Solution


x 4So $

$


(or zero) So y 0$


59 Exercises


(a) 4 3 y x= +

(b) f x 4= -] g

(c) 3x =



(f) f x xx 12 2= - -] g

(g) 64x y 2 2+ =

(h) f t t 4

3=

-] g

(i) ( ) g 2

5zz

= +

(j) | |f x x=] g


(a) y x=

(b) 2 y x= -

(c) | |f x x2 3= -] g

(d) | | 2 y x= -

(e) f x x2 5= - +] g

(f) | | y x5= -

(g) 2 y x=

(h) y 5x= -

(i) f x xx 1

= +] g

(j)2

4 3 y x

x= -


(a) y x x 5 2= -] g


= +-

(d) g x x x164 2= -] g

(e) 49x y 2 2+ =

You may like to


by dividing by x




4 (a) Solve x1 02$-


f x x1 2= -] g


(a) 2 y x x2= - -

(b) g t t t 62= +] g



in each case


x3 3 -


x2 3 -


x2 1 - (d)


x1 5


0 4x


x3 3 -


x1 1 -


x2 3 -


x4 4 -

(j) y x x7 62= - + - in the

domain 0 7x


function y x 1

3=

+





x=] g


function

find its range(b)


plane

(a) f x x4=] g

(b) y x3= -

(c) y x 34= -

(d) 2 p x x3=] g

(e) 1 g x x3= +] g

(f) 100x y 2 2+ =

(g) 2 1 y x= +


y x 1= -









(b) f x x 22= -] g

(c)1

y x=

(d) f x x3=] g

(e) f x 3x=] g

13 (a) Solve x4 02$-


(i) 4 y x2= -

(ii) y x4

2= - -




DID YOU KNOW




a light



Limits






a a1

1

03

Z

=

=

3

3

-


lim a 0x=

x 3-



asymptote

EXAMPLES

1 Find lim x

x x5x 0

2+

Solution


which is undefined


( )

lim lim

lim

xx x

x

x x

x

5 5

5

5

x x

x

0

2

0 1

1

0

+=

+

= +

=

] g




2 Find limx

x

4

22

-

-

x 2

Solution


lim lim

lim

x

x

x x

x

x

4

2

2 2

2

21

41

2 1

1

-

-=

+ -

-

=+

=

x x2 2

x 2

^ _h i

3 Find limh

h x hx h2 72 2+ -

h 0

Solution

lim lim

lim

h

h x hx h

h

h hx x

hx x

x

2 7 2 7

2 7

7

2 2 2

2

2

+ -=

+ -

= + -

= -

h 0

h h0 0

^ h

Continuity





EXAMPLES

1 Find lim

x

x

1

12

-

-

x 1


11

y x

x2

=-

- Sketch the curve

Solution


xx2

-

- gives

00

CONTINUED




( )

lim lim

limx

x

x

x x

x11

1

1 1

1

2

x x

x

1

2

1

1

-

-=

-

+ -

= +

=

-

] ]g g

11

y xx2

=-





y xx

x

x x

x

11

1

1 1

1

2

=-

-

=

+

= +

-

-^ ^h h


2 Find limx

x x2

2x 2

2

+

+ -

-


x x2

22

= + -

+

Solution


x x2

22

+

+ - gives

00

lim lim

lim

xx x

x

x x

x

22

2

1 2

1

3

x x

x

2

2

2

2

+

+ -=

+

- +

=

= -

-

- -

-

^^ ^

^

hh h

h2 y

x

x xx

y x

x

x

x

22

2

2

1

1


=+

+ -= -

=+

= -

+ -^ ^h h


Remember that x 1




1 Find(a) lim x 52

+x 4

(b) lim t 7-t 3 -

(c) lim x x2 43+ -

x 2

(d) lim xx x32

+

x 0

(e) limh

h h

2

22

-

- -

h 2

(f) lim y

y

5

1253

-

-

y 5

(g) limx

x x

12 12

+

+ +

x 1-

(h) limx

x x4

2 82

+

+ -

x 4 -

(i) limc

c

4

22

-

-

c 2

(j) limx x

x 12

-

-

x 1

(k) lim h

h h h2 73 2+ -

h 0

(l) limh

hx hx h32 2- +

h 0

(m) limh

hx h x hx h2 3 53 2 2- + -

h 0

(n) lim x c x c 3 3

-

-

x c



discontinuous

(a) 3 y x2= -

(b)1

1 y

x=

+

(c) f x x 1= -] g

(d)4

1 y

x2=

+

(e)4

1

y x2=

-



(a)3

y xx x2

= +

(b)33

y x

x x2

=+

+

(c)1

5 4 y

xx x2

=+

+ +

510 Exercises




Regions

Class Investigation


them all







EXAMPLE


Solution



y

x =3

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4







region

EXAMPLE


Solution



included


shown

y

x

1

1

-2 2 3 4

-3

-4

-5

2

3

4

5

-1

-2

-3-4

-1 y = -1



y -intercept 1-






EXAMPLES


1 y x 2$ +

Solution






y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0



2 x y 2 3 61-

Solution


region


side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g








3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g






EXAMPLE


Solution


x x

x x

x x

x x

0 7 8

8 1

8 0 1 0

8 1

2= + -

= + -

+ = - =

= - =

] ]g g


y 0 7 0 8

8

2= + -

= -

] ]g g





Domain and range




domain


image) of f

EXAMPLE


Solution


x

y

CONTINUED






Domain all real x

Range y y 0$















EXAMPLES


x x 3 x 2 x 1

y

Solution


increasing


x 1 x 2

2

x x 3

x 2 x 1

y

Solution



increases again




1 1








=







x

y









x

y




EXAMPLES


Solution

f x x

x

f x

f x x

3

3


2

2

2`

- = - +

= +

=

= +

] ]]

]

g gg

g


Solution

f x x x

x x

x x

f x


3

3

3

3`

- = - - -

= - +

= - -

= -

= -

] ] ]^]

]

g g gh

gg

Investigation






1 f x kxn=] g

2 f x x kn= +] g

3 f x x k n= +

] ]g g






shape of graphs


each function

(a) y x3 2= -

(b) x y 2 5 20 0- + =

(c) x y 3 12 0+ - =

(d) f x x x32= +] g

(e) f x x 42= -] g

(f) p x x x5 62= + +] g

(g) y x x8 152= - +

(h) p x x 53

= +

] g

53 Exercises




(i) y xx

x3

0= + ] g

(j) g x x9 2= -] g


f x x 22= -

] g What type of

function is it


(a) f x2^ h (b) ( )f x 26



an even function


f x x=] g


function


odd function



f x f x- -] ]g g


neither

(a) y x x

x4 2

3

=

-

(b) y x 1

13

=

-

(c) f xx 4

32

=

-

] g

(d) y x

x

33

=+

-

(e) f x x x

x5 2

3

=-] g



f x xn=] g

even(a)

odd(b)


ever be

even(a)

odd(b)



graph is increasing




x

y(a)

x

4

y(b)

2-2

x

y(c)




Investigation


graph







y kx

y kx

y kx

y kx

y xk

(a)

(b)

(c)

(d)

(e)

2

3

4

=

=

=

=

=



y x k

y x k

y x k

y x k

y x k1

(a)

(b)

(c)

(d)

(e)

2

2

3

4

= +

= +

= +

= +

= +

] g



-2

1 2

-4

-1-2

2

4

y

x

(d) y

x

(e)

CONTINUED




Gradient form



Investigation










y x k

y x k

y x k

y x k y

x k

1

(a)

(b)

(c)

(d)(e)

2

3

4

= +

= +

= +

= +

=+

]]

]

gg

g





the right








Linear Function





EXAMPLE



Solution


Find the intercepts


0 3 5

5 3

1

x

x

x32

=

=

=

-


3 5

5

y 0=

= -

-] g

-1

-2

y

5

4

3

2

1 1 23

6

-3

-4

-5

1 4-1-2 32-3-4

x



Range all real y








Special lines






1 y kx=

2 y x k= +






EXAMPLES


Solution




-1

-3

y

4

3

2

1

5

-2

-4

-5

1 4-1-2

x

32-3-4


2 Sketch x 1= -


Solution





-

-

4

3

2

5

-2

-4

-5

1 4-1-2-4

y

x







Domain all real x

Range y y b=

54 Exercises


each function

(a) y x 2= -

(b) f x x2 3= +] g (c) x y 2 1 0+ =-

(d) x y 3 0+ =-

(e) x y 3 6 2 0=- -


line

(a) x 4=

(b) x 3 0=-

(c) y 5=

(d) y 1 0+ =

(e) f x x2 1= -] g (f) y x 4= +

(g) f x x3 2= +] g

(h) x y 3+ =

(i) x y 1 0=- -

(j) x y 2 3 0+ =-


(a) x y 3 2 7 0+ =-

(b) y 2=

(c) x 4= - (d) x 2 0=-

(e) y 3 0=-


are even or odd

(a) y x2=

(b) y 3=

(c) x 4=

(d) y x= -

(e) y x=




meet




Applications






The pronumeral

a is called the


Quadratic Function










EXAMPLES



Solution



x

x

x

0 1

1

1

2

2

= -

=

=


0 1

1

y 2= -

= -



Domain xall real

Range y y 1$ -


Solution


parabola will lie




left



0

1 0

1

x

x

x

1 2= +

+ =

= -

] g


1

y 0 1 2= +

=

] g

-1

-

4

3

2

1

5

-2

-4

-5

-

1 41-2 5-4

y

x









Solution



x x

x xx x

x x

0 6

3 23 0 2 0

3 2

2= + -

= + -

+ = - =

= - =

] ]g g


f 0 0 0 66

2= + -

= -

] ] ]g g g



x -intercepts


23 2

21- +

= -


-c m

f 21

21

21

6

41

21

6

641

2

- = - + - -

= - -

= -

c c cm m m


-

CONTINUED

You will learn more


-1

-3

4

3

2

1

5

-2

-4

-5

1 4-1-2 32-3-4

y

x










41



2

11 -





Solution


So f x 0=] g

0 4 54 5 0

0

x xx x

x x5 1

2

2

= - + +

=

+ =

- -

-] ]g g

x x

x x

5 0 1 0

5 1

- = + =

= = -


f 0 0 4 0 5

5

2= - + +

=

] ] ]g g g

-1

-3

4

2

5

-2

-4

-5

-

y

1 41-2 5-4

x

-2 4






21 5

2- +

=

f 2 2 4 2 59

= - + +

=

2] ] ]g g g










each function

(a) 2 y x x2= +

(b) 3 y x x2= - +

(c) f x x 12= -] g

(d) y x x 22= - -

(e) y x x9 82= +-

2 Sketch

(a) 2 y x2= +

(b) y x 12= - +

(c) f x x 42= -] g

(d) 2 y x x2= +

(e) y x x2= - -

(f) f x x 3= -2] ]g g

55 Exercises

-1

9

8

7

5

4

3

2

6

1

-2

-3

-4

-5

y

2 51 643-1-2-3-4

x




EXAMPLES


Solution









(g) f x x 1 2= +] ]g g

(h) y x x3 42= + -

(i) y x x2 5 32= - +

(j) f x x x3 22= - + -] g



(a) ndash y x x7 122= +

(b) f x x x42= +] g

(c) y x x2 82= - -

(d) y x x6 92= +-

(e) f t t 4 2= -] g


(a) y x 52= -

(b) f x x x6

2= -

] g (c) f x x x 22= - -] g

(d) y x2= -

(e) f x x 7 2= -] ]g g



(a) y x2= for x0 3

(b) y x 42= - + for x1 2 -

(c) f x x 12= -] g for x2 5 -

(d) y x x2 32= + - for x2 4 -

(e) y x x

22= - +- for

x0 4


function is

increasing(i)

decreasing(ii)

(a) y x2=

(b) y x2

= - (c) f x x 92

= -] g

(d) y x x42= - +

(e) f x x 5 2= +] ]g g


function



(a) y x 12= +

(b) f x x 32= -

] g

(c) y x2 2= -

(d) f x x x32= -] g

(e) f x x x2= +] g

(f) y x 42= -

(g) y x x2 32= - -

(h) y x x5 42= +-

(i) p x x 1 2= +] ]g g

(j) y x 2= -2] g




CONTINUED

eg When x 3= -

| | y 3 13 12

= - -

= -

=

x -3 -2 -1 0 1 2 3

y 2 1 0 -1 0 1 2


y

-2

4

3

2

1

5

-1

-3

-4

-5

1 4-1-2 32-3-4

x


| | y x

x x

x x1

1 0

1 0

when

when 1

$= - =

-


y x x1 0$= - ] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x - 1




y x 1= - - x 01] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y =- x - 1



-3

4

3

2

1

5

-2

-1

-4

-5

y y

3-1-2 421-3-4

x

y = - x - 1

y = x - 1


intercepts


So f x 0=

] g

| |

| |

x

x

x

0 1

1

1`

= -

=

=


( ) | |f 0 0 11

= -

= -





-3

4

3

2

1

5

-2

-1

-4

-5

y

4-2 5321-1-3-4

x



Domain all real x

Range y y 1$ -

2 Sketch | | y x 2= +

Solution


| | ( ) y x x xx x2 2 2 0

2 2 0whenwhen 1

$= + = + +

- + +


2 y x= + when x 2 0$+

x 2$ -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x + 2






y-intercept of 1-

CONTINUED




2 y x= - +] g when x 2 01+

ie y x 2= - - when x 21 -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2


-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2

y = x + 2



So 0f x =] g

0 | 2 |

0 2

2

x

x

x

= +

= +

- =


(0) | 0 2 |

2

f = +

=

There is only one

solution for the


Can you see why





-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

If you know how to

sketch the graph of




Investigation




are their equations





1 | |f x k x=] g

2 | |f x x k= +] g

3 | |f x x k= +] g







EXAMPLES

Solve

1 |2 1 | 3x - =

Solution



is x 1 2= -

2 |2 1 | 3 2x x= -+

Solution



3 | 1 | 2x 1+

Solution






and then check them







each function

(a) | | y x=

(b) | |f x x 7= +] g

(c) | |f x x 2= -] g

(d) 5 | | y x=

(e) | |f x x 3= - +] g

(f) | 6 | y x= +

(g) | |f x x3 2= -] g

(h) | 5 4 | y x= +

(i) | 7 1 | y x= -

(j) | |f x x2 9= +] g


plane

(a) | | y x=

(b) | |f x x 1= +] g

(c) | |f x x 3= -] g

(d) 2 | | y x=

(e) | |f x x= -] g

(f) | 1 | y x= +

(g) | |f x x 1= - -] g

(h) | 2 3 | y x= -

(i) | 4 2 | y x= +

(j) | |f x x3 1= +] g


each function

(a) | 1 | y x= -

(b) | |f x x 8= -] g

(c) | |f x x2 5= +] g

(d) 2 | | 3 y x= -

(e) | |f x x 3= - -] g


function is

increasing(i)

decreasing(ii)

(a) | 2 | y x= -

(b) | |f x x 2= +

] g

(c) | |f x x2 3= -] g

(d) 4 | | 1 y x= -

(e) | |f x x= -] g


of each function

(a) | | y x= for x2 2 -

(b) | |f x x 4= - -] g for

x4 3 -

(c) | |f x x 4= +] g for x7 2 -

(d) | 2 5 | y x= - for x3 3 -

(e) | |f x x= -] g for x1 1 -


function increasing

(a) | 3 | y x= +

(b) | |f x x 4= - +] g

(c) | |f x x 9= -] g

(d) | | y x 2 1= - -

(e) | |f x x 2= - +] g

56 Exercises




7 Solve graphically

(a) | | 3x =

(b) | |x 12

(c) | |x 2

(d) | 2 | 1x + =

(e) | 3 | 0x- =

(f) |2 3 | 1x - =

(g) | |x 1 41-

(h) | |x 1 3+

(i) | |x 2 22-

(j) | |x 3 1$-

(k) | |x2 3 5+

(l) | |x2 1 1$-

(m) |3 1 | 3x x- = +

(n) |3 2 | 4x x- = -

(o) | 1 | 1x x- = +

(p) | 3 | 2 2x x+ = + (q) |2 1 | 1x x+ = -

(r) |2 5 | 3x x- = -

(s) | 1 | 2x x- =

(t) |2 3 | 3x x- = +

The Hyperbola


or= =

EXAMPLE

Sketch1

y x=

Solution



x -3 -2 -121-

41- 0

41

21 1 2 3

y 3

1-

2

1- -1 -2 -4 mdash 4 2 1

2

1

3

1

Class Discussion



y is never 0 Why






but never touches)



Investigation




equations




1 y xk

=

21

y x k= +

31

y x k

=+


EXAMPLES


3=

-

] g


Solution



So x 3






be zero

CONTINUED







1

y 0 3

3=

-

= -

-3

4

3

2

1

5

-2

-1

-4

-5

y

-1-2 4 521-3-4

x

x = 3

y = 0

Asymptotes

3

2 Sketch y x2 4

1= -

+

Solution




x

x

2 4

2

-

-






( ) y

2 0 41

41

= -+

= -


a translation of3

y x

=





y

-2

x

-

1

4


a=

+


domain x xb



1 For each graph



exists


(a)2

y x=

(b)1

y x= -

(c) f xx 1

1=

+] g

(d) f xx 2

3=

-

] g

(e)3 6

1 y

x=

+

(f) f xx 3

2= -

-

] g

(g) f xx 1

4=

-

] g

(h)1

2 y

x= -

+

(i) f xx6 3

2=

-

] g

(j)2

6 y

x= -

+

2 Show that f x x2

=] g is an odd

function



(a) f xx2 5

1=

+

] g for x2 2 -

(b)3

1 y

x=

+ for x2 0 -

(c) f xx2 4

5=

-

] g for x3 1 -

57 Exercises




(d) f xx 4

3= -

-

] g for x3 3 -

(e)3 1

2 y

x= -

+ for x0 5



(a)3

y x= for y 1 3

(b)2

y x= - for y 221

- -

(c) f xx 1

1=

-

] g for y 171

- -

(d) f xx2 1

3= -

+

] g for

y 131

- -

(e)3 2

6 y

x=

- for y 1

21

6



Circle gate





Proof

r y

x

( x y)

y

x








c a b

r x y

2 2 2

2 2 2

`

= +

= +

EXAMPLE



Solution


y

x

-2

-2 2

2


one place

y

x

2

2

2

-2

The radius is 4

CONTINUED






Domain 2 2 x x -

Range 2 2 y y -


range y r y r -



Proof


shown

y

x

(a b)

x

y

r

( x y)

a

b x - a

y - b




ndash ndash

c a b

r x a y b

2 2 2

2 2 2

= +

= +] ^g h




EXAMPLES



Solution



y

x 9

9

-9

-9



Domain 9 9 x x -

Range 9 9 y y -



Solution


ndash

ndash ndash

x y

x y

1 2 4

1 2 2

2

2 2

+ + =

+ - =

2

2

] ^

] ]_

g h

g gi

So 1 2a b= = - and 2r =

CONTINUED







y

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

(1 -2)



Domain 1 3 x x -

Range 4 0 y y -


Solution


a b2 1= - = and r 3=

Substituting

ndash ndash

ndash

ndash

x a y b r

x y

x y

2 1 3

2 1 9

2 2 2

2 2 2

2 2

+ =

- - + =

+ + =

] ^]] ^

] ^

g hg g hg h


ndash

ndash

a b a ab b

x x x

x x

a b a ab b

y y y

y y

2

2 2 2 2

4 4

2

1 2 1 1

2 1

So

So

2 2 2

2 2 2

2

2 2 2

2 2 2

2

+ = + +

+ = + +

= + +

= - +

= - +

= - +

]] ] ]]^ ^ ]

gg g ggh h g


ndash

x x y y

x x y y

x x y y

x x y y

4 4 2 1 9

4 2 5 9

4 2 5 9

4 2 4 0

9 9

2

2

2

2

+ + + - + =

+ + - + =

+ + + =

+ + - - =

- -


in Chapter 2




Investigation


two functions





circle






for all values

y

x r

r

-r


Proof

ndash

x y r

y r x y r x

2 2 2

2 2 2

2 2

+ =

=

= -



semi-circles





y 0 for all values

y

x r

-

r

-r


EXAMPLES


1 f x x92

= -] g

Solution



y

x 3

3

-3

Domain 3 3 x x -

Range 0 3 y y




2 y x4 2= - -

Solution



y

x 2

-2

-2

Domain 2 2 x x -

Range 2 0 y y -




range

(a) 9x y 2 2+ =

(b) x y 16 02 2+ =-


(d) 1 9x y 2 2+ + =

] g

(e) ndashx y 2 1 12 2+ + =] ^g h



below the x -axis



range

(a) 25 y x2= - -

(b) 1 y x2= -

(c) 36 y x2= -

(d) 64 y x2= - -

(e) 7 y x2= - -



each circle

(a) 100x y 2 2+ =

(b) 5x y 2 2+ =


(d) ndashx y 5 6 492 2+ + =] ^g h

(e) ndashx y 3 812 2+ =^ h

58 Exercises






grouping symbols)



Centre(c) 1 5-







Centre(j) 4 7- -

^ h and radius 3

Other Graphs




EXAMPLES


Solution



eg When 0x =

y 3

1

c=

=

x

y

1

3

3

1

31

When1

1

= -

=

=

=

-

x 3- 2- 1- 0 1 2 3

y 271

91

31

1 3 9 27



You will meet these

graphs again in the

HSC Course





other point

When

x

y

1

3

3

1

=

=

=

y

x

1

2

1

3







Solution



x minus2 minus1 0 05 1 2 3 4

y minus03 0 03 05 06

y

x

1

2

1 2 3 4

-1




You learned about


stages of maths





range y y 02



range all real y

Cubic function


EXAMPLE



Solution



y minus25 minus6 1 2 3 10 29

y

x

1

1

-2 2 3 4

-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4



If you already

know the shape of


3






Domain and range


EXAMPLE


Solution




function are

f

f

1 1 2

1 21

4 4 2

64 2

66

3

3

- = - +

= - +

=

= +

= +

=

] ]

] ]

g g

g g


these points

y

x

(-1 1)

(4 66)

Range 1 66 y y










positive number

EXAMPLE


Solution


x 4So $

$


(or zero) So y 0$


59 Exercises


(a) 4 3 y x= +

(b) f x 4= -] g

(c) 3x =



(f) f x xx 12 2= - -] g

(g) 64x y 2 2+ =

(h) f t t 4

3=

-] g

(i) ( ) g 2

5zz

= +

(j) | |f x x=] g


(a) y x=

(b) 2 y x= -

(c) | |f x x2 3= -] g

(d) | | 2 y x= -

(e) f x x2 5= - +] g

(f) | | y x5= -

(g) 2 y x=

(h) y 5x= -

(i) f x xx 1

= +] g

(j)2

4 3 y x

x= -


(a) y x x 5 2= -] g


= +-

(d) g x x x164 2= -] g

(e) 49x y 2 2+ =

You may like to


by dividing by x




4 (a) Solve x1 02$-


f x x1 2= -] g


(a) 2 y x x2= - -

(b) g t t t 62= +] g



in each case


x3 3 -


x2 3 -


x2 1 - (d)


x1 5


0 4x


x3 3 -


x1 1 -


x2 3 -


x4 4 -

(j) y x x7 62= - + - in the

domain 0 7x


function y x 1

3=

+





x=] g


function

find its range(b)


plane

(a) f x x4=] g

(b) y x3= -

(c) y x 34= -

(d) 2 p x x3=] g

(e) 1 g x x3= +] g

(f) 100x y 2 2+ =

(g) 2 1 y x= +


y x 1= -









(b) f x x 22= -] g

(c)1

y x=

(d) f x x3=] g

(e) f x 3x=] g

13 (a) Solve x4 02$-


(i) 4 y x2= -

(ii) y x4

2= - -




DID YOU KNOW




a light



Limits






a a1

1

03

Z

=

=

3

3

-


lim a 0x=

x 3-



asymptote

EXAMPLES

1 Find lim x

x x5x 0

2+

Solution


which is undefined


( )

lim lim

lim

xx x

x

x x

x

5 5

5

5

x x

x

0

2

0 1

1

0

+=

+

= +

=

] g




2 Find limx

x

4

22

-

-

x 2

Solution


lim lim

lim

x

x

x x

x

x

4

2

2 2

2

21

41

2 1

1

-

-=

+ -

-

=+

=

x x2 2

x 2

^ _h i

3 Find limh

h x hx h2 72 2+ -

h 0

Solution

lim lim

lim

h

h x hx h

h

h hx x

hx x

x

2 7 2 7

2 7

7

2 2 2

2

2

+ -=

+ -

= + -

= -

h 0

h h0 0

^ h

Continuity





EXAMPLES

1 Find lim

x

x

1

12

-

-

x 1


11

y x

x2

=-

- Sketch the curve

Solution


xx2

-

- gives

00

CONTINUED




( )

lim lim

limx

x

x

x x

x11

1

1 1

1

2

x x

x

1

2

1

1

-

-=

-

+ -

= +

=

-

] ]g g

11

y xx2

=-





y xx

x

x x

x

11

1

1 1

1

2

=-

-

=

+

= +

-

-^ ^h h


2 Find limx

x x2

2x 2

2

+

+ -

-


x x2

22

= + -

+

Solution


x x2

22

+

+ - gives

00

lim lim

lim

xx x

x

x x

x

22

2

1 2

1

3

x x

x

2

2

2

2

+

+ -=

+

- +

=

= -

-

- -

-

^^ ^

^

hh h

h2 y

x

x xx

y x

x

x

x

22

2

2

1

1


=+

+ -= -

=+

= -

+ -^ ^h h


Remember that x 1




1 Find(a) lim x 52

+x 4

(b) lim t 7-t 3 -

(c) lim x x2 43+ -

x 2

(d) lim xx x32

+

x 0

(e) limh

h h

2

22

-

- -

h 2

(f) lim y

y

5

1253

-

-

y 5

(g) limx

x x

12 12

+

+ +

x 1-

(h) limx

x x4

2 82

+

+ -

x 4 -

(i) limc

c

4

22

-

-

c 2

(j) limx x

x 12

-

-

x 1

(k) lim h

h h h2 73 2+ -

h 0

(l) limh

hx hx h32 2- +

h 0

(m) limh

hx h x hx h2 3 53 2 2- + -

h 0

(n) lim x c x c 3 3

-

-

x c



discontinuous

(a) 3 y x2= -

(b)1

1 y

x=

+

(c) f x x 1= -] g

(d)4

1 y

x2=

+

(e)4

1

y x2=

-



(a)3

y xx x2

= +

(b)33

y x

x x2

=+

+

(c)1

5 4 y

xx x2

=+

+ +

510 Exercises




Regions

Class Investigation


them all







EXAMPLE


Solution



y

x =3

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4







region

EXAMPLE


Solution



included


shown

y

x

1

1

-2 2 3 4

-3

-4

-5

2

3

4

5

-1

-2

-3-4

-1 y = -1



y -intercept 1-






EXAMPLES


1 y x 2$ +

Solution






y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0



2 x y 2 3 61-

Solution


region


side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g








3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g








Domain all real x

Range y y 0$















EXAMPLES


x x 3 x 2 x 1

y

Solution


increasing


x 1 x 2

2

x x 3

x 2 x 1

y

Solution



increases again




1 1








=







x

y









x

y




EXAMPLES


Solution

f x x

x

f x

f x x

3

3


2

2

2`

- = - +

= +

=

= +

] ]]

]

g gg

g


Solution

f x x x

x x

x x

f x


3

3

3

3`

- = - - -

= - +

= - -

= -

= -

] ] ]^]

]

g g gh

gg

Investigation






1 f x kxn=] g

2 f x x kn= +] g

3 f x x k n= +

] ]g g






shape of graphs


each function

(a) y x3 2= -

(b) x y 2 5 20 0- + =

(c) x y 3 12 0+ - =

(d) f x x x32= +] g

(e) f x x 42= -] g

(f) p x x x5 62= + +] g

(g) y x x8 152= - +

(h) p x x 53

= +

] g

53 Exercises




(i) y xx

x3

0= + ] g

(j) g x x9 2= -] g


f x x 22= -

] g What type of

function is it


(a) f x2^ h (b) ( )f x 26



an even function


f x x=] g


function


odd function



f x f x- -] ]g g


neither

(a) y x x

x4 2

3

=

-

(b) y x 1

13

=

-

(c) f xx 4

32

=

-

] g

(d) y x

x

33

=+

-

(e) f x x x

x5 2

3

=-] g



f x xn=] g

even(a)

odd(b)


ever be

even(a)

odd(b)



graph is increasing




x

y(a)

x

4

y(b)

2-2

x

y(c)




Investigation


graph







y kx

y kx

y kx

y kx

y xk

(a)

(b)

(c)

(d)

(e)

2

3

4

=

=

=

=

=



y x k

y x k

y x k

y x k

y x k1

(a)

(b)

(c)

(d)

(e)

2

2

3

4

= +

= +

= +

= +

= +

] g



-2

1 2

-4

-1-2

2

4

y

x

(d) y

x

(e)

CONTINUED




Gradient form



Investigation










y x k

y x k

y x k

y x k y

x k

1

(a)

(b)

(c)

(d)(e)

2

3

4

= +

= +

= +

= +

=+

]]

]

gg

g





the right








Linear Function





EXAMPLE



Solution


Find the intercepts


0 3 5

5 3

1

x

x

x32

=

=

=

-


3 5

5

y 0=

= -

-] g

-1

-2

y

5

4

3

2

1 1 23

6

-3

-4

-5

1 4-1-2 32-3-4

x



Range all real y








Special lines






1 y kx=

2 y x k= +






EXAMPLES


Solution




-1

-3

y

4

3

2

1

5

-2

-4

-5

1 4-1-2

x

32-3-4


2 Sketch x 1= -


Solution





-

-

4

3

2

5

-2

-4

-5

1 4-1-2-4

y

x







Domain all real x

Range y y b=

54 Exercises


each function

(a) y x 2= -

(b) f x x2 3= +] g (c) x y 2 1 0+ =-

(d) x y 3 0+ =-

(e) x y 3 6 2 0=- -


line

(a) x 4=

(b) x 3 0=-

(c) y 5=

(d) y 1 0+ =

(e) f x x2 1= -] g (f) y x 4= +

(g) f x x3 2= +] g

(h) x y 3+ =

(i) x y 1 0=- -

(j) x y 2 3 0+ =-


(a) x y 3 2 7 0+ =-

(b) y 2=

(c) x 4= - (d) x 2 0=-

(e) y 3 0=-


are even or odd

(a) y x2=

(b) y 3=

(c) x 4=

(d) y x= -

(e) y x=




meet




Applications






The pronumeral

a is called the


Quadratic Function










EXAMPLES



Solution



x

x

x

0 1

1

1

2

2

= -

=

=


0 1

1

y 2= -

= -



Domain xall real

Range y y 1$ -


Solution


parabola will lie




left



0

1 0

1

x

x

x

1 2= +

+ =

= -

] g


1

y 0 1 2= +

=

] g

-1

-

4

3

2

1

5

-2

-4

-5

-

1 41-2 5-4

y

x









Solution



x x

x xx x

x x

0 6

3 23 0 2 0

3 2

2= + -

= + -

+ = - =

= - =

] ]g g


f 0 0 0 66

2= + -

= -

] ] ]g g g



x -intercepts


23 2

21- +

= -


-c m

f 21

21

21

6

41

21

6

641

2

- = - + - -

= - -

= -

c c cm m m


-

CONTINUED

You will learn more


-1

-3

4

3

2

1

5

-2

-4

-5

1 4-1-2 32-3-4

y

x










41



2

11 -





Solution


So f x 0=] g

0 4 54 5 0

0

x xx x

x x5 1

2

2

= - + +

=

+ =

- -

-] ]g g

x x

x x

5 0 1 0

5 1

- = + =

= = -


f 0 0 4 0 5

5

2= - + +

=

] ] ]g g g

-1

-3

4

2

5

-2

-4

-5

-

y

1 41-2 5-4

x

-2 4






21 5

2- +

=

f 2 2 4 2 59

= - + +

=

2] ] ]g g g










each function

(a) 2 y x x2= +

(b) 3 y x x2= - +

(c) f x x 12= -] g

(d) y x x 22= - -

(e) y x x9 82= +-

2 Sketch

(a) 2 y x2= +

(b) y x 12= - +

(c) f x x 42= -] g

(d) 2 y x x2= +

(e) y x x2= - -

(f) f x x 3= -2] ]g g

55 Exercises

-1

9

8

7

5

4

3

2

6

1

-2

-3

-4

-5

y

2 51 643-1-2-3-4

x




EXAMPLES


Solution









(g) f x x 1 2= +] ]g g

(h) y x x3 42= + -

(i) y x x2 5 32= - +

(j) f x x x3 22= - + -] g



(a) ndash y x x7 122= +

(b) f x x x42= +] g

(c) y x x2 82= - -

(d) y x x6 92= +-

(e) f t t 4 2= -] g


(a) y x 52= -

(b) f x x x6

2= -

] g (c) f x x x 22= - -] g

(d) y x2= -

(e) f x x 7 2= -] ]g g



(a) y x2= for x0 3

(b) y x 42= - + for x1 2 -

(c) f x x 12= -] g for x2 5 -

(d) y x x2 32= + - for x2 4 -

(e) y x x

22= - +- for

x0 4


function is

increasing(i)

decreasing(ii)

(a) y x2=

(b) y x2

= - (c) f x x 92

= -] g

(d) y x x42= - +

(e) f x x 5 2= +] ]g g


function



(a) y x 12= +

(b) f x x 32= -

] g

(c) y x2 2= -

(d) f x x x32= -] g

(e) f x x x2= +] g

(f) y x 42= -

(g) y x x2 32= - -

(h) y x x5 42= +-

(i) p x x 1 2= +] ]g g

(j) y x 2= -2] g




CONTINUED

eg When x 3= -

| | y 3 13 12

= - -

= -

=

x -3 -2 -1 0 1 2 3

y 2 1 0 -1 0 1 2


y

-2

4

3

2

1

5

-1

-3

-4

-5

1 4-1-2 32-3-4

x


| | y x

x x

x x1

1 0

1 0

when

when 1

$= - =

-


y x x1 0$= - ] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x - 1




y x 1= - - x 01] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y =- x - 1



-3

4

3

2

1

5

-2

-1

-4

-5

y y

3-1-2 421-3-4

x

y = - x - 1

y = x - 1


intercepts


So f x 0=

] g

| |

| |

x

x

x

0 1

1

1`

= -

=

=


( ) | |f 0 0 11

= -

= -





-3

4

3

2

1

5

-2

-1

-4

-5

y

4-2 5321-1-3-4

x



Domain all real x

Range y y 1$ -

2 Sketch | | y x 2= +

Solution


| | ( ) y x x xx x2 2 2 0

2 2 0whenwhen 1

$= + = + +

- + +


2 y x= + when x 2 0$+

x 2$ -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x + 2






y-intercept of 1-

CONTINUED




2 y x= - +] g when x 2 01+

ie y x 2= - - when x 21 -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2


-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2

y = x + 2



So 0f x =] g

0 | 2 |

0 2

2

x

x

x

= +

= +

- =


(0) | 0 2 |

2

f = +

=

There is only one

solution for the


Can you see why





-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

If you know how to

sketch the graph of




Investigation




are their equations





1 | |f x k x=] g

2 | |f x x k= +] g

3 | |f x x k= +] g







EXAMPLES

Solve

1 |2 1 | 3x - =

Solution



is x 1 2= -

2 |2 1 | 3 2x x= -+

Solution



3 | 1 | 2x 1+

Solution






and then check them







each function

(a) | | y x=

(b) | |f x x 7= +] g

(c) | |f x x 2= -] g

(d) 5 | | y x=

(e) | |f x x 3= - +] g

(f) | 6 | y x= +

(g) | |f x x3 2= -] g

(h) | 5 4 | y x= +

(i) | 7 1 | y x= -

(j) | |f x x2 9= +] g


plane

(a) | | y x=

(b) | |f x x 1= +] g

(c) | |f x x 3= -] g

(d) 2 | | y x=

(e) | |f x x= -] g

(f) | 1 | y x= +

(g) | |f x x 1= - -] g

(h) | 2 3 | y x= -

(i) | 4 2 | y x= +

(j) | |f x x3 1= +] g


each function

(a) | 1 | y x= -

(b) | |f x x 8= -] g

(c) | |f x x2 5= +] g

(d) 2 | | 3 y x= -

(e) | |f x x 3= - -] g


function is

increasing(i)

decreasing(ii)

(a) | 2 | y x= -

(b) | |f x x 2= +

] g

(c) | |f x x2 3= -] g

(d) 4 | | 1 y x= -

(e) | |f x x= -] g


of each function

(a) | | y x= for x2 2 -

(b) | |f x x 4= - -] g for

x4 3 -

(c) | |f x x 4= +] g for x7 2 -

(d) | 2 5 | y x= - for x3 3 -

(e) | |f x x= -] g for x1 1 -


function increasing

(a) | 3 | y x= +

(b) | |f x x 4= - +] g

(c) | |f x x 9= -] g

(d) | | y x 2 1= - -

(e) | |f x x 2= - +] g

56 Exercises




7 Solve graphically

(a) | | 3x =

(b) | |x 12

(c) | |x 2

(d) | 2 | 1x + =

(e) | 3 | 0x- =

(f) |2 3 | 1x - =

(g) | |x 1 41-

(h) | |x 1 3+

(i) | |x 2 22-

(j) | |x 3 1$-

(k) | |x2 3 5+

(l) | |x2 1 1$-

(m) |3 1 | 3x x- = +

(n) |3 2 | 4x x- = -

(o) | 1 | 1x x- = +

(p) | 3 | 2 2x x+ = + (q) |2 1 | 1x x+ = -

(r) |2 5 | 3x x- = -

(s) | 1 | 2x x- =

(t) |2 3 | 3x x- = +

The Hyperbola


or= =

EXAMPLE

Sketch1

y x=

Solution



x -3 -2 -121-

41- 0

41

21 1 2 3

y 3

1-

2

1- -1 -2 -4 mdash 4 2 1

2

1

3

1

Class Discussion



y is never 0 Why






but never touches)



Investigation




equations




1 y xk

=

21

y x k= +

31

y x k

=+


EXAMPLES


3=

-

] g


Solution



So x 3






be zero

CONTINUED







1

y 0 3

3=

-

= -

-3

4

3

2

1

5

-2

-1

-4

-5

y

-1-2 4 521-3-4

x

x = 3

y = 0

Asymptotes

3

2 Sketch y x2 4

1= -

+

Solution




x

x

2 4

2

-

-






( ) y

2 0 41

41

= -+

= -


a translation of3

y x

=





y

-2

x

-

1

4


a=

+


domain x xb



1 For each graph



exists


(a)2

y x=

(b)1

y x= -

(c) f xx 1

1=

+] g

(d) f xx 2

3=

-

] g

(e)3 6

1 y

x=

+

(f) f xx 3

2= -

-

] g

(g) f xx 1

4=

-

] g

(h)1

2 y

x= -

+

(i) f xx6 3

2=

-

] g

(j)2

6 y

x= -

+

2 Show that f x x2

=] g is an odd

function



(a) f xx2 5

1=

+

] g for x2 2 -

(b)3

1 y

x=

+ for x2 0 -

(c) f xx2 4

5=

-

] g for x3 1 -

57 Exercises




(d) f xx 4

3= -

-

] g for x3 3 -

(e)3 1

2 y

x= -

+ for x0 5



(a)3

y x= for y 1 3

(b)2

y x= - for y 221

- -

(c) f xx 1

1=

-

] g for y 171

- -

(d) f xx2 1

3= -

+

] g for

y 131

- -

(e)3 2

6 y

x=

- for y 1

21

6



Circle gate





Proof

r y

x

( x y)

y

x








c a b

r x y

2 2 2

2 2 2

`

= +

= +

EXAMPLE



Solution


y

x

-2

-2 2

2


one place

y

x

2

2

2

-2

The radius is 4

CONTINUED






Domain 2 2 x x -

Range 2 2 y y -


range y r y r -



Proof


shown

y

x

(a b)

x

y

r

( x y)

a

b x - a

y - b




ndash ndash

c a b

r x a y b

2 2 2

2 2 2

= +

= +] ^g h




EXAMPLES



Solution



y

x 9

9

-9

-9



Domain 9 9 x x -

Range 9 9 y y -



Solution


ndash

ndash ndash

x y

x y

1 2 4

1 2 2

2

2 2

+ + =

+ - =

2

2

] ^

] ]_

g h

g gi

So 1 2a b= = - and 2r =

CONTINUED







y

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

(1 -2)



Domain 1 3 x x -

Range 4 0 y y -


Solution


a b2 1= - = and r 3=

Substituting

ndash ndash

ndash

ndash

x a y b r

x y

x y

2 1 3

2 1 9

2 2 2

2 2 2

2 2

+ =

- - + =

+ + =

] ^]] ^

] ^

g hg g hg h


ndash

ndash

a b a ab b

x x x

x x

a b a ab b

y y y

y y

2

2 2 2 2

4 4

2

1 2 1 1

2 1

So

So

2 2 2

2 2 2

2

2 2 2

2 2 2

2

+ = + +

+ = + +

= + +

= - +

= - +

= - +

]] ] ]]^ ^ ]

gg g ggh h g


ndash

x x y y

x x y y

x x y y

x x y y

4 4 2 1 9

4 2 5 9

4 2 5 9

4 2 4 0

9 9

2

2

2

2

+ + + - + =

+ + - + =

+ + + =

+ + - - =

- -


in Chapter 2




Investigation


two functions





circle






for all values

y

x r

r

-r


Proof

ndash

x y r

y r x y r x

2 2 2

2 2 2

2 2

+ =

=

= -



semi-circles





y 0 for all values

y

x r

-

r

-r


EXAMPLES


1 f x x92

= -] g

Solution



y

x 3

3

-3

Domain 3 3 x x -

Range 0 3 y y




2 y x4 2= - -

Solution



y

x 2

-2

-2

Domain 2 2 x x -

Range 2 0 y y -




range

(a) 9x y 2 2+ =

(b) x y 16 02 2+ =-


(d) 1 9x y 2 2+ + =

] g

(e) ndashx y 2 1 12 2+ + =] ^g h



below the x -axis



range

(a) 25 y x2= - -

(b) 1 y x2= -

(c) 36 y x2= -

(d) 64 y x2= - -

(e) 7 y x2= - -



each circle

(a) 100x y 2 2+ =

(b) 5x y 2 2+ =


(d) ndashx y 5 6 492 2+ + =] ^g h

(e) ndashx y 3 812 2+ =^ h

58 Exercises






grouping symbols)



Centre(c) 1 5-







Centre(j) 4 7- -

^ h and radius 3

Other Graphs




EXAMPLES


Solution



eg When 0x =

y 3

1

c=

=

x

y

1

3

3

1

31

When1

1

= -

=

=

=

-

x 3- 2- 1- 0 1 2 3

y 271

91

31

1 3 9 27



You will meet these

graphs again in the

HSC Course





other point

When

x

y

1

3

3

1

=

=

=

y

x

1

2

1

3







Solution



x minus2 minus1 0 05 1 2 3 4

y minus03 0 03 05 06

y

x

1

2

1 2 3 4

-1




You learned about


stages of maths





range y y 02



range all real y

Cubic function


EXAMPLE



Solution



y minus25 minus6 1 2 3 10 29

y

x

1

1

-2 2 3 4

-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4



If you already

know the shape of


3






Domain and range


EXAMPLE


Solution




function are

f

f

1 1 2

1 21

4 4 2

64 2

66

3

3

- = - +

= - +

=

= +

= +

=

] ]

] ]

g g

g g


these points

y

x

(-1 1)

(4 66)

Range 1 66 y y










positive number

EXAMPLE


Solution


x 4So $

$


(or zero) So y 0$


59 Exercises


(a) 4 3 y x= +

(b) f x 4= -] g

(c) 3x =



(f) f x xx 12 2= - -] g

(g) 64x y 2 2+ =

(h) f t t 4

3=

-] g

(i) ( ) g 2

5zz

= +

(j) | |f x x=] g


(a) y x=

(b) 2 y x= -

(c) | |f x x2 3= -] g

(d) | | 2 y x= -

(e) f x x2 5= - +] g

(f) | | y x5= -

(g) 2 y x=

(h) y 5x= -

(i) f x xx 1

= +] g

(j)2

4 3 y x

x= -


(a) y x x 5 2= -] g


= +-

(d) g x x x164 2= -] g

(e) 49x y 2 2+ =

You may like to


by dividing by x




4 (a) Solve x1 02$-


f x x1 2= -] g


(a) 2 y x x2= - -

(b) g t t t 62= +] g



in each case


x3 3 -


x2 3 -


x2 1 - (d)


x1 5


0 4x


x3 3 -


x1 1 -


x2 3 -


x4 4 -

(j) y x x7 62= - + - in the

domain 0 7x


function y x 1

3=

+





x=] g


function

find its range(b)


plane

(a) f x x4=] g

(b) y x3= -

(c) y x 34= -

(d) 2 p x x3=] g

(e) 1 g x x3= +] g

(f) 100x y 2 2+ =

(g) 2 1 y x= +


y x 1= -









(b) f x x 22= -] g

(c)1

y x=

(d) f x x3=] g

(e) f x 3x=] g

13 (a) Solve x4 02$-


(i) 4 y x2= -

(ii) y x4

2= - -




DID YOU KNOW




a light



Limits






a a1

1

03

Z

=

=

3

3

-


lim a 0x=

x 3-



asymptote

EXAMPLES

1 Find lim x

x x5x 0

2+

Solution


which is undefined


( )

lim lim

lim

xx x

x

x x

x

5 5

5

5

x x

x

0

2

0 1

1

0

+=

+

= +

=

] g




2 Find limx

x

4

22

-

-

x 2

Solution


lim lim

lim

x

x

x x

x

x

4

2

2 2

2

21

41

2 1

1

-

-=

+ -

-

=+

=

x x2 2

x 2

^ _h i

3 Find limh

h x hx h2 72 2+ -

h 0

Solution

lim lim

lim

h

h x hx h

h

h hx x

hx x

x

2 7 2 7

2 7

7

2 2 2

2

2

+ -=

+ -

= + -

= -

h 0

h h0 0

^ h

Continuity





EXAMPLES

1 Find lim

x

x

1

12

-

-

x 1


11

y x

x2

=-

- Sketch the curve

Solution


xx2

-

- gives

00

CONTINUED




( )

lim lim

limx

x

x

x x

x11

1

1 1

1

2

x x

x

1

2

1

1

-

-=

-

+ -

= +

=

-

] ]g g

11

y xx2

=-





y xx

x

x x

x

11

1

1 1

1

2

=-

-

=

+

= +

-

-^ ^h h


2 Find limx

x x2

2x 2

2

+

+ -

-


x x2

22

= + -

+

Solution


x x2

22

+

+ - gives

00

lim lim

lim

xx x

x

x x

x

22

2

1 2

1

3

x x

x

2

2

2

2

+

+ -=

+

- +

=

= -

-

- -

-

^^ ^

^

hh h

h2 y

x

x xx

y x

x

x

x

22

2

2

1

1


=+

+ -= -

=+

= -

+ -^ ^h h


Remember that x 1




1 Find(a) lim x 52

+x 4

(b) lim t 7-t 3 -

(c) lim x x2 43+ -

x 2

(d) lim xx x32

+

x 0

(e) limh

h h

2

22

-

- -

h 2

(f) lim y

y

5

1253

-

-

y 5

(g) limx

x x

12 12

+

+ +

x 1-

(h) limx

x x4

2 82

+

+ -

x 4 -

(i) limc

c

4

22

-

-

c 2

(j) limx x

x 12

-

-

x 1

(k) lim h

h h h2 73 2+ -

h 0

(l) limh

hx hx h32 2- +

h 0

(m) limh

hx h x hx h2 3 53 2 2- + -

h 0

(n) lim x c x c 3 3

-

-

x c



discontinuous

(a) 3 y x2= -

(b)1

1 y

x=

+

(c) f x x 1= -] g

(d)4

1 y

x2=

+

(e)4

1

y x2=

-



(a)3

y xx x2

= +

(b)33

y x

x x2

=+

+

(c)1

5 4 y

xx x2

=+

+ +

510 Exercises




Regions

Class Investigation


them all







EXAMPLE


Solution



y

x =3

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4







region

EXAMPLE


Solution



included


shown

y

x

1

1

-2 2 3 4

-3

-4

-5

2

3

4

5

-1

-2

-3-4

-1 y = -1



y -intercept 1-






EXAMPLES


1 y x 2$ +

Solution






y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0



2 x y 2 3 61-

Solution


region


side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g








3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g






EXAMPLES


x x 3 x 2 x 1

y

Solution


increasing


x 1 x 2

2

x x 3

x 2 x 1

y

Solution



increases again




1 1








=







x

y









x

y




EXAMPLES


Solution

f x x

x

f x

f x x

3

3


2

2

2`

- = - +

= +

=

= +

] ]]

]

g gg

g


Solution

f x x x

x x

x x

f x


3

3

3

3`

- = - - -

= - +

= - -

= -

= -

] ] ]^]

]

g g gh

gg

Investigation






1 f x kxn=] g

2 f x x kn= +] g

3 f x x k n= +

] ]g g






shape of graphs


each function

(a) y x3 2= -

(b) x y 2 5 20 0- + =

(c) x y 3 12 0+ - =

(d) f x x x32= +] g

(e) f x x 42= -] g

(f) p x x x5 62= + +] g

(g) y x x8 152= - +

(h) p x x 53

= +

] g

53 Exercises




(i) y xx

x3

0= + ] g

(j) g x x9 2= -] g


f x x 22= -

] g What type of

function is it


(a) f x2^ h (b) ( )f x 26



an even function


f x x=] g


function


odd function



f x f x- -] ]g g


neither

(a) y x x

x4 2

3

=

-

(b) y x 1

13

=

-

(c) f xx 4

32

=

-

] g

(d) y x

x

33

=+

-

(e) f x x x

x5 2

3

=-] g



f x xn=] g

even(a)

odd(b)


ever be

even(a)

odd(b)



graph is increasing




x

y(a)

x

4

y(b)

2-2

x

y(c)




Investigation


graph







y kx

y kx

y kx

y kx

y xk

(a)

(b)

(c)

(d)

(e)

2

3

4

=

=

=

=

=



y x k

y x k

y x k

y x k

y x k1

(a)

(b)

(c)

(d)

(e)

2

2

3

4

= +

= +

= +

= +

= +

] g



-2

1 2

-4

-1-2

2

4

y

x

(d) y

x

(e)

CONTINUED




Gradient form



Investigation










y x k

y x k

y x k

y x k y

x k

1

(a)

(b)

(c)

(d)(e)

2

3

4

= +

= +

= +

= +

=+

]]

]

gg

g





the right








Linear Function





EXAMPLE



Solution


Find the intercepts


0 3 5

5 3

1

x

x

x32

=

=

=

-


3 5

5

y 0=

= -

-] g

-1

-2

y

5

4

3

2

1 1 23

6

-3

-4

-5

1 4-1-2 32-3-4

x



Range all real y








Special lines






1 y kx=

2 y x k= +






EXAMPLES


Solution




-1

-3

y

4

3

2

1

5

-2

-4

-5

1 4-1-2

x

32-3-4


2 Sketch x 1= -


Solution





-

-

4

3

2

5

-2

-4

-5

1 4-1-2-4

y

x







Domain all real x

Range y y b=

54 Exercises


each function

(a) y x 2= -

(b) f x x2 3= +] g (c) x y 2 1 0+ =-

(d) x y 3 0+ =-

(e) x y 3 6 2 0=- -


line

(a) x 4=

(b) x 3 0=-

(c) y 5=

(d) y 1 0+ =

(e) f x x2 1= -] g (f) y x 4= +

(g) f x x3 2= +] g

(h) x y 3+ =

(i) x y 1 0=- -

(j) x y 2 3 0+ =-


(a) x y 3 2 7 0+ =-

(b) y 2=

(c) x 4= - (d) x 2 0=-

(e) y 3 0=-


are even or odd

(a) y x2=

(b) y 3=

(c) x 4=

(d) y x= -

(e) y x=




meet




Applications






The pronumeral

a is called the


Quadratic Function










EXAMPLES



Solution



x

x

x

0 1

1

1

2

2

= -

=

=


0 1

1

y 2= -

= -



Domain xall real

Range y y 1$ -


Solution


parabola will lie




left



0

1 0

1

x

x

x

1 2= +

+ =

= -

] g


1

y 0 1 2= +

=

] g

-1

-

4

3

2

1

5

-2

-4

-5

-

1 41-2 5-4

y

x









Solution



x x

x xx x

x x

0 6

3 23 0 2 0

3 2

2= + -

= + -

+ = - =

= - =

] ]g g


f 0 0 0 66

2= + -

= -

] ] ]g g g



x -intercepts


23 2

21- +

= -


-c m

f 21

21

21

6

41

21

6

641

2

- = - + - -

= - -

= -

c c cm m m


-

CONTINUED

You will learn more


-1

-3

4

3

2

1

5

-2

-4

-5

1 4-1-2 32-3-4

y

x










41



2

11 -





Solution


So f x 0=] g

0 4 54 5 0

0

x xx x

x x5 1

2

2

= - + +

=

+ =

- -

-] ]g g

x x

x x

5 0 1 0

5 1

- = + =

= = -


f 0 0 4 0 5

5

2= - + +

=

] ] ]g g g

-1

-3

4

2

5

-2

-4

-5

-

y

1 41-2 5-4

x

-2 4






21 5

2- +

=

f 2 2 4 2 59

= - + +

=

2] ] ]g g g










each function

(a) 2 y x x2= +

(b) 3 y x x2= - +

(c) f x x 12= -] g

(d) y x x 22= - -

(e) y x x9 82= +-

2 Sketch

(a) 2 y x2= +

(b) y x 12= - +

(c) f x x 42= -] g

(d) 2 y x x2= +

(e) y x x2= - -

(f) f x x 3= -2] ]g g

55 Exercises

-1

9

8

7

5

4

3

2

6

1

-2

-3

-4

-5

y

2 51 643-1-2-3-4

x




EXAMPLES


Solution









(g) f x x 1 2= +] ]g g

(h) y x x3 42= + -

(i) y x x2 5 32= - +

(j) f x x x3 22= - + -] g



(a) ndash y x x7 122= +

(b) f x x x42= +] g

(c) y x x2 82= - -

(d) y x x6 92= +-

(e) f t t 4 2= -] g


(a) y x 52= -

(b) f x x x6

2= -

] g (c) f x x x 22= - -] g

(d) y x2= -

(e) f x x 7 2= -] ]g g



(a) y x2= for x0 3

(b) y x 42= - + for x1 2 -

(c) f x x 12= -] g for x2 5 -

(d) y x x2 32= + - for x2 4 -

(e) y x x

22= - +- for

x0 4


function is

increasing(i)

decreasing(ii)

(a) y x2=

(b) y x2

= - (c) f x x 92

= -] g

(d) y x x42= - +

(e) f x x 5 2= +] ]g g


function



(a) y x 12= +

(b) f x x 32= -

] g

(c) y x2 2= -

(d) f x x x32= -] g

(e) f x x x2= +] g

(f) y x 42= -

(g) y x x2 32= - -

(h) y x x5 42= +-

(i) p x x 1 2= +] ]g g

(j) y x 2= -2] g




CONTINUED

eg When x 3= -

| | y 3 13 12

= - -

= -

=

x -3 -2 -1 0 1 2 3

y 2 1 0 -1 0 1 2


y

-2

4

3

2

1

5

-1

-3

-4

-5

1 4-1-2 32-3-4

x


| | y x

x x

x x1

1 0

1 0

when

when 1

$= - =

-


y x x1 0$= - ] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x - 1




y x 1= - - x 01] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y =- x - 1



-3

4

3

2

1

5

-2

-1

-4

-5

y y

3-1-2 421-3-4

x

y = - x - 1

y = x - 1


intercepts


So f x 0=

] g

| |

| |

x

x

x

0 1

1

1`

= -

=

=


( ) | |f 0 0 11

= -

= -





-3

4

3

2

1

5

-2

-1

-4

-5

y

4-2 5321-1-3-4

x



Domain all real x

Range y y 1$ -

2 Sketch | | y x 2= +

Solution


| | ( ) y x x xx x2 2 2 0

2 2 0whenwhen 1

$= + = + +

- + +


2 y x= + when x 2 0$+

x 2$ -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x + 2






y-intercept of 1-

CONTINUED




2 y x= - +] g when x 2 01+

ie y x 2= - - when x 21 -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2


-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2

y = x + 2



So 0f x =] g

0 | 2 |

0 2

2

x

x

x

= +

= +

- =


(0) | 0 2 |

2

f = +

=

There is only one

solution for the


Can you see why





-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

If you know how to

sketch the graph of




Investigation




are their equations





1 | |f x k x=] g

2 | |f x x k= +] g

3 | |f x x k= +] g







EXAMPLES

Solve

1 |2 1 | 3x - =

Solution



is x 1 2= -

2 |2 1 | 3 2x x= -+

Solution



3 | 1 | 2x 1+

Solution






and then check them







each function

(a) | | y x=

(b) | |f x x 7= +] g

(c) | |f x x 2= -] g

(d) 5 | | y x=

(e) | |f x x 3= - +] g

(f) | 6 | y x= +

(g) | |f x x3 2= -] g

(h) | 5 4 | y x= +

(i) | 7 1 | y x= -

(j) | |f x x2 9= +] g


plane

(a) | | y x=

(b) | |f x x 1= +] g

(c) | |f x x 3= -] g

(d) 2 | | y x=

(e) | |f x x= -] g

(f) | 1 | y x= +

(g) | |f x x 1= - -] g

(h) | 2 3 | y x= -

(i) | 4 2 | y x= +

(j) | |f x x3 1= +] g


each function

(a) | 1 | y x= -

(b) | |f x x 8= -] g

(c) | |f x x2 5= +] g

(d) 2 | | 3 y x= -

(e) | |f x x 3= - -] g


function is

increasing(i)

decreasing(ii)

(a) | 2 | y x= -

(b) | |f x x 2= +

] g

(c) | |f x x2 3= -] g

(d) 4 | | 1 y x= -

(e) | |f x x= -] g


of each function

(a) | | y x= for x2 2 -

(b) | |f x x 4= - -] g for

x4 3 -

(c) | |f x x 4= +] g for x7 2 -

(d) | 2 5 | y x= - for x3 3 -

(e) | |f x x= -] g for x1 1 -


function increasing

(a) | 3 | y x= +

(b) | |f x x 4= - +] g

(c) | |f x x 9= -] g

(d) | | y x 2 1= - -

(e) | |f x x 2= - +] g

56 Exercises




7 Solve graphically

(a) | | 3x =

(b) | |x 12

(c) | |x 2

(d) | 2 | 1x + =

(e) | 3 | 0x- =

(f) |2 3 | 1x - =

(g) | |x 1 41-

(h) | |x 1 3+

(i) | |x 2 22-

(j) | |x 3 1$-

(k) | |x2 3 5+

(l) | |x2 1 1$-

(m) |3 1 | 3x x- = +

(n) |3 2 | 4x x- = -

(o) | 1 | 1x x- = +

(p) | 3 | 2 2x x+ = + (q) |2 1 | 1x x+ = -

(r) |2 5 | 3x x- = -

(s) | 1 | 2x x- =

(t) |2 3 | 3x x- = +

The Hyperbola


or= =

EXAMPLE

Sketch1

y x=

Solution



x -3 -2 -121-

41- 0

41

21 1 2 3

y 3

1-

2

1- -1 -2 -4 mdash 4 2 1

2

1

3

1

Class Discussion



y is never 0 Why






but never touches)



Investigation




equations




1 y xk

=

21

y x k= +

31

y x k

=+


EXAMPLES


3=

-

] g


Solution



So x 3






be zero

CONTINUED







1

y 0 3

3=

-

= -

-3

4

3

2

1

5

-2

-1

-4

-5

y

-1-2 4 521-3-4

x

x = 3

y = 0

Asymptotes

3

2 Sketch y x2 4

1= -

+

Solution




x

x

2 4

2

-

-






( ) y

2 0 41

41

= -+

= -


a translation of3

y x

=





y

-2

x

-

1

4


a=

+


domain x xb



1 For each graph



exists


(a)2

y x=

(b)1

y x= -

(c) f xx 1

1=

+] g

(d) f xx 2

3=

-

] g

(e)3 6

1 y

x=

+

(f) f xx 3

2= -

-

] g

(g) f xx 1

4=

-

] g

(h)1

2 y

x= -

+

(i) f xx6 3

2=

-

] g

(j)2

6 y

x= -

+

2 Show that f x x2

=] g is an odd

function



(a) f xx2 5

1=

+

] g for x2 2 -

(b)3

1 y

x=

+ for x2 0 -

(c) f xx2 4

5=

-

] g for x3 1 -

57 Exercises




(d) f xx 4

3= -

-

] g for x3 3 -

(e)3 1

2 y

x= -

+ for x0 5



(a)3

y x= for y 1 3

(b)2

y x= - for y 221

- -

(c) f xx 1

1=

-

] g for y 171

- -

(d) f xx2 1

3= -

+

] g for

y 131

- -

(e)3 2

6 y

x=

- for y 1

21

6



Circle gate





Proof

r y

x

( x y)

y

x








c a b

r x y

2 2 2

2 2 2

`

= +

= +

EXAMPLE



Solution


y

x

-2

-2 2

2


one place

y

x

2

2

2

-2

The radius is 4

CONTINUED






Domain 2 2 x x -

Range 2 2 y y -


range y r y r -



Proof


shown

y

x

(a b)

x

y

r

( x y)

a

b x - a

y - b




ndash ndash

c a b

r x a y b

2 2 2

2 2 2

= +

= +] ^g h




EXAMPLES



Solution



y

x 9

9

-9

-9



Domain 9 9 x x -

Range 9 9 y y -



Solution


ndash

ndash ndash

x y

x y

1 2 4

1 2 2

2

2 2

+ + =

+ - =

2

2

] ^

] ]_

g h

g gi

So 1 2a b= = - and 2r =

CONTINUED







y

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

(1 -2)



Domain 1 3 x x -

Range 4 0 y y -


Solution


a b2 1= - = and r 3=

Substituting

ndash ndash

ndash

ndash

x a y b r

x y

x y

2 1 3

2 1 9

2 2 2

2 2 2

2 2

+ =

- - + =

+ + =

] ^]] ^

] ^

g hg g hg h


ndash

ndash

a b a ab b

x x x

x x

a b a ab b

y y y

y y

2

2 2 2 2

4 4

2

1 2 1 1

2 1

So

So

2 2 2

2 2 2

2

2 2 2

2 2 2

2

+ = + +

+ = + +

= + +

= - +

= - +

= - +

]] ] ]]^ ^ ]

gg g ggh h g


ndash

x x y y

x x y y

x x y y

x x y y

4 4 2 1 9

4 2 5 9

4 2 5 9

4 2 4 0

9 9

2

2

2

2

+ + + - + =

+ + - + =

+ + + =

+ + - - =

- -


in Chapter 2




Investigation


two functions





circle






for all values

y

x r

r

-r


Proof

ndash

x y r

y r x y r x

2 2 2

2 2 2

2 2

+ =

=

= -



semi-circles





y 0 for all values

y

x r

-

r

-r


EXAMPLES


1 f x x92

= -] g

Solution



y

x 3

3

-3

Domain 3 3 x x -

Range 0 3 y y




2 y x4 2= - -

Solution



y

x 2

-2

-2

Domain 2 2 x x -

Range 2 0 y y -




range

(a) 9x y 2 2+ =

(b) x y 16 02 2+ =-


(d) 1 9x y 2 2+ + =

] g

(e) ndashx y 2 1 12 2+ + =] ^g h



below the x -axis



range

(a) 25 y x2= - -

(b) 1 y x2= -

(c) 36 y x2= -

(d) 64 y x2= - -

(e) 7 y x2= - -



each circle

(a) 100x y 2 2+ =

(b) 5x y 2 2+ =


(d) ndashx y 5 6 492 2+ + =] ^g h

(e) ndashx y 3 812 2+ =^ h

58 Exercises






grouping symbols)



Centre(c) 1 5-







Centre(j) 4 7- -

^ h and radius 3

Other Graphs




EXAMPLES


Solution



eg When 0x =

y 3

1

c=

=

x

y

1

3

3

1

31

When1

1

= -

=

=

=

-

x 3- 2- 1- 0 1 2 3

y 271

91

31

1 3 9 27



You will meet these

graphs again in the

HSC Course





other point

When

x

y

1

3

3

1

=

=

=

y

x

1

2

1

3







Solution



x minus2 minus1 0 05 1 2 3 4

y minus03 0 03 05 06

y

x

1

2

1 2 3 4

-1




You learned about


stages of maths





range y y 02



range all real y

Cubic function


EXAMPLE



Solution



y minus25 minus6 1 2 3 10 29

y

x

1

1

-2 2 3 4

-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4



If you already

know the shape of


3






Domain and range


EXAMPLE


Solution




function are

f

f

1 1 2

1 21

4 4 2

64 2

66

3

3

- = - +

= - +

=

= +

= +

=

] ]

] ]

g g

g g


these points

y

x

(-1 1)

(4 66)

Range 1 66 y y










positive number

EXAMPLE


Solution


x 4So $

$


(or zero) So y 0$


59 Exercises


(a) 4 3 y x= +

(b) f x 4= -] g

(c) 3x =



(f) f x xx 12 2= - -] g

(g) 64x y 2 2+ =

(h) f t t 4

3=

-] g

(i) ( ) g 2

5zz

= +

(j) | |f x x=] g


(a) y x=

(b) 2 y x= -

(c) | |f x x2 3= -] g

(d) | | 2 y x= -

(e) f x x2 5= - +] g

(f) | | y x5= -

(g) 2 y x=

(h) y 5x= -

(i) f x xx 1

= +] g

(j)2

4 3 y x

x= -


(a) y x x 5 2= -] g


= +-

(d) g x x x164 2= -] g

(e) 49x y 2 2+ =

You may like to


by dividing by x




4 (a) Solve x1 02$-


f x x1 2= -] g


(a) 2 y x x2= - -

(b) g t t t 62= +] g



in each case


x3 3 -


x2 3 -


x2 1 - (d)


x1 5


0 4x


x3 3 -


x1 1 -


x2 3 -


x4 4 -

(j) y x x7 62= - + - in the

domain 0 7x


function y x 1

3=

+





x=] g


function

find its range(b)


plane

(a) f x x4=] g

(b) y x3= -

(c) y x 34= -

(d) 2 p x x3=] g

(e) 1 g x x3= +] g

(f) 100x y 2 2+ =

(g) 2 1 y x= +


y x 1= -









(b) f x x 22= -] g

(c)1

y x=

(d) f x x3=] g

(e) f x 3x=] g

13 (a) Solve x4 02$-


(i) 4 y x2= -

(ii) y x4

2= - -




DID YOU KNOW




a light



Limits






a a1

1

03

Z

=

=

3

3

-


lim a 0x=

x 3-



asymptote

EXAMPLES

1 Find lim x

x x5x 0

2+

Solution


which is undefined


( )

lim lim

lim

xx x

x

x x

x

5 5

5

5

x x

x

0

2

0 1

1

0

+=

+

= +

=

] g




2 Find limx

x

4

22

-

-

x 2

Solution


lim lim

lim

x

x

x x

x

x

4

2

2 2

2

21

41

2 1

1

-

-=

+ -

-

=+

=

x x2 2

x 2

^ _h i

3 Find limh

h x hx h2 72 2+ -

h 0

Solution

lim lim

lim

h

h x hx h

h

h hx x

hx x

x

2 7 2 7

2 7

7

2 2 2

2

2

+ -=

+ -

= + -

= -

h 0

h h0 0

^ h

Continuity





EXAMPLES

1 Find lim

x

x

1

12

-

-

x 1


11

y x

x2

=-

- Sketch the curve

Solution


xx2

-

- gives

00

CONTINUED




( )

lim lim

limx

x

x

x x

x11

1

1 1

1

2

x x

x

1

2

1

1

-

-=

-

+ -

= +

=

-

] ]g g

11

y xx2

=-





y xx

x

x x

x

11

1

1 1

1

2

=-

-

=

+

= +

-

-^ ^h h


2 Find limx

x x2

2x 2

2

+

+ -

-


x x2

22

= + -

+

Solution


x x2

22

+

+ - gives

00

lim lim

lim

xx x

x

x x

x

22

2

1 2

1

3

x x

x

2

2

2

2

+

+ -=

+

- +

=

= -

-

- -

-

^^ ^

^

hh h

h2 y

x

x xx

y x

x

x

x

22

2

2

1

1


=+

+ -= -

=+

= -

+ -^ ^h h


Remember that x 1




1 Find(a) lim x 52

+x 4

(b) lim t 7-t 3 -

(c) lim x x2 43+ -

x 2

(d) lim xx x32

+

x 0

(e) limh

h h

2

22

-

- -

h 2

(f) lim y

y

5

1253

-

-

y 5

(g) limx

x x

12 12

+

+ +

x 1-

(h) limx

x x4

2 82

+

+ -

x 4 -

(i) limc

c

4

22

-

-

c 2

(j) limx x

x 12

-

-

x 1

(k) lim h

h h h2 73 2+ -

h 0

(l) limh

hx hx h32 2- +

h 0

(m) limh

hx h x hx h2 3 53 2 2- + -

h 0

(n) lim x c x c 3 3

-

-

x c



discontinuous

(a) 3 y x2= -

(b)1

1 y

x=

+

(c) f x x 1= -] g

(d)4

1 y

x2=

+

(e)4

1

y x2=

-



(a)3

y xx x2

= +

(b)33

y x

x x2

=+

+

(c)1

5 4 y

xx x2

=+

+ +

510 Exercises




Regions

Class Investigation


them all







EXAMPLE


Solution



y

x =3

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4







region

EXAMPLE


Solution



included


shown

y

x

1

1

-2 2 3 4

-3

-4

-5

2

3

4

5

-1

-2

-3-4

-1 y = -1



y -intercept 1-






EXAMPLES


1 y x 2$ +

Solution






y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0



2 x y 2 3 61-

Solution


region


side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g








3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g









x

y









x

y




EXAMPLES


Solution

f x x

x

f x

f x x

3

3


2

2

2`

- = - +

= +

=

= +

] ]]

]

g gg

g


Solution

f x x x

x x

x x

f x


3

3

3

3`

- = - - -

= - +

= - -

= -

= -

] ] ]^]

]

g g gh

gg

Investigation






1 f x kxn=] g

2 f x x kn= +] g

3 f x x k n= +

] ]g g






shape of graphs


each function

(a) y x3 2= -

(b) x y 2 5 20 0- + =

(c) x y 3 12 0+ - =

(d) f x x x32= +] g

(e) f x x 42= -] g

(f) p x x x5 62= + +] g

(g) y x x8 152= - +

(h) p x x 53

= +

] g

53 Exercises




(i) y xx

x3

0= + ] g

(j) g x x9 2= -] g


f x x 22= -

] g What type of

function is it


(a) f x2^ h (b) ( )f x 26



an even function


f x x=] g


function


odd function



f x f x- -] ]g g


neither

(a) y x x

x4 2

3

=

-

(b) y x 1

13

=

-

(c) f xx 4

32

=

-

] g

(d) y x

x

33

=+

-

(e) f x x x

x5 2

3

=-] g



f x xn=] g

even(a)

odd(b)


ever be

even(a)

odd(b)



graph is increasing




x

y(a)

x

4

y(b)

2-2

x

y(c)




Investigation


graph







y kx

y kx

y kx

y kx

y xk

(a)

(b)

(c)

(d)

(e)

2

3

4

=

=

=

=

=



y x k

y x k

y x k

y x k

y x k1

(a)

(b)

(c)

(d)

(e)

2

2

3

4

= +

= +

= +

= +

= +

] g



-2

1 2

-4

-1-2

2

4

y

x

(d) y

x

(e)

CONTINUED




Gradient form



Investigation










y x k

y x k

y x k

y x k y

x k

1

(a)

(b)

(c)

(d)(e)

2

3

4

= +

= +

= +

= +

=+

]]

]

gg

g





the right








Linear Function





EXAMPLE



Solution


Find the intercepts


0 3 5

5 3

1

x

x

x32

=

=

=

-


3 5

5

y 0=

= -

-] g

-1

-2

y

5

4

3

2

1 1 23

6

-3

-4

-5

1 4-1-2 32-3-4

x



Range all real y








Special lines






1 y kx=

2 y x k= +






EXAMPLES


Solution




-1

-3

y

4

3

2

1

5

-2

-4

-5

1 4-1-2

x

32-3-4


2 Sketch x 1= -


Solution





-

-

4

3

2

5

-2

-4

-5

1 4-1-2-4

y

x







Domain all real x

Range y y b=

54 Exercises


each function

(a) y x 2= -

(b) f x x2 3= +] g (c) x y 2 1 0+ =-

(d) x y 3 0+ =-

(e) x y 3 6 2 0=- -


line

(a) x 4=

(b) x 3 0=-

(c) y 5=

(d) y 1 0+ =

(e) f x x2 1= -] g (f) y x 4= +

(g) f x x3 2= +] g

(h) x y 3+ =

(i) x y 1 0=- -

(j) x y 2 3 0+ =-


(a) x y 3 2 7 0+ =-

(b) y 2=

(c) x 4= - (d) x 2 0=-

(e) y 3 0=-


are even or odd

(a) y x2=

(b) y 3=

(c) x 4=

(d) y x= -

(e) y x=




meet




Applications






The pronumeral

a is called the


Quadratic Function










EXAMPLES



Solution



x

x

x

0 1

1

1

2

2

= -

=

=


0 1

1

y 2= -

= -



Domain xall real

Range y y 1$ -


Solution


parabola will lie




left



0

1 0

1

x

x

x

1 2= +

+ =

= -

] g


1

y 0 1 2= +

=

] g

-1

-

4

3

2

1

5

-2

-4

-5

-

1 41-2 5-4

y

x









Solution



x x

x xx x

x x

0 6

3 23 0 2 0

3 2

2= + -

= + -

+ = - =

= - =

] ]g g


f 0 0 0 66

2= + -

= -

] ] ]g g g



x -intercepts


23 2

21- +

= -


-c m

f 21

21

21

6

41

21

6

641

2

- = - + - -

= - -

= -

c c cm m m


-

CONTINUED

You will learn more


-1

-3

4

3

2

1

5

-2

-4

-5

1 4-1-2 32-3-4

y

x










41



2

11 -





Solution


So f x 0=] g

0 4 54 5 0

0

x xx x

x x5 1

2

2

= - + +

=

+ =

- -

-] ]g g

x x

x x

5 0 1 0

5 1

- = + =

= = -


f 0 0 4 0 5

5

2= - + +

=

] ] ]g g g

-1

-3

4

2

5

-2

-4

-5

-

y

1 41-2 5-4

x

-2 4






21 5

2- +

=

f 2 2 4 2 59

= - + +

=

2] ] ]g g g










each function

(a) 2 y x x2= +

(b) 3 y x x2= - +

(c) f x x 12= -] g

(d) y x x 22= - -

(e) y x x9 82= +-

2 Sketch

(a) 2 y x2= +

(b) y x 12= - +

(c) f x x 42= -] g

(d) 2 y x x2= +

(e) y x x2= - -

(f) f x x 3= -2] ]g g

55 Exercises

-1

9

8

7

5

4

3

2

6

1

-2

-3

-4

-5

y

2 51 643-1-2-3-4

x




EXAMPLES


Solution









(g) f x x 1 2= +] ]g g

(h) y x x3 42= + -

(i) y x x2 5 32= - +

(j) f x x x3 22= - + -] g



(a) ndash y x x7 122= +

(b) f x x x42= +] g

(c) y x x2 82= - -

(d) y x x6 92= +-

(e) f t t 4 2= -] g


(a) y x 52= -

(b) f x x x6

2= -

] g (c) f x x x 22= - -] g

(d) y x2= -

(e) f x x 7 2= -] ]g g



(a) y x2= for x0 3

(b) y x 42= - + for x1 2 -

(c) f x x 12= -] g for x2 5 -

(d) y x x2 32= + - for x2 4 -

(e) y x x

22= - +- for

x0 4


function is

increasing(i)

decreasing(ii)

(a) y x2=

(b) y x2

= - (c) f x x 92

= -] g

(d) y x x42= - +

(e) f x x 5 2= +] ]g g


function



(a) y x 12= +

(b) f x x 32= -

] g

(c) y x2 2= -

(d) f x x x32= -] g

(e) f x x x2= +] g

(f) y x 42= -

(g) y x x2 32= - -

(h) y x x5 42= +-

(i) p x x 1 2= +] ]g g

(j) y x 2= -2] g




CONTINUED

eg When x 3= -

| | y 3 13 12

= - -

= -

=

x -3 -2 -1 0 1 2 3

y 2 1 0 -1 0 1 2


y

-2

4

3

2

1

5

-1

-3

-4

-5

1 4-1-2 32-3-4

x


| | y x

x x

x x1

1 0

1 0

when

when 1

$= - =

-


y x x1 0$= - ] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x - 1




y x 1= - - x 01] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y =- x - 1



-3

4

3

2

1

5

-2

-1

-4

-5

y y

3-1-2 421-3-4

x

y = - x - 1

y = x - 1


intercepts


So f x 0=

] g

| |

| |

x

x

x

0 1

1

1`

= -

=

=


( ) | |f 0 0 11

= -

= -





-3

4

3

2

1

5

-2

-1

-4

-5

y

4-2 5321-1-3-4

x



Domain all real x

Range y y 1$ -

2 Sketch | | y x 2= +

Solution


| | ( ) y x x xx x2 2 2 0

2 2 0whenwhen 1

$= + = + +

- + +


2 y x= + when x 2 0$+

x 2$ -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x + 2






y-intercept of 1-

CONTINUED




2 y x= - +] g when x 2 01+

ie y x 2= - - when x 21 -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2


-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2

y = x + 2



So 0f x =] g

0 | 2 |

0 2

2

x

x

x

= +

= +

- =


(0) | 0 2 |

2

f = +

=

There is only one

solution for the


Can you see why





-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

If you know how to

sketch the graph of




Investigation




are their equations





1 | |f x k x=] g

2 | |f x x k= +] g

3 | |f x x k= +] g







EXAMPLES

Solve

1 |2 1 | 3x - =

Solution



is x 1 2= -

2 |2 1 | 3 2x x= -+

Solution



3 | 1 | 2x 1+

Solution






and then check them







each function

(a) | | y x=

(b) | |f x x 7= +] g

(c) | |f x x 2= -] g

(d) 5 | | y x=

(e) | |f x x 3= - +] g

(f) | 6 | y x= +

(g) | |f x x3 2= -] g

(h) | 5 4 | y x= +

(i) | 7 1 | y x= -

(j) | |f x x2 9= +] g


plane

(a) | | y x=

(b) | |f x x 1= +] g

(c) | |f x x 3= -] g

(d) 2 | | y x=

(e) | |f x x= -] g

(f) | 1 | y x= +

(g) | |f x x 1= - -] g

(h) | 2 3 | y x= -

(i) | 4 2 | y x= +

(j) | |f x x3 1= +] g


each function

(a) | 1 | y x= -

(b) | |f x x 8= -] g

(c) | |f x x2 5= +] g

(d) 2 | | 3 y x= -

(e) | |f x x 3= - -] g


function is

increasing(i)

decreasing(ii)

(a) | 2 | y x= -

(b) | |f x x 2= +

] g

(c) | |f x x2 3= -] g

(d) 4 | | 1 y x= -

(e) | |f x x= -] g


of each function

(a) | | y x= for x2 2 -

(b) | |f x x 4= - -] g for

x4 3 -

(c) | |f x x 4= +] g for x7 2 -

(d) | 2 5 | y x= - for x3 3 -

(e) | |f x x= -] g for x1 1 -


function increasing

(a) | 3 | y x= +

(b) | |f x x 4= - +] g

(c) | |f x x 9= -] g

(d) | | y x 2 1= - -

(e) | |f x x 2= - +] g

56 Exercises




7 Solve graphically

(a) | | 3x =

(b) | |x 12

(c) | |x 2

(d) | 2 | 1x + =

(e) | 3 | 0x- =

(f) |2 3 | 1x - =

(g) | |x 1 41-

(h) | |x 1 3+

(i) | |x 2 22-

(j) | |x 3 1$-

(k) | |x2 3 5+

(l) | |x2 1 1$-

(m) |3 1 | 3x x- = +

(n) |3 2 | 4x x- = -

(o) | 1 | 1x x- = +

(p) | 3 | 2 2x x+ = + (q) |2 1 | 1x x+ = -

(r) |2 5 | 3x x- = -

(s) | 1 | 2x x- =

(t) |2 3 | 3x x- = +

The Hyperbola


or= =

EXAMPLE

Sketch1

y x=

Solution



x -3 -2 -121-

41- 0

41

21 1 2 3

y 3

1-

2

1- -1 -2 -4 mdash 4 2 1

2

1

3

1

Class Discussion



y is never 0 Why






but never touches)



Investigation




equations




1 y xk

=

21

y x k= +

31

y x k

=+


EXAMPLES


3=

-

] g


Solution



So x 3






be zero

CONTINUED







1

y 0 3

3=

-

= -

-3

4

3

2

1

5

-2

-1

-4

-5

y

-1-2 4 521-3-4

x

x = 3

y = 0

Asymptotes

3

2 Sketch y x2 4

1= -

+

Solution




x

x

2 4

2

-

-






( ) y

2 0 41

41

= -+

= -


a translation of3

y x

=





y

-2

x

-

1

4


a=

+


domain x xb



1 For each graph



exists


(a)2

y x=

(b)1

y x= -

(c) f xx 1

1=

+] g

(d) f xx 2

3=

-

] g

(e)3 6

1 y

x=

+

(f) f xx 3

2= -

-

] g

(g) f xx 1

4=

-

] g

(h)1

2 y

x= -

+

(i) f xx6 3

2=

-

] g

(j)2

6 y

x= -

+

2 Show that f x x2

=] g is an odd

function



(a) f xx2 5

1=

+

] g for x2 2 -

(b)3

1 y

x=

+ for x2 0 -

(c) f xx2 4

5=

-

] g for x3 1 -

57 Exercises




(d) f xx 4

3= -

-

] g for x3 3 -

(e)3 1

2 y

x= -

+ for x0 5



(a)3

y x= for y 1 3

(b)2

y x= - for y 221

- -

(c) f xx 1

1=

-

] g for y 171

- -

(d) f xx2 1

3= -

+

] g for

y 131

- -

(e)3 2

6 y

x=

- for y 1

21

6



Circle gate





Proof

r y

x

( x y)

y

x








c a b

r x y

2 2 2

2 2 2

`

= +

= +

EXAMPLE



Solution


y

x

-2

-2 2

2


one place

y

x

2

2

2

-2

The radius is 4

CONTINUED






Domain 2 2 x x -

Range 2 2 y y -


range y r y r -



Proof


shown

y

x

(a b)

x

y

r

( x y)

a

b x - a

y - b




ndash ndash

c a b

r x a y b

2 2 2

2 2 2

= +

= +] ^g h




EXAMPLES



Solution



y

x 9

9

-9

-9



Domain 9 9 x x -

Range 9 9 y y -



Solution


ndash

ndash ndash

x y

x y

1 2 4

1 2 2

2

2 2

+ + =

+ - =

2

2

] ^

] ]_

g h

g gi

So 1 2a b= = - and 2r =

CONTINUED







y

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

(1 -2)



Domain 1 3 x x -

Range 4 0 y y -


Solution


a b2 1= - = and r 3=

Substituting

ndash ndash

ndash

ndash

x a y b r

x y

x y

2 1 3

2 1 9

2 2 2

2 2 2

2 2

+ =

- - + =

+ + =

] ^]] ^

] ^

g hg g hg h


ndash

ndash

a b a ab b

x x x

x x

a b a ab b

y y y

y y

2

2 2 2 2

4 4

2

1 2 1 1

2 1

So

So

2 2 2

2 2 2

2

2 2 2

2 2 2

2

+ = + +

+ = + +

= + +

= - +

= - +

= - +

]] ] ]]^ ^ ]

gg g ggh h g


ndash

x x y y

x x y y

x x y y

x x y y

4 4 2 1 9

4 2 5 9

4 2 5 9

4 2 4 0

9 9

2

2

2

2

+ + + - + =

+ + - + =

+ + + =

+ + - - =

- -


in Chapter 2




Investigation


two functions





circle






for all values

y

x r

r

-r


Proof

ndash

x y r

y r x y r x

2 2 2

2 2 2

2 2

+ =

=

= -



semi-circles





y 0 for all values

y

x r

-

r

-r


EXAMPLES


1 f x x92

= -] g

Solution



y

x 3

3

-3

Domain 3 3 x x -

Range 0 3 y y




2 y x4 2= - -

Solution



y

x 2

-2

-2

Domain 2 2 x x -

Range 2 0 y y -




range

(a) 9x y 2 2+ =

(b) x y 16 02 2+ =-


(d) 1 9x y 2 2+ + =

] g

(e) ndashx y 2 1 12 2+ + =] ^g h



below the x -axis



range

(a) 25 y x2= - -

(b) 1 y x2= -

(c) 36 y x2= -

(d) 64 y x2= - -

(e) 7 y x2= - -



each circle

(a) 100x y 2 2+ =

(b) 5x y 2 2+ =


(d) ndashx y 5 6 492 2+ + =] ^g h

(e) ndashx y 3 812 2+ =^ h

58 Exercises






grouping symbols)



Centre(c) 1 5-







Centre(j) 4 7- -

^ h and radius 3

Other Graphs




EXAMPLES


Solution



eg When 0x =

y 3

1

c=

=

x

y

1

3

3

1

31

When1

1

= -

=

=

=

-

x 3- 2- 1- 0 1 2 3

y 271

91

31

1 3 9 27



You will meet these

graphs again in the

HSC Course





other point

When

x

y

1

3

3

1

=

=

=

y

x

1

2

1

3







Solution



x minus2 minus1 0 05 1 2 3 4

y minus03 0 03 05 06

y

x

1

2

1 2 3 4

-1




You learned about


stages of maths





range y y 02



range all real y

Cubic function


EXAMPLE



Solution



y minus25 minus6 1 2 3 10 29

y

x

1

1

-2 2 3 4

-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4



If you already

know the shape of


3






Domain and range


EXAMPLE


Solution




function are

f

f

1 1 2

1 21

4 4 2

64 2

66

3

3

- = - +

= - +

=

= +

= +

=

] ]

] ]

g g

g g


these points

y

x

(-1 1)

(4 66)

Range 1 66 y y










positive number

EXAMPLE


Solution


x 4So $

$


(or zero) So y 0$


59 Exercises


(a) 4 3 y x= +

(b) f x 4= -] g

(c) 3x =



(f) f x xx 12 2= - -] g

(g) 64x y 2 2+ =

(h) f t t 4

3=

-] g

(i) ( ) g 2

5zz

= +

(j) | |f x x=] g


(a) y x=

(b) 2 y x= -

(c) | |f x x2 3= -] g

(d) | | 2 y x= -

(e) f x x2 5= - +] g

(f) | | y x5= -

(g) 2 y x=

(h) y 5x= -

(i) f x xx 1

= +] g

(j)2

4 3 y x

x= -


(a) y x x 5 2= -] g


= +-

(d) g x x x164 2= -] g

(e) 49x y 2 2+ =

You may like to


by dividing by x




4 (a) Solve x1 02$-


f x x1 2= -] g


(a) 2 y x x2= - -

(b) g t t t 62= +] g



in each case


x3 3 -


x2 3 -


x2 1 - (d)


x1 5


0 4x


x3 3 -


x1 1 -


x2 3 -


x4 4 -

(j) y x x7 62= - + - in the

domain 0 7x


function y x 1

3=

+





x=] g


function

find its range(b)


plane

(a) f x x4=] g

(b) y x3= -

(c) y x 34= -

(d) 2 p x x3=] g

(e) 1 g x x3= +] g

(f) 100x y 2 2+ =

(g) 2 1 y x= +


y x 1= -









(b) f x x 22= -] g

(c)1

y x=

(d) f x x3=] g

(e) f x 3x=] g

13 (a) Solve x4 02$-


(i) 4 y x2= -

(ii) y x4

2= - -




DID YOU KNOW




a light



Limits






a a1

1

03

Z

=

=

3

3

-


lim a 0x=

x 3-



asymptote

EXAMPLES

1 Find lim x

x x5x 0

2+

Solution


which is undefined


( )

lim lim

lim

xx x

x

x x

x

5 5

5

5

x x

x

0

2

0 1

1

0

+=

+

= +

=

] g




2 Find limx

x

4

22

-

-

x 2

Solution


lim lim

lim

x

x

x x

x

x

4

2

2 2

2

21

41

2 1

1

-

-=

+ -

-

=+

=

x x2 2

x 2

^ _h i

3 Find limh

h x hx h2 72 2+ -

h 0

Solution

lim lim

lim

h

h x hx h

h

h hx x

hx x

x

2 7 2 7

2 7

7

2 2 2

2

2

+ -=

+ -

= + -

= -

h 0

h h0 0

^ h

Continuity





EXAMPLES

1 Find lim

x

x

1

12

-

-

x 1


11

y x

x2

=-

- Sketch the curve

Solution


xx2

-

- gives

00

CONTINUED




( )

lim lim

limx

x

x

x x

x11

1

1 1

1

2

x x

x

1

2

1

1

-

-=

-

+ -

= +

=

-

] ]g g

11

y xx2

=-





y xx

x

x x

x

11

1

1 1

1

2

=-

-

=

+

= +

-

-^ ^h h


2 Find limx

x x2

2x 2

2

+

+ -

-


x x2

22

= + -

+

Solution


x x2

22

+

+ - gives

00

lim lim

lim

xx x

x

x x

x

22

2

1 2

1

3

x x

x

2

2

2

2

+

+ -=

+

- +

=

= -

-

- -

-

^^ ^

^

hh h

h2 y

x

x xx

y x

x

x

x

22

2

2

1

1


=+

+ -= -

=+

= -

+ -^ ^h h


Remember that x 1




1 Find(a) lim x 52

+x 4

(b) lim t 7-t 3 -

(c) lim x x2 43+ -

x 2

(d) lim xx x32

+

x 0

(e) limh

h h

2

22

-

- -

h 2

(f) lim y

y

5

1253

-

-

y 5

(g) limx

x x

12 12

+

+ +

x 1-

(h) limx

x x4

2 82

+

+ -

x 4 -

(i) limc

c

4

22

-

-

c 2

(j) limx x

x 12

-

-

x 1

(k) lim h

h h h2 73 2+ -

h 0

(l) limh

hx hx h32 2- +

h 0

(m) limh

hx h x hx h2 3 53 2 2- + -

h 0

(n) lim x c x c 3 3

-

-

x c



discontinuous

(a) 3 y x2= -

(b)1

1 y

x=

+

(c) f x x 1= -] g

(d)4

1 y

x2=

+

(e)4

1

y x2=

-



(a)3

y xx x2

= +

(b)33

y x

x x2

=+

+

(c)1

5 4 y

xx x2

=+

+ +

510 Exercises




Regions

Class Investigation


them all







EXAMPLE


Solution



y

x =3

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4







region

EXAMPLE


Solution



included


shown

y

x

1

1

-2 2 3 4

-3

-4

-5

2

3

4

5

-1

-2

-3-4

-1 y = -1



y -intercept 1-






EXAMPLES


1 y x 2$ +

Solution






y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0



2 x y 2 3 61-

Solution


region


side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g








3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g






EXAMPLES


Solution

f x x

x

f x

f x x

3

3


2

2

2`

- = - +

= +

=

= +

] ]]

]

g gg

g


Solution

f x x x

x x

x x

f x


3

3

3

3`

- = - - -

= - +

= - -

= -

= -

] ] ]^]

]

g g gh

gg

Investigation






1 f x kxn=] g

2 f x x kn= +] g

3 f x x k n= +

] ]g g






shape of graphs


each function

(a) y x3 2= -

(b) x y 2 5 20 0- + =

(c) x y 3 12 0+ - =

(d) f x x x32= +] g

(e) f x x 42= -] g

(f) p x x x5 62= + +] g

(g) y x x8 152= - +

(h) p x x 53

= +

] g

53 Exercises




(i) y xx

x3

0= + ] g

(j) g x x9 2= -] g


f x x 22= -

] g What type of

function is it


(a) f x2^ h (b) ( )f x 26



an even function


f x x=] g


function


odd function



f x f x- -] ]g g


neither

(a) y x x

x4 2

3

=

-

(b) y x 1

13

=

-

(c) f xx 4

32

=

-

] g

(d) y x

x

33

=+

-

(e) f x x x

x5 2

3

=-] g



f x xn=] g

even(a)

odd(b)


ever be

even(a)

odd(b)



graph is increasing




x

y(a)

x

4

y(b)

2-2

x

y(c)




Investigation


graph







y kx

y kx

y kx

y kx

y xk

(a)

(b)

(c)

(d)

(e)

2

3

4

=

=

=

=

=



y x k

y x k

y x k

y x k

y x k1

(a)

(b)

(c)

(d)

(e)

2

2

3

4

= +

= +

= +

= +

= +

] g



-2

1 2

-4

-1-2

2

4

y

x

(d) y

x

(e)

CONTINUED




Gradient form



Investigation










y x k

y x k

y x k

y x k y

x k

1

(a)

(b)

(c)

(d)(e)

2

3

4

= +

= +

= +

= +

=+

]]

]

gg

g





the right








Linear Function





EXAMPLE



Solution


Find the intercepts


0 3 5

5 3

1

x

x

x32

=

=

=

-


3 5

5

y 0=

= -

-] g

-1

-2

y

5

4

3

2

1 1 23

6

-3

-4

-5

1 4-1-2 32-3-4

x



Range all real y








Special lines






1 y kx=

2 y x k= +






EXAMPLES


Solution




-1

-3

y

4

3

2

1

5

-2

-4

-5

1 4-1-2

x

32-3-4


2 Sketch x 1= -


Solution





-

-

4

3

2

5

-2

-4

-5

1 4-1-2-4

y

x







Domain all real x

Range y y b=

54 Exercises


each function

(a) y x 2= -

(b) f x x2 3= +] g (c) x y 2 1 0+ =-

(d) x y 3 0+ =-

(e) x y 3 6 2 0=- -


line

(a) x 4=

(b) x 3 0=-

(c) y 5=

(d) y 1 0+ =

(e) f x x2 1= -] g (f) y x 4= +

(g) f x x3 2= +] g

(h) x y 3+ =

(i) x y 1 0=- -

(j) x y 2 3 0+ =-


(a) x y 3 2 7 0+ =-

(b) y 2=

(c) x 4= - (d) x 2 0=-

(e) y 3 0=-


are even or odd

(a) y x2=

(b) y 3=

(c) x 4=

(d) y x= -

(e) y x=




meet




Applications






The pronumeral

a is called the


Quadratic Function










EXAMPLES



Solution



x

x

x

0 1

1

1

2

2

= -

=

=


0 1

1

y 2= -

= -



Domain xall real

Range y y 1$ -


Solution


parabola will lie




left



0

1 0

1

x

x

x

1 2= +

+ =

= -

] g


1

y 0 1 2= +

=

] g

-1

-

4

3

2

1

5

-2

-4

-5

-

1 41-2 5-4

y

x









Solution



x x

x xx x

x x

0 6

3 23 0 2 0

3 2

2= + -

= + -

+ = - =

= - =

] ]g g


f 0 0 0 66

2= + -

= -

] ] ]g g g



x -intercepts


23 2

21- +

= -


-c m

f 21

21

21

6

41

21

6

641

2

- = - + - -

= - -

= -

c c cm m m


-

CONTINUED

You will learn more


-1

-3

4

3

2

1

5

-2

-4

-5

1 4-1-2 32-3-4

y

x










41



2

11 -





Solution


So f x 0=] g

0 4 54 5 0

0

x xx x

x x5 1

2

2

= - + +

=

+ =

- -

-] ]g g

x x

x x

5 0 1 0

5 1

- = + =

= = -


f 0 0 4 0 5

5

2= - + +

=

] ] ]g g g

-1

-3

4

2

5

-2

-4

-5

-

y

1 41-2 5-4

x

-2 4






21 5

2- +

=

f 2 2 4 2 59

= - + +

=

2] ] ]g g g










each function

(a) 2 y x x2= +

(b) 3 y x x2= - +

(c) f x x 12= -] g

(d) y x x 22= - -

(e) y x x9 82= +-

2 Sketch

(a) 2 y x2= +

(b) y x 12= - +

(c) f x x 42= -] g

(d) 2 y x x2= +

(e) y x x2= - -

(f) f x x 3= -2] ]g g

55 Exercises

-1

9

8

7

5

4

3

2

6

1

-2

-3

-4

-5

y

2 51 643-1-2-3-4

x




EXAMPLES


Solution









(g) f x x 1 2= +] ]g g

(h) y x x3 42= + -

(i) y x x2 5 32= - +

(j) f x x x3 22= - + -] g



(a) ndash y x x7 122= +

(b) f x x x42= +] g

(c) y x x2 82= - -

(d) y x x6 92= +-

(e) f t t 4 2= -] g


(a) y x 52= -

(b) f x x x6

2= -

] g (c) f x x x 22= - -] g

(d) y x2= -

(e) f x x 7 2= -] ]g g



(a) y x2= for x0 3

(b) y x 42= - + for x1 2 -

(c) f x x 12= -] g for x2 5 -

(d) y x x2 32= + - for x2 4 -

(e) y x x

22= - +- for

x0 4


function is

increasing(i)

decreasing(ii)

(a) y x2=

(b) y x2

= - (c) f x x 92

= -] g

(d) y x x42= - +

(e) f x x 5 2= +] ]g g


function



(a) y x 12= +

(b) f x x 32= -

] g

(c) y x2 2= -

(d) f x x x32= -] g

(e) f x x x2= +] g

(f) y x 42= -

(g) y x x2 32= - -

(h) y x x5 42= +-

(i) p x x 1 2= +] ]g g

(j) y x 2= -2] g




CONTINUED

eg When x 3= -

| | y 3 13 12

= - -

= -

=

x -3 -2 -1 0 1 2 3

y 2 1 0 -1 0 1 2


y

-2

4

3

2

1

5

-1

-3

-4

-5

1 4-1-2 32-3-4

x


| | y x

x x

x x1

1 0

1 0

when

when 1

$= - =

-


y x x1 0$= - ] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x - 1




y x 1= - - x 01] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y =- x - 1



-3

4

3

2

1

5

-2

-1

-4

-5

y y

3-1-2 421-3-4

x

y = - x - 1

y = x - 1


intercepts


So f x 0=

] g

| |

| |

x

x

x

0 1

1

1`

= -

=

=


( ) | |f 0 0 11

= -

= -





-3

4

3

2

1

5

-2

-1

-4

-5

y

4-2 5321-1-3-4

x



Domain all real x

Range y y 1$ -

2 Sketch | | y x 2= +

Solution


| | ( ) y x x xx x2 2 2 0

2 2 0whenwhen 1

$= + = + +

- + +


2 y x= + when x 2 0$+

x 2$ -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x + 2






y-intercept of 1-

CONTINUED




2 y x= - +] g when x 2 01+

ie y x 2= - - when x 21 -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2


-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2

y = x + 2



So 0f x =] g

0 | 2 |

0 2

2

x

x

x

= +

= +

- =


(0) | 0 2 |

2

f = +

=

There is only one

solution for the


Can you see why





-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

If you know how to

sketch the graph of




Investigation




are their equations





1 | |f x k x=] g

2 | |f x x k= +] g

3 | |f x x k= +] g







EXAMPLES

Solve

1 |2 1 | 3x - =

Solution



is x 1 2= -

2 |2 1 | 3 2x x= -+

Solution



3 | 1 | 2x 1+

Solution






and then check them







each function

(a) | | y x=

(b) | |f x x 7= +] g

(c) | |f x x 2= -] g

(d) 5 | | y x=

(e) | |f x x 3= - +] g

(f) | 6 | y x= +

(g) | |f x x3 2= -] g

(h) | 5 4 | y x= +

(i) | 7 1 | y x= -

(j) | |f x x2 9= +] g


plane

(a) | | y x=

(b) | |f x x 1= +] g

(c) | |f x x 3= -] g

(d) 2 | | y x=

(e) | |f x x= -] g

(f) | 1 | y x= +

(g) | |f x x 1= - -] g

(h) | 2 3 | y x= -

(i) | 4 2 | y x= +

(j) | |f x x3 1= +] g


each function

(a) | 1 | y x= -

(b) | |f x x 8= -] g

(c) | |f x x2 5= +] g

(d) 2 | | 3 y x= -

(e) | |f x x 3= - -] g


function is

increasing(i)

decreasing(ii)

(a) | 2 | y x= -

(b) | |f x x 2= +

] g

(c) | |f x x2 3= -] g

(d) 4 | | 1 y x= -

(e) | |f x x= -] g


of each function

(a) | | y x= for x2 2 -

(b) | |f x x 4= - -] g for

x4 3 -

(c) | |f x x 4= +] g for x7 2 -

(d) | 2 5 | y x= - for x3 3 -

(e) | |f x x= -] g for x1 1 -


function increasing

(a) | 3 | y x= +

(b) | |f x x 4= - +] g

(c) | |f x x 9= -] g

(d) | | y x 2 1= - -

(e) | |f x x 2= - +] g

56 Exercises




7 Solve graphically

(a) | | 3x =

(b) | |x 12

(c) | |x 2

(d) | 2 | 1x + =

(e) | 3 | 0x- =

(f) |2 3 | 1x - =

(g) | |x 1 41-

(h) | |x 1 3+

(i) | |x 2 22-

(j) | |x 3 1$-

(k) | |x2 3 5+

(l) | |x2 1 1$-

(m) |3 1 | 3x x- = +

(n) |3 2 | 4x x- = -

(o) | 1 | 1x x- = +

(p) | 3 | 2 2x x+ = + (q) |2 1 | 1x x+ = -

(r) |2 5 | 3x x- = -

(s) | 1 | 2x x- =

(t) |2 3 | 3x x- = +

The Hyperbola


or= =

EXAMPLE

Sketch1

y x=

Solution



x -3 -2 -121-

41- 0

41

21 1 2 3

y 3

1-

2

1- -1 -2 -4 mdash 4 2 1

2

1

3

1

Class Discussion



y is never 0 Why






but never touches)



Investigation




equations




1 y xk

=

21

y x k= +

31

y x k

=+


EXAMPLES


3=

-

] g


Solution



So x 3






be zero

CONTINUED







1

y 0 3

3=

-

= -

-3

4

3

2

1

5

-2

-1

-4

-5

y

-1-2 4 521-3-4

x

x = 3

y = 0

Asymptotes

3

2 Sketch y x2 4

1= -

+

Solution




x

x

2 4

2

-

-






( ) y

2 0 41

41

= -+

= -


a translation of3

y x

=





y

-2

x

-

1

4


a=

+


domain x xb



1 For each graph



exists


(a)2

y x=

(b)1

y x= -

(c) f xx 1

1=

+] g

(d) f xx 2

3=

-

] g

(e)3 6

1 y

x=

+

(f) f xx 3

2= -

-

] g

(g) f xx 1

4=

-

] g

(h)1

2 y

x= -

+

(i) f xx6 3

2=

-

] g

(j)2

6 y

x= -

+

2 Show that f x x2

=] g is an odd

function



(a) f xx2 5

1=

+

] g for x2 2 -

(b)3

1 y

x=

+ for x2 0 -

(c) f xx2 4

5=

-

] g for x3 1 -

57 Exercises




(d) f xx 4

3= -

-

] g for x3 3 -

(e)3 1

2 y

x= -

+ for x0 5



(a)3

y x= for y 1 3

(b)2

y x= - for y 221

- -

(c) f xx 1

1=

-

] g for y 171

- -

(d) f xx2 1

3= -

+

] g for

y 131

- -

(e)3 2

6 y

x=

- for y 1

21

6



Circle gate





Proof

r y

x

( x y)

y

x








c a b

r x y

2 2 2

2 2 2

`

= +

= +

EXAMPLE



Solution


y

x

-2

-2 2

2


one place

y

x

2

2

2

-2

The radius is 4

CONTINUED






Domain 2 2 x x -

Range 2 2 y y -


range y r y r -



Proof


shown

y

x

(a b)

x

y

r

( x y)

a

b x - a

y - b




ndash ndash

c a b

r x a y b

2 2 2

2 2 2

= +

= +] ^g h




EXAMPLES



Solution



y

x 9

9

-9

-9



Domain 9 9 x x -

Range 9 9 y y -



Solution


ndash

ndash ndash

x y

x y

1 2 4

1 2 2

2

2 2

+ + =

+ - =

2

2

] ^

] ]_

g h

g gi

So 1 2a b= = - and 2r =

CONTINUED







y

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

(1 -2)



Domain 1 3 x x -

Range 4 0 y y -


Solution


a b2 1= - = and r 3=

Substituting

ndash ndash

ndash

ndash

x a y b r

x y

x y

2 1 3

2 1 9

2 2 2

2 2 2

2 2

+ =

- - + =

+ + =

] ^]] ^

] ^

g hg g hg h


ndash

ndash

a b a ab b

x x x

x x

a b a ab b

y y y

y y

2

2 2 2 2

4 4

2

1 2 1 1

2 1

So

So

2 2 2

2 2 2

2

2 2 2

2 2 2

2

+ = + +

+ = + +

= + +

= - +

= - +

= - +

]] ] ]]^ ^ ]

gg g ggh h g


ndash

x x y y

x x y y

x x y y

x x y y

4 4 2 1 9

4 2 5 9

4 2 5 9

4 2 4 0

9 9

2

2

2

2

+ + + - + =

+ + - + =

+ + + =

+ + - - =

- -


in Chapter 2




Investigation


two functions





circle






for all values

y

x r

r

-r


Proof

ndash

x y r

y r x y r x

2 2 2

2 2 2

2 2

+ =

=

= -



semi-circles





y 0 for all values

y

x r

-

r

-r


EXAMPLES


1 f x x92

= -] g

Solution



y

x 3

3

-3

Domain 3 3 x x -

Range 0 3 y y




2 y x4 2= - -

Solution



y

x 2

-2

-2

Domain 2 2 x x -

Range 2 0 y y -




range

(a) 9x y 2 2+ =

(b) x y 16 02 2+ =-


(d) 1 9x y 2 2+ + =

] g

(e) ndashx y 2 1 12 2+ + =] ^g h



below the x -axis



range

(a) 25 y x2= - -

(b) 1 y x2= -

(c) 36 y x2= -

(d) 64 y x2= - -

(e) 7 y x2= - -



each circle

(a) 100x y 2 2+ =

(b) 5x y 2 2+ =


(d) ndashx y 5 6 492 2+ + =] ^g h

(e) ndashx y 3 812 2+ =^ h

58 Exercises






grouping symbols)



Centre(c) 1 5-







Centre(j) 4 7- -

^ h and radius 3

Other Graphs




EXAMPLES


Solution



eg When 0x =

y 3

1

c=

=

x

y

1

3

3

1

31

When1

1

= -

=

=

=

-

x 3- 2- 1- 0 1 2 3

y 271

91

31

1 3 9 27



You will meet these

graphs again in the

HSC Course





other point

When

x

y

1

3

3

1

=

=

=

y

x

1

2

1

3







Solution



x minus2 minus1 0 05 1 2 3 4

y minus03 0 03 05 06

y

x

1

2

1 2 3 4

-1




You learned about


stages of maths





range y y 02



range all real y

Cubic function


EXAMPLE



Solution



y minus25 minus6 1 2 3 10 29

y

x

1

1

-2 2 3 4

-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4



If you already

know the shape of


3






Domain and range


EXAMPLE


Solution




function are

f

f

1 1 2

1 21

4 4 2

64 2

66

3

3

- = - +

= - +

=

= +

= +

=

] ]

] ]

g g

g g


these points

y

x

(-1 1)

(4 66)

Range 1 66 y y










positive number

EXAMPLE


Solution


x 4So $

$


(or zero) So y 0$


59 Exercises


(a) 4 3 y x= +

(b) f x 4= -] g

(c) 3x =



(f) f x xx 12 2= - -] g

(g) 64x y 2 2+ =

(h) f t t 4

3=

-] g

(i) ( ) g 2

5zz

= +

(j) | |f x x=] g


(a) y x=

(b) 2 y x= -

(c) | |f x x2 3= -] g

(d) | | 2 y x= -

(e) f x x2 5= - +] g

(f) | | y x5= -

(g) 2 y x=

(h) y 5x= -

(i) f x xx 1

= +] g

(j)2

4 3 y x

x= -


(a) y x x 5 2= -] g


= +-

(d) g x x x164 2= -] g

(e) 49x y 2 2+ =

You may like to


by dividing by x




4 (a) Solve x1 02$-


f x x1 2= -] g


(a) 2 y x x2= - -

(b) g t t t 62= +] g



in each case


x3 3 -


x2 3 -


x2 1 - (d)


x1 5


0 4x


x3 3 -


x1 1 -


x2 3 -


x4 4 -

(j) y x x7 62= - + - in the

domain 0 7x


function y x 1

3=

+





x=] g


function

find its range(b)


plane

(a) f x x4=] g

(b) y x3= -

(c) y x 34= -

(d) 2 p x x3=] g

(e) 1 g x x3= +] g

(f) 100x y 2 2+ =

(g) 2 1 y x= +


y x 1= -









(b) f x x 22= -] g

(c)1

y x=

(d) f x x3=] g

(e) f x 3x=] g

13 (a) Solve x4 02$-


(i) 4 y x2= -

(ii) y x4

2= - -




DID YOU KNOW




a light



Limits






a a1

1

03

Z

=

=

3

3

-


lim a 0x=

x 3-



asymptote

EXAMPLES

1 Find lim x

x x5x 0

2+

Solution


which is undefined


( )

lim lim

lim

xx x

x

x x

x

5 5

5

5

x x

x

0

2

0 1

1

0

+=

+

= +

=

] g




2 Find limx

x

4

22

-

-

x 2

Solution


lim lim

lim

x

x

x x

x

x

4

2

2 2

2

21

41

2 1

1

-

-=

+ -

-

=+

=

x x2 2

x 2

^ _h i

3 Find limh

h x hx h2 72 2+ -

h 0

Solution

lim lim

lim

h

h x hx h

h

h hx x

hx x

x

2 7 2 7

2 7

7

2 2 2

2

2

+ -=

+ -

= + -

= -

h 0

h h0 0

^ h

Continuity





EXAMPLES

1 Find lim

x

x

1

12

-

-

x 1


11

y x

x2

=-

- Sketch the curve

Solution


xx2

-

- gives

00

CONTINUED




( )

lim lim

limx

x

x

x x

x11

1

1 1

1

2

x x

x

1

2

1

1

-

-=

-

+ -

= +

=

-

] ]g g

11

y xx2

=-





y xx

x

x x

x

11

1

1 1

1

2

=-

-

=

+

= +

-

-^ ^h h


2 Find limx

x x2

2x 2

2

+

+ -

-


x x2

22

= + -

+

Solution


x x2

22

+

+ - gives

00

lim lim

lim

xx x

x

x x

x

22

2

1 2

1

3

x x

x

2

2

2

2

+

+ -=

+

- +

=

= -

-

- -

-

^^ ^

^

hh h

h2 y

x

x xx

y x

x

x

x

22

2

2

1

1


=+

+ -= -

=+

= -

+ -^ ^h h


Remember that x 1




1 Find(a) lim x 52

+x 4

(b) lim t 7-t 3 -

(c) lim x x2 43+ -

x 2

(d) lim xx x32

+

x 0

(e) limh

h h

2

22

-

- -

h 2

(f) lim y

y

5

1253

-

-

y 5

(g) limx

x x

12 12

+

+ +

x 1-

(h) limx

x x4

2 82

+

+ -

x 4 -

(i) limc

c

4

22

-

-

c 2

(j) limx x

x 12

-

-

x 1

(k) lim h

h h h2 73 2+ -

h 0

(l) limh

hx hx h32 2- +

h 0

(m) limh

hx h x hx h2 3 53 2 2- + -

h 0

(n) lim x c x c 3 3

-

-

x c



discontinuous

(a) 3 y x2= -

(b)1

1 y

x=

+

(c) f x x 1= -] g

(d)4

1 y

x2=

+

(e)4

1

y x2=

-



(a)3

y xx x2

= +

(b)33

y x

x x2

=+

+

(c)1

5 4 y

xx x2

=+

+ +

510 Exercises




Regions

Class Investigation


them all







EXAMPLE


Solution



y

x =3

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4







region

EXAMPLE


Solution



included


shown

y

x

1

1

-2 2 3 4

-3

-4

-5

2

3

4

5

-1

-2

-3-4

-1 y = -1



y -intercept 1-






EXAMPLES


1 y x 2$ +

Solution






y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0



2 x y 2 3 61-

Solution


region


side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g








3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g






(i) y xx

x3

0= + ] g

(j) g x x9 2= -] g


f x x 22= -

] g What type of

function is it


(a) f x2^ h (b) ( )f x 26



an even function


f x x=] g


function


odd function



f x f x- -] ]g g


neither

(a) y x x

x4 2

3

=

-

(b) y x 1

13

=

-

(c) f xx 4

32

=

-

] g

(d) y x

x

33

=+

-

(e) f x x x

x5 2

3

=-] g



f x xn=] g

even(a)

odd(b)


ever be

even(a)

odd(b)



graph is increasing




x

y(a)

x

4

y(b)

2-2

x

y(c)




Investigation


graph







y kx

y kx

y kx

y kx

y xk

(a)

(b)

(c)

(d)

(e)

2

3

4

=

=

=

=

=



y x k

y x k

y x k

y x k

y x k1

(a)

(b)

(c)

(d)

(e)

2

2

3

4

= +

= +

= +

= +

= +

] g



-2

1 2

-4

-1-2

2

4

y

x

(d) y

x

(e)

CONTINUED




Gradient form



Investigation










y x k

y x k

y x k

y x k y

x k

1

(a)

(b)

(c)

(d)(e)

2

3

4

= +

= +

= +

= +

=+

]]

]

gg

g





the right








Linear Function





EXAMPLE



Solution


Find the intercepts


0 3 5

5 3

1

x

x

x32

=

=

=

-


3 5

5

y 0=

= -

-] g

-1

-2

y

5

4

3

2

1 1 23

6

-3

-4

-5

1 4-1-2 32-3-4

x



Range all real y








Special lines






1 y kx=

2 y x k= +






EXAMPLES


Solution




-1

-3

y

4

3

2

1

5

-2

-4

-5

1 4-1-2

x

32-3-4


2 Sketch x 1= -


Solution





-

-

4

3

2

5

-2

-4

-5

1 4-1-2-4

y

x







Domain all real x

Range y y b=

54 Exercises


each function

(a) y x 2= -

(b) f x x2 3= +] g (c) x y 2 1 0+ =-

(d) x y 3 0+ =-

(e) x y 3 6 2 0=- -


line

(a) x 4=

(b) x 3 0=-

(c) y 5=

(d) y 1 0+ =

(e) f x x2 1= -] g (f) y x 4= +

(g) f x x3 2= +] g

(h) x y 3+ =

(i) x y 1 0=- -

(j) x y 2 3 0+ =-


(a) x y 3 2 7 0+ =-

(b) y 2=

(c) x 4= - (d) x 2 0=-

(e) y 3 0=-


are even or odd

(a) y x2=

(b) y 3=

(c) x 4=

(d) y x= -

(e) y x=




meet




Applications






The pronumeral

a is called the


Quadratic Function










EXAMPLES



Solution



x

x

x

0 1

1

1

2

2

= -

=

=


0 1

1

y 2= -

= -



Domain xall real

Range y y 1$ -


Solution


parabola will lie




left



0

1 0

1

x

x

x

1 2= +

+ =

= -

] g


1

y 0 1 2= +

=

] g

-1

-

4

3

2

1

5

-2

-4

-5

-

1 41-2 5-4

y

x









Solution



x x

x xx x

x x

0 6

3 23 0 2 0

3 2

2= + -

= + -

+ = - =

= - =

] ]g g


f 0 0 0 66

2= + -

= -

] ] ]g g g



x -intercepts


23 2

21- +

= -


-c m

f 21

21

21

6

41

21

6

641

2

- = - + - -

= - -

= -

c c cm m m


-

CONTINUED

You will learn more


-1

-3

4

3

2

1

5

-2

-4

-5

1 4-1-2 32-3-4

y

x










41



2

11 -





Solution


So f x 0=] g

0 4 54 5 0

0

x xx x

x x5 1

2

2

= - + +

=

+ =

- -

-] ]g g

x x

x x

5 0 1 0

5 1

- = + =

= = -


f 0 0 4 0 5

5

2= - + +

=

] ] ]g g g

-1

-3

4

2

5

-2

-4

-5

-

y

1 41-2 5-4

x

-2 4






21 5

2- +

=

f 2 2 4 2 59

= - + +

=

2] ] ]g g g










each function

(a) 2 y x x2= +

(b) 3 y x x2= - +

(c) f x x 12= -] g

(d) y x x 22= - -

(e) y x x9 82= +-

2 Sketch

(a) 2 y x2= +

(b) y x 12= - +

(c) f x x 42= -] g

(d) 2 y x x2= +

(e) y x x2= - -

(f) f x x 3= -2] ]g g

55 Exercises

-1

9

8

7

5

4

3

2

6

1

-2

-3

-4

-5

y

2 51 643-1-2-3-4

x




EXAMPLES


Solution









(g) f x x 1 2= +] ]g g

(h) y x x3 42= + -

(i) y x x2 5 32= - +

(j) f x x x3 22= - + -] g



(a) ndash y x x7 122= +

(b) f x x x42= +] g

(c) y x x2 82= - -

(d) y x x6 92= +-

(e) f t t 4 2= -] g


(a) y x 52= -

(b) f x x x6

2= -

] g (c) f x x x 22= - -] g

(d) y x2= -

(e) f x x 7 2= -] ]g g



(a) y x2= for x0 3

(b) y x 42= - + for x1 2 -

(c) f x x 12= -] g for x2 5 -

(d) y x x2 32= + - for x2 4 -

(e) y x x

22= - +- for

x0 4


function is

increasing(i)

decreasing(ii)

(a) y x2=

(b) y x2

= - (c) f x x 92

= -] g

(d) y x x42= - +

(e) f x x 5 2= +] ]g g


function



(a) y x 12= +

(b) f x x 32= -

] g

(c) y x2 2= -

(d) f x x x32= -] g

(e) f x x x2= +] g

(f) y x 42= -

(g) y x x2 32= - -

(h) y x x5 42= +-

(i) p x x 1 2= +] ]g g

(j) y x 2= -2] g




CONTINUED

eg When x 3= -

| | y 3 13 12

= - -

= -

=

x -3 -2 -1 0 1 2 3

y 2 1 0 -1 0 1 2


y

-2

4

3

2

1

5

-1

-3

-4

-5

1 4-1-2 32-3-4

x


| | y x

x x

x x1

1 0

1 0

when

when 1

$= - =

-


y x x1 0$= - ] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x - 1




y x 1= - - x 01] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y =- x - 1



-3

4

3

2

1

5

-2

-1

-4

-5

y y

3-1-2 421-3-4

x

y = - x - 1

y = x - 1


intercepts


So f x 0=

] g

| |

| |

x

x

x

0 1

1

1`

= -

=

=


( ) | |f 0 0 11

= -

= -





-3

4

3

2

1

5

-2

-1

-4

-5

y

4-2 5321-1-3-4

x



Domain all real x

Range y y 1$ -

2 Sketch | | y x 2= +

Solution


| | ( ) y x x xx x2 2 2 0

2 2 0whenwhen 1

$= + = + +

- + +


2 y x= + when x 2 0$+

x 2$ -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x + 2






y-intercept of 1-

CONTINUED




2 y x= - +] g when x 2 01+

ie y x 2= - - when x 21 -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2


-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2

y = x + 2



So 0f x =] g

0 | 2 |

0 2

2

x

x

x

= +

= +

- =


(0) | 0 2 |

2

f = +

=

There is only one

solution for the


Can you see why





-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

If you know how to

sketch the graph of




Investigation




are their equations





1 | |f x k x=] g

2 | |f x x k= +] g

3 | |f x x k= +] g







EXAMPLES

Solve

1 |2 1 | 3x - =

Solution



is x 1 2= -

2 |2 1 | 3 2x x= -+

Solution



3 | 1 | 2x 1+

Solution






and then check them







each function

(a) | | y x=

(b) | |f x x 7= +] g

(c) | |f x x 2= -] g

(d) 5 | | y x=

(e) | |f x x 3= - +] g

(f) | 6 | y x= +

(g) | |f x x3 2= -] g

(h) | 5 4 | y x= +

(i) | 7 1 | y x= -

(j) | |f x x2 9= +] g


plane

(a) | | y x=

(b) | |f x x 1= +] g

(c) | |f x x 3= -] g

(d) 2 | | y x=

(e) | |f x x= -] g

(f) | 1 | y x= +

(g) | |f x x 1= - -] g

(h) | 2 3 | y x= -

(i) | 4 2 | y x= +

(j) | |f x x3 1= +] g


each function

(a) | 1 | y x= -

(b) | |f x x 8= -] g

(c) | |f x x2 5= +] g

(d) 2 | | 3 y x= -

(e) | |f x x 3= - -] g


function is

increasing(i)

decreasing(ii)

(a) | 2 | y x= -

(b) | |f x x 2= +

] g

(c) | |f x x2 3= -] g

(d) 4 | | 1 y x= -

(e) | |f x x= -] g


of each function

(a) | | y x= for x2 2 -

(b) | |f x x 4= - -] g for

x4 3 -

(c) | |f x x 4= +] g for x7 2 -

(d) | 2 5 | y x= - for x3 3 -

(e) | |f x x= -] g for x1 1 -


function increasing

(a) | 3 | y x= +

(b) | |f x x 4= - +] g

(c) | |f x x 9= -] g

(d) | | y x 2 1= - -

(e) | |f x x 2= - +] g

56 Exercises




7 Solve graphically

(a) | | 3x =

(b) | |x 12

(c) | |x 2

(d) | 2 | 1x + =

(e) | 3 | 0x- =

(f) |2 3 | 1x - =

(g) | |x 1 41-

(h) | |x 1 3+

(i) | |x 2 22-

(j) | |x 3 1$-

(k) | |x2 3 5+

(l) | |x2 1 1$-

(m) |3 1 | 3x x- = +

(n) |3 2 | 4x x- = -

(o) | 1 | 1x x- = +

(p) | 3 | 2 2x x+ = + (q) |2 1 | 1x x+ = -

(r) |2 5 | 3x x- = -

(s) | 1 | 2x x- =

(t) |2 3 | 3x x- = +

The Hyperbola


or= =

EXAMPLE

Sketch1

y x=

Solution



x -3 -2 -121-

41- 0

41

21 1 2 3

y 3

1-

2

1- -1 -2 -4 mdash 4 2 1

2

1

3

1

Class Discussion



y is never 0 Why






but never touches)



Investigation




equations




1 y xk

=

21

y x k= +

31

y x k

=+


EXAMPLES


3=

-

] g


Solution



So x 3






be zero

CONTINUED







1

y 0 3

3=

-

= -

-3

4

3

2

1

5

-2

-1

-4

-5

y

-1-2 4 521-3-4

x

x = 3

y = 0

Asymptotes

3

2 Sketch y x2 4

1= -

+

Solution




x

x

2 4

2

-

-






( ) y

2 0 41

41

= -+

= -


a translation of3

y x

=





y

-2

x

-

1

4


a=

+


domain x xb



1 For each graph



exists


(a)2

y x=

(b)1

y x= -

(c) f xx 1

1=

+] g

(d) f xx 2

3=

-

] g

(e)3 6

1 y

x=

+

(f) f xx 3

2= -

-

] g

(g) f xx 1

4=

-

] g

(h)1

2 y

x= -

+

(i) f xx6 3

2=

-

] g

(j)2

6 y

x= -

+

2 Show that f x x2

=] g is an odd

function



(a) f xx2 5

1=

+

] g for x2 2 -

(b)3

1 y

x=

+ for x2 0 -

(c) f xx2 4

5=

-

] g for x3 1 -

57 Exercises




(d) f xx 4

3= -

-

] g for x3 3 -

(e)3 1

2 y

x= -

+ for x0 5



(a)3

y x= for y 1 3

(b)2

y x= - for y 221

- -

(c) f xx 1

1=

-

] g for y 171

- -

(d) f xx2 1

3= -

+

] g for

y 131

- -

(e)3 2

6 y

x=

- for y 1

21

6



Circle gate





Proof

r y

x

( x y)

y

x








c a b

r x y

2 2 2

2 2 2

`

= +

= +

EXAMPLE



Solution


y

x

-2

-2 2

2


one place

y

x

2

2

2

-2

The radius is 4

CONTINUED






Domain 2 2 x x -

Range 2 2 y y -


range y r y r -



Proof


shown

y

x

(a b)

x

y

r

( x y)

a

b x - a

y - b




ndash ndash

c a b

r x a y b

2 2 2

2 2 2

= +

= +] ^g h




EXAMPLES



Solution



y

x 9

9

-9

-9



Domain 9 9 x x -

Range 9 9 y y -



Solution


ndash

ndash ndash

x y

x y

1 2 4

1 2 2

2

2 2

+ + =

+ - =

2

2

] ^

] ]_

g h

g gi

So 1 2a b= = - and 2r =

CONTINUED







y

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

(1 -2)



Domain 1 3 x x -

Range 4 0 y y -


Solution


a b2 1= - = and r 3=

Substituting

ndash ndash

ndash

ndash

x a y b r

x y

x y

2 1 3

2 1 9

2 2 2

2 2 2

2 2

+ =

- - + =

+ + =

] ^]] ^

] ^

g hg g hg h


ndash

ndash

a b a ab b

x x x

x x

a b a ab b

y y y

y y

2

2 2 2 2

4 4

2

1 2 1 1

2 1

So

So

2 2 2

2 2 2

2

2 2 2

2 2 2

2

+ = + +

+ = + +

= + +

= - +

= - +

= - +

]] ] ]]^ ^ ]

gg g ggh h g


ndash

x x y y

x x y y

x x y y

x x y y

4 4 2 1 9

4 2 5 9

4 2 5 9

4 2 4 0

9 9

2

2

2

2

+ + + - + =

+ + - + =

+ + + =

+ + - - =

- -


in Chapter 2




Investigation


two functions





circle






for all values

y

x r

r

-r


Proof

ndash

x y r

y r x y r x

2 2 2

2 2 2

2 2

+ =

=

= -



semi-circles





y 0 for all values

y

x r

-

r

-r


EXAMPLES


1 f x x92

= -] g

Solution



y

x 3

3

-3

Domain 3 3 x x -

Range 0 3 y y




2 y x4 2= - -

Solution



y

x 2

-2

-2

Domain 2 2 x x -

Range 2 0 y y -




range

(a) 9x y 2 2+ =

(b) x y 16 02 2+ =-


(d) 1 9x y 2 2+ + =

] g

(e) ndashx y 2 1 12 2+ + =] ^g h



below the x -axis



range

(a) 25 y x2= - -

(b) 1 y x2= -

(c) 36 y x2= -

(d) 64 y x2= - -

(e) 7 y x2= - -



each circle

(a) 100x y 2 2+ =

(b) 5x y 2 2+ =


(d) ndashx y 5 6 492 2+ + =] ^g h

(e) ndashx y 3 812 2+ =^ h

58 Exercises






grouping symbols)



Centre(c) 1 5-







Centre(j) 4 7- -

^ h and radius 3

Other Graphs




EXAMPLES


Solution



eg When 0x =

y 3

1

c=

=

x

y

1

3

3

1

31

When1

1

= -

=

=

=

-

x 3- 2- 1- 0 1 2 3

y 271

91

31

1 3 9 27



You will meet these

graphs again in the

HSC Course





other point

When

x

y

1

3

3

1

=

=

=

y

x

1

2

1

3







Solution



x minus2 minus1 0 05 1 2 3 4

y minus03 0 03 05 06

y

x

1

2

1 2 3 4

-1




You learned about


stages of maths





range y y 02



range all real y

Cubic function


EXAMPLE



Solution



y minus25 minus6 1 2 3 10 29

y

x

1

1

-2 2 3 4

-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4



If you already

know the shape of


3






Domain and range


EXAMPLE


Solution




function are

f

f

1 1 2

1 21

4 4 2

64 2

66

3

3

- = - +

= - +

=

= +

= +

=

] ]

] ]

g g

g g


these points

y

x

(-1 1)

(4 66)

Range 1 66 y y










positive number

EXAMPLE


Solution


x 4So $

$


(or zero) So y 0$


59 Exercises


(a) 4 3 y x= +

(b) f x 4= -] g

(c) 3x =



(f) f x xx 12 2= - -] g

(g) 64x y 2 2+ =

(h) f t t 4

3=

-] g

(i) ( ) g 2

5zz

= +

(j) | |f x x=] g


(a) y x=

(b) 2 y x= -

(c) | |f x x2 3= -] g

(d) | | 2 y x= -

(e) f x x2 5= - +] g

(f) | | y x5= -

(g) 2 y x=

(h) y 5x= -

(i) f x xx 1

= +] g

(j)2

4 3 y x

x= -


(a) y x x 5 2= -] g


= +-

(d) g x x x164 2= -] g

(e) 49x y 2 2+ =

You may like to


by dividing by x




4 (a) Solve x1 02$-


f x x1 2= -] g


(a) 2 y x x2= - -

(b) g t t t 62= +] g



in each case


x3 3 -


x2 3 -


x2 1 - (d)


x1 5


0 4x


x3 3 -


x1 1 -


x2 3 -


x4 4 -

(j) y x x7 62= - + - in the

domain 0 7x


function y x 1

3=

+





x=] g


function

find its range(b)


plane

(a) f x x4=] g

(b) y x3= -

(c) y x 34= -

(d) 2 p x x3=] g

(e) 1 g x x3= +] g

(f) 100x y 2 2+ =

(g) 2 1 y x= +


y x 1= -









(b) f x x 22= -] g

(c)1

y x=

(d) f x x3=] g

(e) f x 3x=] g

13 (a) Solve x4 02$-


(i) 4 y x2= -

(ii) y x4

2= - -




DID YOU KNOW




a light



Limits






a a1

1

03

Z

=

=

3

3

-


lim a 0x=

x 3-



asymptote

EXAMPLES

1 Find lim x

x x5x 0

2+

Solution


which is undefined


( )

lim lim

lim

xx x

x

x x

x

5 5

5

5

x x

x

0

2

0 1

1

0

+=

+

= +

=

] g




2 Find limx

x

4

22

-

-

x 2

Solution


lim lim

lim

x

x

x x

x

x

4

2

2 2

2

21

41

2 1

1

-

-=

+ -

-

=+

=

x x2 2

x 2

^ _h i

3 Find limh

h x hx h2 72 2+ -

h 0

Solution

lim lim

lim

h

h x hx h

h

h hx x

hx x

x

2 7 2 7

2 7

7

2 2 2

2

2

+ -=

+ -

= + -

= -

h 0

h h0 0

^ h

Continuity





EXAMPLES

1 Find lim

x

x

1

12

-

-

x 1


11

y x

x2

=-

- Sketch the curve

Solution


xx2

-

- gives

00

CONTINUED




( )

lim lim

limx

x

x

x x

x11

1

1 1

1

2

x x

x

1

2

1

1

-

-=

-

+ -

= +

=

-

] ]g g

11

y xx2

=-





y xx

x

x x

x

11

1

1 1

1

2

=-

-

=

+

= +

-

-^ ^h h


2 Find limx

x x2

2x 2

2

+

+ -

-


x x2

22

= + -

+

Solution


x x2

22

+

+ - gives

00

lim lim

lim

xx x

x

x x

x

22

2

1 2

1

3

x x

x

2

2

2

2

+

+ -=

+

- +

=

= -

-

- -

-

^^ ^

^

hh h

h2 y

x

x xx

y x

x

x

x

22

2

2

1

1


=+

+ -= -

=+

= -

+ -^ ^h h


Remember that x 1




1 Find(a) lim x 52

+x 4

(b) lim t 7-t 3 -

(c) lim x x2 43+ -

x 2

(d) lim xx x32

+

x 0

(e) limh

h h

2

22

-

- -

h 2

(f) lim y

y

5

1253

-

-

y 5

(g) limx

x x

12 12

+

+ +

x 1-

(h) limx

x x4

2 82

+

+ -

x 4 -

(i) limc

c

4

22

-

-

c 2

(j) limx x

x 12

-

-

x 1

(k) lim h

h h h2 73 2+ -

h 0

(l) limh

hx hx h32 2- +

h 0

(m) limh

hx h x hx h2 3 53 2 2- + -

h 0

(n) lim x c x c 3 3

-

-

x c



discontinuous

(a) 3 y x2= -

(b)1

1 y

x=

+

(c) f x x 1= -] g

(d)4

1 y

x2=

+

(e)4

1

y x2=

-



(a)3

y xx x2

= +

(b)33

y x

x x2

=+

+

(c)1

5 4 y

xx x2

=+

+ +

510 Exercises




Regions

Class Investigation


them all







EXAMPLE


Solution



y

x =3

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4







region

EXAMPLE


Solution



included


shown

y

x

1

1

-2 2 3 4

-3

-4

-5

2

3

4

5

-1

-2

-3-4

-1 y = -1



y -intercept 1-






EXAMPLES


1 y x 2$ +

Solution






y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0



2 x y 2 3 61-

Solution


region


side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g








3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g






Investigation


graph







y kx

y kx

y kx

y kx

y xk

(a)

(b)

(c)

(d)

(e)

2

3

4

=

=

=

=

=



y x k

y x k

y x k

y x k

y x k1

(a)

(b)

(c)

(d)

(e)

2

2

3

4

= +

= +

= +

= +

= +

] g



-2

1 2

-4

-1-2

2

4

y

x

(d) y

x

(e)

CONTINUED




Gradient form



Investigation










y x k

y x k

y x k

y x k y

x k

1

(a)

(b)

(c)

(d)(e)

2

3

4

= +

= +

= +

= +

=+

]]

]

gg

g





the right








Linear Function





EXAMPLE



Solution


Find the intercepts


0 3 5

5 3

1

x

x

x32

=

=

=

-


3 5

5

y 0=

= -

-] g

-1

-2

y

5

4

3

2

1 1 23

6

-3

-4

-5

1 4-1-2 32-3-4

x



Range all real y








Special lines






1 y kx=

2 y x k= +






EXAMPLES


Solution




-1

-3

y

4

3

2

1

5

-2

-4

-5

1 4-1-2

x

32-3-4


2 Sketch x 1= -


Solution





-

-

4

3

2

5

-2

-4

-5

1 4-1-2-4

y

x







Domain all real x

Range y y b=

54 Exercises


each function

(a) y x 2= -

(b) f x x2 3= +] g (c) x y 2 1 0+ =-

(d) x y 3 0+ =-

(e) x y 3 6 2 0=- -


line

(a) x 4=

(b) x 3 0=-

(c) y 5=

(d) y 1 0+ =

(e) f x x2 1= -] g (f) y x 4= +

(g) f x x3 2= +] g

(h) x y 3+ =

(i) x y 1 0=- -

(j) x y 2 3 0+ =-


(a) x y 3 2 7 0+ =-

(b) y 2=

(c) x 4= - (d) x 2 0=-

(e) y 3 0=-


are even or odd

(a) y x2=

(b) y 3=

(c) x 4=

(d) y x= -

(e) y x=




meet




Applications






The pronumeral

a is called the


Quadratic Function










EXAMPLES



Solution



x

x

x

0 1

1

1

2

2

= -

=

=


0 1

1

y 2= -

= -



Domain xall real

Range y y 1$ -


Solution


parabola will lie




left



0

1 0

1

x

x

x

1 2= +

+ =

= -

] g


1

y 0 1 2= +

=

] g

-1

-

4

3

2

1

5

-2

-4

-5

-

1 41-2 5-4

y

x









Solution



x x

x xx x

x x

0 6

3 23 0 2 0

3 2

2= + -

= + -

+ = - =

= - =

] ]g g


f 0 0 0 66

2= + -

= -

] ] ]g g g



x -intercepts


23 2

21- +

= -


-c m

f 21

21

21

6

41

21

6

641

2

- = - + - -

= - -

= -

c c cm m m


-

CONTINUED

You will learn more


-1

-3

4

3

2

1

5

-2

-4

-5

1 4-1-2 32-3-4

y

x










41



2

11 -





Solution


So f x 0=] g

0 4 54 5 0

0

x xx x

x x5 1

2

2

= - + +

=

+ =

- -

-] ]g g

x x

x x

5 0 1 0

5 1

- = + =

= = -


f 0 0 4 0 5

5

2= - + +

=

] ] ]g g g

-1

-3

4

2

5

-2

-4

-5

-

y

1 41-2 5-4

x

-2 4






21 5

2- +

=

f 2 2 4 2 59

= - + +

=

2] ] ]g g g










each function

(a) 2 y x x2= +

(b) 3 y x x2= - +

(c) f x x 12= -] g

(d) y x x 22= - -

(e) y x x9 82= +-

2 Sketch

(a) 2 y x2= +

(b) y x 12= - +

(c) f x x 42= -] g

(d) 2 y x x2= +

(e) y x x2= - -

(f) f x x 3= -2] ]g g

55 Exercises

-1

9

8

7

5

4

3

2

6

1

-2

-3

-4

-5

y

2 51 643-1-2-3-4

x




EXAMPLES


Solution









(g) f x x 1 2= +] ]g g

(h) y x x3 42= + -

(i) y x x2 5 32= - +

(j) f x x x3 22= - + -] g



(a) ndash y x x7 122= +

(b) f x x x42= +] g

(c) y x x2 82= - -

(d) y x x6 92= +-

(e) f t t 4 2= -] g


(a) y x 52= -

(b) f x x x6

2= -

] g (c) f x x x 22= - -] g

(d) y x2= -

(e) f x x 7 2= -] ]g g



(a) y x2= for x0 3

(b) y x 42= - + for x1 2 -

(c) f x x 12= -] g for x2 5 -

(d) y x x2 32= + - for x2 4 -

(e) y x x

22= - +- for

x0 4


function is

increasing(i)

decreasing(ii)

(a) y x2=

(b) y x2

= - (c) f x x 92

= -] g

(d) y x x42= - +

(e) f x x 5 2= +] ]g g


function



(a) y x 12= +

(b) f x x 32= -

] g

(c) y x2 2= -

(d) f x x x32= -] g

(e) f x x x2= +] g

(f) y x 42= -

(g) y x x2 32= - -

(h) y x x5 42= +-

(i) p x x 1 2= +] ]g g

(j) y x 2= -2] g




CONTINUED

eg When x 3= -

| | y 3 13 12

= - -

= -

=

x -3 -2 -1 0 1 2 3

y 2 1 0 -1 0 1 2


y

-2

4

3

2

1

5

-1

-3

-4

-5

1 4-1-2 32-3-4

x


| | y x

x x

x x1

1 0

1 0

when

when 1

$= - =

-


y x x1 0$= - ] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x - 1




y x 1= - - x 01] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y =- x - 1



-3

4

3

2

1

5

-2

-1

-4

-5

y y

3-1-2 421-3-4

x

y = - x - 1

y = x - 1


intercepts


So f x 0=

] g

| |

| |

x

x

x

0 1

1

1`

= -

=

=


( ) | |f 0 0 11

= -

= -





-3

4

3

2

1

5

-2

-1

-4

-5

y

4-2 5321-1-3-4

x



Domain all real x

Range y y 1$ -

2 Sketch | | y x 2= +

Solution


| | ( ) y x x xx x2 2 2 0

2 2 0whenwhen 1

$= + = + +

- + +


2 y x= + when x 2 0$+

x 2$ -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x + 2






y-intercept of 1-

CONTINUED




2 y x= - +] g when x 2 01+

ie y x 2= - - when x 21 -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2


-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2

y = x + 2



So 0f x =] g

0 | 2 |

0 2

2

x

x

x

= +

= +

- =


(0) | 0 2 |

2

f = +

=

There is only one

solution for the


Can you see why





-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

If you know how to

sketch the graph of




Investigation




are their equations





1 | |f x k x=] g

2 | |f x x k= +] g

3 | |f x x k= +] g







EXAMPLES

Solve

1 |2 1 | 3x - =

Solution



is x 1 2= -

2 |2 1 | 3 2x x= -+

Solution



3 | 1 | 2x 1+

Solution






and then check them







each function

(a) | | y x=

(b) | |f x x 7= +] g

(c) | |f x x 2= -] g

(d) 5 | | y x=

(e) | |f x x 3= - +] g

(f) | 6 | y x= +

(g) | |f x x3 2= -] g

(h) | 5 4 | y x= +

(i) | 7 1 | y x= -

(j) | |f x x2 9= +] g


plane

(a) | | y x=

(b) | |f x x 1= +] g

(c) | |f x x 3= -] g

(d) 2 | | y x=

(e) | |f x x= -] g

(f) | 1 | y x= +

(g) | |f x x 1= - -] g

(h) | 2 3 | y x= -

(i) | 4 2 | y x= +

(j) | |f x x3 1= +] g


each function

(a) | 1 | y x= -

(b) | |f x x 8= -] g

(c) | |f x x2 5= +] g

(d) 2 | | 3 y x= -

(e) | |f x x 3= - -] g


function is

increasing(i)

decreasing(ii)

(a) | 2 | y x= -

(b) | |f x x 2= +

] g

(c) | |f x x2 3= -] g

(d) 4 | | 1 y x= -

(e) | |f x x= -] g


of each function

(a) | | y x= for x2 2 -

(b) | |f x x 4= - -] g for

x4 3 -

(c) | |f x x 4= +] g for x7 2 -

(d) | 2 5 | y x= - for x3 3 -

(e) | |f x x= -] g for x1 1 -


function increasing

(a) | 3 | y x= +

(b) | |f x x 4= - +] g

(c) | |f x x 9= -] g

(d) | | y x 2 1= - -

(e) | |f x x 2= - +] g

56 Exercises




7 Solve graphically

(a) | | 3x =

(b) | |x 12

(c) | |x 2

(d) | 2 | 1x + =

(e) | 3 | 0x- =

(f) |2 3 | 1x - =

(g) | |x 1 41-

(h) | |x 1 3+

(i) | |x 2 22-

(j) | |x 3 1$-

(k) | |x2 3 5+

(l) | |x2 1 1$-

(m) |3 1 | 3x x- = +

(n) |3 2 | 4x x- = -

(o) | 1 | 1x x- = +

(p) | 3 | 2 2x x+ = + (q) |2 1 | 1x x+ = -

(r) |2 5 | 3x x- = -

(s) | 1 | 2x x- =

(t) |2 3 | 3x x- = +

The Hyperbola


or= =

EXAMPLE

Sketch1

y x=

Solution



x -3 -2 -121-

41- 0

41

21 1 2 3

y 3

1-

2

1- -1 -2 -4 mdash 4 2 1

2

1

3

1

Class Discussion



y is never 0 Why






but never touches)



Investigation




equations




1 y xk

=

21

y x k= +

31

y x k

=+


EXAMPLES


3=

-

] g


Solution



So x 3






be zero

CONTINUED







1

y 0 3

3=

-

= -

-3

4

3

2

1

5

-2

-1

-4

-5

y

-1-2 4 521-3-4

x

x = 3

y = 0

Asymptotes

3

2 Sketch y x2 4

1= -

+

Solution




x

x

2 4

2

-

-






( ) y

2 0 41

41

= -+

= -


a translation of3

y x

=





y

-2

x

-

1

4


a=

+


domain x xb



1 For each graph



exists


(a)2

y x=

(b)1

y x= -

(c) f xx 1

1=

+] g

(d) f xx 2

3=

-

] g

(e)3 6

1 y

x=

+

(f) f xx 3

2= -

-

] g

(g) f xx 1

4=

-

] g

(h)1

2 y

x= -

+

(i) f xx6 3

2=

-

] g

(j)2

6 y

x= -

+

2 Show that f x x2

=] g is an odd

function



(a) f xx2 5

1=

+

] g for x2 2 -

(b)3

1 y

x=

+ for x2 0 -

(c) f xx2 4

5=

-

] g for x3 1 -

57 Exercises




(d) f xx 4

3= -

-

] g for x3 3 -

(e)3 1

2 y

x= -

+ for x0 5



(a)3

y x= for y 1 3

(b)2

y x= - for y 221

- -

(c) f xx 1

1=

-

] g for y 171

- -

(d) f xx2 1

3= -

+

] g for

y 131

- -

(e)3 2

6 y

x=

- for y 1

21

6



Circle gate





Proof

r y

x

( x y)

y

x








c a b

r x y

2 2 2

2 2 2

`

= +

= +

EXAMPLE



Solution


y

x

-2

-2 2

2


one place

y

x

2

2

2

-2

The radius is 4

CONTINUED






Domain 2 2 x x -

Range 2 2 y y -


range y r y r -



Proof


shown

y

x

(a b)

x

y

r

( x y)

a

b x - a

y - b




ndash ndash

c a b

r x a y b

2 2 2

2 2 2

= +

= +] ^g h




EXAMPLES



Solution



y

x 9

9

-9

-9



Domain 9 9 x x -

Range 9 9 y y -



Solution


ndash

ndash ndash

x y

x y

1 2 4

1 2 2

2

2 2

+ + =

+ - =

2

2

] ^

] ]_

g h

g gi

So 1 2a b= = - and 2r =

CONTINUED







y

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

(1 -2)



Domain 1 3 x x -

Range 4 0 y y -


Solution


a b2 1= - = and r 3=

Substituting

ndash ndash

ndash

ndash

x a y b r

x y

x y

2 1 3

2 1 9

2 2 2

2 2 2

2 2

+ =

- - + =

+ + =

] ^]] ^

] ^

g hg g hg h


ndash

ndash

a b a ab b

x x x

x x

a b a ab b

y y y

y y

2

2 2 2 2

4 4

2

1 2 1 1

2 1

So

So

2 2 2

2 2 2

2

2 2 2

2 2 2

2

+ = + +

+ = + +

= + +

= - +

= - +

= - +

]] ] ]]^ ^ ]

gg g ggh h g


ndash

x x y y

x x y y

x x y y

x x y y

4 4 2 1 9

4 2 5 9

4 2 5 9

4 2 4 0

9 9

2

2

2

2

+ + + - + =

+ + - + =

+ + + =

+ + - - =

- -


in Chapter 2




Investigation


two functions





circle






for all values

y

x r

r

-r


Proof

ndash

x y r

y r x y r x

2 2 2

2 2 2

2 2

+ =

=

= -



semi-circles





y 0 for all values

y

x r

-

r

-r


EXAMPLES


1 f x x92

= -] g

Solution



y

x 3

3

-3

Domain 3 3 x x -

Range 0 3 y y




2 y x4 2= - -

Solution



y

x 2

-2

-2

Domain 2 2 x x -

Range 2 0 y y -




range

(a) 9x y 2 2+ =

(b) x y 16 02 2+ =-


(d) 1 9x y 2 2+ + =

] g

(e) ndashx y 2 1 12 2+ + =] ^g h



below the x -axis



range

(a) 25 y x2= - -

(b) 1 y x2= -

(c) 36 y x2= -

(d) 64 y x2= - -

(e) 7 y x2= - -



each circle

(a) 100x y 2 2+ =

(b) 5x y 2 2+ =


(d) ndashx y 5 6 492 2+ + =] ^g h

(e) ndashx y 3 812 2+ =^ h

58 Exercises






grouping symbols)



Centre(c) 1 5-







Centre(j) 4 7- -

^ h and radius 3

Other Graphs




EXAMPLES


Solution



eg When 0x =

y 3

1

c=

=

x

y

1

3

3

1

31

When1

1

= -

=

=

=

-

x 3- 2- 1- 0 1 2 3

y 271

91

31

1 3 9 27



You will meet these

graphs again in the

HSC Course





other point

When

x

y

1

3

3

1

=

=

=

y

x

1

2

1

3







Solution



x minus2 minus1 0 05 1 2 3 4

y minus03 0 03 05 06

y

x

1

2

1 2 3 4

-1




You learned about


stages of maths





range y y 02



range all real y

Cubic function


EXAMPLE



Solution



y minus25 minus6 1 2 3 10 29

y

x

1

1

-2 2 3 4

-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4



If you already

know the shape of


3






Domain and range


EXAMPLE


Solution




function are

f

f

1 1 2

1 21

4 4 2

64 2

66

3

3

- = - +

= - +

=

= +

= +

=

] ]

] ]

g g

g g


these points

y

x

(-1 1)

(4 66)

Range 1 66 y y










positive number

EXAMPLE


Solution


x 4So $

$


(or zero) So y 0$


59 Exercises


(a) 4 3 y x= +

(b) f x 4= -] g

(c) 3x =



(f) f x xx 12 2= - -] g

(g) 64x y 2 2+ =

(h) f t t 4

3=

-] g

(i) ( ) g 2

5zz

= +

(j) | |f x x=] g


(a) y x=

(b) 2 y x= -

(c) | |f x x2 3= -] g

(d) | | 2 y x= -

(e) f x x2 5= - +] g

(f) | | y x5= -

(g) 2 y x=

(h) y 5x= -

(i) f x xx 1

= +] g

(j)2

4 3 y x

x= -


(a) y x x 5 2= -] g


= +-

(d) g x x x164 2= -] g

(e) 49x y 2 2+ =

You may like to


by dividing by x




4 (a) Solve x1 02$-


f x x1 2= -] g


(a) 2 y x x2= - -

(b) g t t t 62= +] g



in each case


x3 3 -


x2 3 -


x2 1 - (d)


x1 5


0 4x


x3 3 -


x1 1 -


x2 3 -


x4 4 -

(j) y x x7 62= - + - in the

domain 0 7x


function y x 1

3=

+





x=] g


function

find its range(b)


plane

(a) f x x4=] g

(b) y x3= -

(c) y x 34= -

(d) 2 p x x3=] g

(e) 1 g x x3= +] g

(f) 100x y 2 2+ =

(g) 2 1 y x= +


y x 1= -









(b) f x x 22= -] g

(c)1

y x=

(d) f x x3=] g

(e) f x 3x=] g

13 (a) Solve x4 02$-


(i) 4 y x2= -

(ii) y x4

2= - -




DID YOU KNOW




a light



Limits






a a1

1

03

Z

=

=

3

3

-


lim a 0x=

x 3-



asymptote

EXAMPLES

1 Find lim x

x x5x 0

2+

Solution


which is undefined


( )

lim lim

lim

xx x

x

x x

x

5 5

5

5

x x

x

0

2

0 1

1

0

+=

+

= +

=

] g




2 Find limx

x

4

22

-

-

x 2

Solution


lim lim

lim

x

x

x x

x

x

4

2

2 2

2

21

41

2 1

1

-

-=

+ -

-

=+

=

x x2 2

x 2

^ _h i

3 Find limh

h x hx h2 72 2+ -

h 0

Solution

lim lim

lim

h

h x hx h

h

h hx x

hx x

x

2 7 2 7

2 7

7

2 2 2

2

2

+ -=

+ -

= + -

= -

h 0

h h0 0

^ h

Continuity





EXAMPLES

1 Find lim

x

x

1

12

-

-

x 1


11

y x

x2

=-

- Sketch the curve

Solution


xx2

-

- gives

00

CONTINUED




( )

lim lim

limx

x

x

x x

x11

1

1 1

1

2

x x

x

1

2

1

1

-

-=

-

+ -

= +

=

-

] ]g g

11

y xx2

=-





y xx

x

x x

x

11

1

1 1

1

2

=-

-

=

+

= +

-

-^ ^h h


2 Find limx

x x2

2x 2

2

+

+ -

-


x x2

22

= + -

+

Solution


x x2

22

+

+ - gives

00

lim lim

lim

xx x

x

x x

x

22

2

1 2

1

3

x x

x

2

2

2

2

+

+ -=

+

- +

=

= -

-

- -

-

^^ ^

^

hh h

h2 y

x

x xx

y x

x

x

x

22

2

2

1

1


=+

+ -= -

=+

= -

+ -^ ^h h


Remember that x 1




1 Find(a) lim x 52

+x 4

(b) lim t 7-t 3 -

(c) lim x x2 43+ -

x 2

(d) lim xx x32

+

x 0

(e) limh

h h

2

22

-

- -

h 2

(f) lim y

y

5

1253

-

-

y 5

(g) limx

x x

12 12

+

+ +

x 1-

(h) limx

x x4

2 82

+

+ -

x 4 -

(i) limc

c

4

22

-

-

c 2

(j) limx x

x 12

-

-

x 1

(k) lim h

h h h2 73 2+ -

h 0

(l) limh

hx hx h32 2- +

h 0

(m) limh

hx h x hx h2 3 53 2 2- + -

h 0

(n) lim x c x c 3 3

-

-

x c



discontinuous

(a) 3 y x2= -

(b)1

1 y

x=

+

(c) f x x 1= -] g

(d)4

1 y

x2=

+

(e)4

1

y x2=

-



(a)3

y xx x2

= +

(b)33

y x

x x2

=+

+

(c)1

5 4 y

xx x2

=+

+ +

510 Exercises




Regions

Class Investigation


them all







EXAMPLE


Solution



y

x =3

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4







region

EXAMPLE


Solution



included


shown

y

x

1

1

-2 2 3 4

-3

-4

-5

2

3

4

5

-1

-2

-3-4

-1 y = -1



y -intercept 1-






EXAMPLES


1 y x 2$ +

Solution






y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0



2 x y 2 3 61-

Solution


region


side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g








3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g






Gradient form



Investigation










y x k

y x k

y x k

y x k y

x k

1

(a)

(b)

(c)

(d)(e)

2

3

4

= +

= +

= +

= +

=+

]]

]

gg

g





the right








Linear Function





EXAMPLE



Solution


Find the intercepts


0 3 5

5 3

1

x

x

x32

=

=

=

-


3 5

5

y 0=

= -

-] g

-1

-2

y

5

4

3

2

1 1 23

6

-3

-4

-5

1 4-1-2 32-3-4

x



Range all real y








Special lines






1 y kx=

2 y x k= +






EXAMPLES


Solution




-1

-3

y

4

3

2

1

5

-2

-4

-5

1 4-1-2

x

32-3-4


2 Sketch x 1= -


Solution





-

-

4

3

2

5

-2

-4

-5

1 4-1-2-4

y

x







Domain all real x

Range y y b=

54 Exercises


each function

(a) y x 2= -

(b) f x x2 3= +] g (c) x y 2 1 0+ =-

(d) x y 3 0+ =-

(e) x y 3 6 2 0=- -


line

(a) x 4=

(b) x 3 0=-

(c) y 5=

(d) y 1 0+ =

(e) f x x2 1= -] g (f) y x 4= +

(g) f x x3 2= +] g

(h) x y 3+ =

(i) x y 1 0=- -

(j) x y 2 3 0+ =-


(a) x y 3 2 7 0+ =-

(b) y 2=

(c) x 4= - (d) x 2 0=-

(e) y 3 0=-


are even or odd

(a) y x2=

(b) y 3=

(c) x 4=

(d) y x= -

(e) y x=




meet




Applications






The pronumeral

a is called the


Quadratic Function










EXAMPLES



Solution



x

x

x

0 1

1

1

2

2

= -

=

=


0 1

1

y 2= -

= -



Domain xall real

Range y y 1$ -


Solution


parabola will lie




left



0

1 0

1

x

x

x

1 2= +

+ =

= -

] g


1

y 0 1 2= +

=

] g

-1

-

4

3

2

1

5

-2

-4

-5

-

1 41-2 5-4

y

x









Solution



x x

x xx x

x x

0 6

3 23 0 2 0

3 2

2= + -

= + -

+ = - =

= - =

] ]g g


f 0 0 0 66

2= + -

= -

] ] ]g g g



x -intercepts


23 2

21- +

= -


-c m

f 21

21

21

6

41

21

6

641

2

- = - + - -

= - -

= -

c c cm m m


-

CONTINUED

You will learn more


-1

-3

4

3

2

1

5

-2

-4

-5

1 4-1-2 32-3-4

y

x










41



2

11 -





Solution


So f x 0=] g

0 4 54 5 0

0

x xx x

x x5 1

2

2

= - + +

=

+ =

- -

-] ]g g

x x

x x

5 0 1 0

5 1

- = + =

= = -


f 0 0 4 0 5

5

2= - + +

=

] ] ]g g g

-1

-3

4

2

5

-2

-4

-5

-

y

1 41-2 5-4

x

-2 4






21 5

2- +

=

f 2 2 4 2 59

= - + +

=

2] ] ]g g g










each function

(a) 2 y x x2= +

(b) 3 y x x2= - +

(c) f x x 12= -] g

(d) y x x 22= - -

(e) y x x9 82= +-

2 Sketch

(a) 2 y x2= +

(b) y x 12= - +

(c) f x x 42= -] g

(d) 2 y x x2= +

(e) y x x2= - -

(f) f x x 3= -2] ]g g

55 Exercises

-1

9

8

7

5

4

3

2

6

1

-2

-3

-4

-5

y

2 51 643-1-2-3-4

x




EXAMPLES


Solution









(g) f x x 1 2= +] ]g g

(h) y x x3 42= + -

(i) y x x2 5 32= - +

(j) f x x x3 22= - + -] g



(a) ndash y x x7 122= +

(b) f x x x42= +] g

(c) y x x2 82= - -

(d) y x x6 92= +-

(e) f t t 4 2= -] g


(a) y x 52= -

(b) f x x x6

2= -

] g (c) f x x x 22= - -] g

(d) y x2= -

(e) f x x 7 2= -] ]g g



(a) y x2= for x0 3

(b) y x 42= - + for x1 2 -

(c) f x x 12= -] g for x2 5 -

(d) y x x2 32= + - for x2 4 -

(e) y x x

22= - +- for

x0 4


function is

increasing(i)

decreasing(ii)

(a) y x2=

(b) y x2

= - (c) f x x 92

= -] g

(d) y x x42= - +

(e) f x x 5 2= +] ]g g


function



(a) y x 12= +

(b) f x x 32= -

] g

(c) y x2 2= -

(d) f x x x32= -] g

(e) f x x x2= +] g

(f) y x 42= -

(g) y x x2 32= - -

(h) y x x5 42= +-

(i) p x x 1 2= +] ]g g

(j) y x 2= -2] g




CONTINUED

eg When x 3= -

| | y 3 13 12

= - -

= -

=

x -3 -2 -1 0 1 2 3

y 2 1 0 -1 0 1 2


y

-2

4

3

2

1

5

-1

-3

-4

-5

1 4-1-2 32-3-4

x


| | y x

x x

x x1

1 0

1 0

when

when 1

$= - =

-


y x x1 0$= - ] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x - 1




y x 1= - - x 01] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y =- x - 1



-3

4

3

2

1

5

-2

-1

-4

-5

y y

3-1-2 421-3-4

x

y = - x - 1

y = x - 1


intercepts


So f x 0=

] g

| |

| |

x

x

x

0 1

1

1`

= -

=

=


( ) | |f 0 0 11

= -

= -





-3

4

3

2

1

5

-2

-1

-4

-5

y

4-2 5321-1-3-4

x



Domain all real x

Range y y 1$ -

2 Sketch | | y x 2= +

Solution


| | ( ) y x x xx x2 2 2 0

2 2 0whenwhen 1

$= + = + +

- + +


2 y x= + when x 2 0$+

x 2$ -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x + 2






y-intercept of 1-

CONTINUED




2 y x= - +] g when x 2 01+

ie y x 2= - - when x 21 -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2


-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2

y = x + 2



So 0f x =] g

0 | 2 |

0 2

2

x

x

x

= +

= +

- =


(0) | 0 2 |

2

f = +

=

There is only one

solution for the


Can you see why





-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

If you know how to

sketch the graph of




Investigation




are their equations





1 | |f x k x=] g

2 | |f x x k= +] g

3 | |f x x k= +] g







EXAMPLES

Solve

1 |2 1 | 3x - =

Solution



is x 1 2= -

2 |2 1 | 3 2x x= -+

Solution



3 | 1 | 2x 1+

Solution






and then check them







each function

(a) | | y x=

(b) | |f x x 7= +] g

(c) | |f x x 2= -] g

(d) 5 | | y x=

(e) | |f x x 3= - +] g

(f) | 6 | y x= +

(g) | |f x x3 2= -] g

(h) | 5 4 | y x= +

(i) | 7 1 | y x= -

(j) | |f x x2 9= +] g


plane

(a) | | y x=

(b) | |f x x 1= +] g

(c) | |f x x 3= -] g

(d) 2 | | y x=

(e) | |f x x= -] g

(f) | 1 | y x= +

(g) | |f x x 1= - -] g

(h) | 2 3 | y x= -

(i) | 4 2 | y x= +

(j) | |f x x3 1= +] g


each function

(a) | 1 | y x= -

(b) | |f x x 8= -] g

(c) | |f x x2 5= +] g

(d) 2 | | 3 y x= -

(e) | |f x x 3= - -] g


function is

increasing(i)

decreasing(ii)

(a) | 2 | y x= -

(b) | |f x x 2= +

] g

(c) | |f x x2 3= -] g

(d) 4 | | 1 y x= -

(e) | |f x x= -] g


of each function

(a) | | y x= for x2 2 -

(b) | |f x x 4= - -] g for

x4 3 -

(c) | |f x x 4= +] g for x7 2 -

(d) | 2 5 | y x= - for x3 3 -

(e) | |f x x= -] g for x1 1 -


function increasing

(a) | 3 | y x= +

(b) | |f x x 4= - +] g

(c) | |f x x 9= -] g

(d) | | y x 2 1= - -

(e) | |f x x 2= - +] g

56 Exercises




7 Solve graphically

(a) | | 3x =

(b) | |x 12

(c) | |x 2

(d) | 2 | 1x + =

(e) | 3 | 0x- =

(f) |2 3 | 1x - =

(g) | |x 1 41-

(h) | |x 1 3+

(i) | |x 2 22-

(j) | |x 3 1$-

(k) | |x2 3 5+

(l) | |x2 1 1$-

(m) |3 1 | 3x x- = +

(n) |3 2 | 4x x- = -

(o) | 1 | 1x x- = +

(p) | 3 | 2 2x x+ = + (q) |2 1 | 1x x+ = -

(r) |2 5 | 3x x- = -

(s) | 1 | 2x x- =

(t) |2 3 | 3x x- = +

The Hyperbola


or= =

EXAMPLE

Sketch1

y x=

Solution



x -3 -2 -121-

41- 0

41

21 1 2 3

y 3

1-

2

1- -1 -2 -4 mdash 4 2 1

2

1

3

1

Class Discussion



y is never 0 Why






but never touches)



Investigation




equations




1 y xk

=

21

y x k= +

31

y x k

=+


EXAMPLES


3=

-

] g


Solution



So x 3






be zero

CONTINUED







1

y 0 3

3=

-

= -

-3

4

3

2

1

5

-2

-1

-4

-5

y

-1-2 4 521-3-4

x

x = 3

y = 0

Asymptotes

3

2 Sketch y x2 4

1= -

+

Solution




x

x

2 4

2

-

-






( ) y

2 0 41

41

= -+

= -


a translation of3

y x

=





y

-2

x

-

1

4


a=

+


domain x xb



1 For each graph



exists


(a)2

y x=

(b)1

y x= -

(c) f xx 1

1=

+] g

(d) f xx 2

3=

-

] g

(e)3 6

1 y

x=

+

(f) f xx 3

2= -

-

] g

(g) f xx 1

4=

-

] g

(h)1

2 y

x= -

+

(i) f xx6 3

2=

-

] g

(j)2

6 y

x= -

+

2 Show that f x x2

=] g is an odd

function



(a) f xx2 5

1=

+

] g for x2 2 -

(b)3

1 y

x=

+ for x2 0 -

(c) f xx2 4

5=

-

] g for x3 1 -

57 Exercises




(d) f xx 4

3= -

-

] g for x3 3 -

(e)3 1

2 y

x= -

+ for x0 5



(a)3

y x= for y 1 3

(b)2

y x= - for y 221

- -

(c) f xx 1

1=

-

] g for y 171

- -

(d) f xx2 1

3= -

+

] g for

y 131

- -

(e)3 2

6 y

x=

- for y 1

21

6



Circle gate





Proof

r y

x

( x y)

y

x








c a b

r x y

2 2 2

2 2 2

`

= +

= +

EXAMPLE



Solution


y

x

-2

-2 2

2


one place

y

x

2

2

2

-2

The radius is 4

CONTINUED






Domain 2 2 x x -

Range 2 2 y y -


range y r y r -



Proof


shown

y

x

(a b)

x

y

r

( x y)

a

b x - a

y - b




ndash ndash

c a b

r x a y b

2 2 2

2 2 2

= +

= +] ^g h




EXAMPLES



Solution



y

x 9

9

-9

-9



Domain 9 9 x x -

Range 9 9 y y -



Solution


ndash

ndash ndash

x y

x y

1 2 4

1 2 2

2

2 2

+ + =

+ - =

2

2

] ^

] ]_

g h

g gi

So 1 2a b= = - and 2r =

CONTINUED







y

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

(1 -2)



Domain 1 3 x x -

Range 4 0 y y -


Solution


a b2 1= - = and r 3=

Substituting

ndash ndash

ndash

ndash

x a y b r

x y

x y

2 1 3

2 1 9

2 2 2

2 2 2

2 2

+ =

- - + =

+ + =

] ^]] ^

] ^

g hg g hg h


ndash

ndash

a b a ab b

x x x

x x

a b a ab b

y y y

y y

2

2 2 2 2

4 4

2

1 2 1 1

2 1

So

So

2 2 2

2 2 2

2

2 2 2

2 2 2

2

+ = + +

+ = + +

= + +

= - +

= - +

= - +

]] ] ]]^ ^ ]

gg g ggh h g


ndash

x x y y

x x y y

x x y y

x x y y

4 4 2 1 9

4 2 5 9

4 2 5 9

4 2 4 0

9 9

2

2

2

2

+ + + - + =

+ + - + =

+ + + =

+ + - - =

- -


in Chapter 2




Investigation


two functions





circle






for all values

y

x r

r

-r


Proof

ndash

x y r

y r x y r x

2 2 2

2 2 2

2 2

+ =

=

= -



semi-circles





y 0 for all values

y

x r

-

r

-r


EXAMPLES


1 f x x92

= -] g

Solution



y

x 3

3

-3

Domain 3 3 x x -

Range 0 3 y y




2 y x4 2= - -

Solution



y

x 2

-2

-2

Domain 2 2 x x -

Range 2 0 y y -




range

(a) 9x y 2 2+ =

(b) x y 16 02 2+ =-


(d) 1 9x y 2 2+ + =

] g

(e) ndashx y 2 1 12 2+ + =] ^g h



below the x -axis



range

(a) 25 y x2= - -

(b) 1 y x2= -

(c) 36 y x2= -

(d) 64 y x2= - -

(e) 7 y x2= - -



each circle

(a) 100x y 2 2+ =

(b) 5x y 2 2+ =


(d) ndashx y 5 6 492 2+ + =] ^g h

(e) ndashx y 3 812 2+ =^ h

58 Exercises






grouping symbols)



Centre(c) 1 5-







Centre(j) 4 7- -

^ h and radius 3

Other Graphs




EXAMPLES


Solution



eg When 0x =

y 3

1

c=

=

x

y

1

3

3

1

31

When1

1

= -

=

=

=

-

x 3- 2- 1- 0 1 2 3

y 271

91

31

1 3 9 27



You will meet these

graphs again in the

HSC Course





other point

When

x

y

1

3

3

1

=

=

=

y

x

1

2

1

3







Solution



x minus2 minus1 0 05 1 2 3 4

y minus03 0 03 05 06

y

x

1

2

1 2 3 4

-1




You learned about


stages of maths





range y y 02



range all real y

Cubic function


EXAMPLE



Solution



y minus25 minus6 1 2 3 10 29

y

x

1

1

-2 2 3 4

-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4



If you already

know the shape of


3






Domain and range


EXAMPLE


Solution




function are

f

f

1 1 2

1 21

4 4 2

64 2

66

3

3

- = - +

= - +

=

= +

= +

=

] ]

] ]

g g

g g


these points

y

x

(-1 1)

(4 66)

Range 1 66 y y










positive number

EXAMPLE


Solution


x 4So $

$


(or zero) So y 0$


59 Exercises


(a) 4 3 y x= +

(b) f x 4= -] g

(c) 3x =



(f) f x xx 12 2= - -] g

(g) 64x y 2 2+ =

(h) f t t 4

3=

-] g

(i) ( ) g 2

5zz

= +

(j) | |f x x=] g


(a) y x=

(b) 2 y x= -

(c) | |f x x2 3= -] g

(d) | | 2 y x= -

(e) f x x2 5= - +] g

(f) | | y x5= -

(g) 2 y x=

(h) y 5x= -

(i) f x xx 1

= +] g

(j)2

4 3 y x

x= -


(a) y x x 5 2= -] g


= +-

(d) g x x x164 2= -] g

(e) 49x y 2 2+ =

You may like to


by dividing by x




4 (a) Solve x1 02$-


f x x1 2= -] g


(a) 2 y x x2= - -

(b) g t t t 62= +] g



in each case


x3 3 -


x2 3 -


x2 1 - (d)


x1 5


0 4x


x3 3 -


x1 1 -


x2 3 -


x4 4 -

(j) y x x7 62= - + - in the

domain 0 7x


function y x 1

3=

+





x=] g


function

find its range(b)


plane

(a) f x x4=] g

(b) y x3= -

(c) y x 34= -

(d) 2 p x x3=] g

(e) 1 g x x3= +] g

(f) 100x y 2 2+ =

(g) 2 1 y x= +


y x 1= -









(b) f x x 22= -] g

(c)1

y x=

(d) f x x3=] g

(e) f x 3x=] g

13 (a) Solve x4 02$-


(i) 4 y x2= -

(ii) y x4

2= - -




DID YOU KNOW




a light



Limits






a a1

1

03

Z

=

=

3

3

-


lim a 0x=

x 3-



asymptote

EXAMPLES

1 Find lim x

x x5x 0

2+

Solution


which is undefined


( )

lim lim

lim

xx x

x

x x

x

5 5

5

5

x x

x

0

2

0 1

1

0

+=

+

= +

=

] g




2 Find limx

x

4

22

-

-

x 2

Solution


lim lim

lim

x

x

x x

x

x

4

2

2 2

2

21

41

2 1

1

-

-=

+ -

-

=+

=

x x2 2

x 2

^ _h i

3 Find limh

h x hx h2 72 2+ -

h 0

Solution

lim lim

lim

h

h x hx h

h

h hx x

hx x

x

2 7 2 7

2 7

7

2 2 2

2

2

+ -=

+ -

= + -

= -

h 0

h h0 0

^ h

Continuity





EXAMPLES

1 Find lim

x

x

1

12

-

-

x 1


11

y x

x2

=-

- Sketch the curve

Solution


xx2

-

- gives

00

CONTINUED




( )

lim lim

limx

x

x

x x

x11

1

1 1

1

2

x x

x

1

2

1

1

-

-=

-

+ -

= +

=

-

] ]g g

11

y xx2

=-





y xx

x

x x

x

11

1

1 1

1

2

=-

-

=

+

= +

-

-^ ^h h


2 Find limx

x x2

2x 2

2

+

+ -

-


x x2

22

= + -

+

Solution


x x2

22

+

+ - gives

00

lim lim

lim

xx x

x

x x

x

22

2

1 2

1

3

x x

x

2

2

2

2

+

+ -=

+

- +

=

= -

-

- -

-

^^ ^

^

hh h

h2 y

x

x xx

y x

x

x

x

22

2

2

1

1


=+

+ -= -

=+

= -

+ -^ ^h h


Remember that x 1




1 Find(a) lim x 52

+x 4

(b) lim t 7-t 3 -

(c) lim x x2 43+ -

x 2

(d) lim xx x32

+

x 0

(e) limh

h h

2

22

-

- -

h 2

(f) lim y

y

5

1253

-

-

y 5

(g) limx

x x

12 12

+

+ +

x 1-

(h) limx

x x4

2 82

+

+ -

x 4 -

(i) limc

c

4

22

-

-

c 2

(j) limx x

x 12

-

-

x 1

(k) lim h

h h h2 73 2+ -

h 0

(l) limh

hx hx h32 2- +

h 0

(m) limh

hx h x hx h2 3 53 2 2- + -

h 0

(n) lim x c x c 3 3

-

-

x c



discontinuous

(a) 3 y x2= -

(b)1

1 y

x=

+

(c) f x x 1= -] g

(d)4

1 y

x2=

+

(e)4

1

y x2=

-



(a)3

y xx x2

= +

(b)33

y x

x x2

=+

+

(c)1

5 4 y

xx x2

=+

+ +

510 Exercises




Regions

Class Investigation


them all







EXAMPLE


Solution



y

x =3

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4







region

EXAMPLE


Solution



included


shown

y

x

1

1

-2 2 3 4

-3

-4

-5

2

3

4

5

-1

-2

-3-4

-1 y = -1



y -intercept 1-






EXAMPLES


1 y x 2$ +

Solution






y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0



2 x y 2 3 61-

Solution


region


side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g








3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g






EXAMPLE



Solution


Find the intercepts


0 3 5

5 3

1

x

x

x32

=

=

=

-


3 5

5

y 0=

= -

-] g

-1

-2

y

5

4

3

2

1 1 23

6

-3

-4

-5

1 4-1-2 32-3-4

x



Range all real y








Special lines






1 y kx=

2 y x k= +






EXAMPLES


Solution




-1

-3

y

4

3

2

1

5

-2

-4

-5

1 4-1-2

x

32-3-4


2 Sketch x 1= -


Solution





-

-

4

3

2

5

-2

-4

-5

1 4-1-2-4

y

x







Domain all real x

Range y y b=

54 Exercises


each function

(a) y x 2= -

(b) f x x2 3= +] g (c) x y 2 1 0+ =-

(d) x y 3 0+ =-

(e) x y 3 6 2 0=- -


line

(a) x 4=

(b) x 3 0=-

(c) y 5=

(d) y 1 0+ =

(e) f x x2 1= -] g (f) y x 4= +

(g) f x x3 2= +] g

(h) x y 3+ =

(i) x y 1 0=- -

(j) x y 2 3 0+ =-


(a) x y 3 2 7 0+ =-

(b) y 2=

(c) x 4= - (d) x 2 0=-

(e) y 3 0=-


are even or odd

(a) y x2=

(b) y 3=

(c) x 4=

(d) y x= -

(e) y x=




meet




Applications






The pronumeral

a is called the


Quadratic Function










EXAMPLES



Solution



x

x

x

0 1

1

1

2

2

= -

=

=


0 1

1

y 2= -

= -



Domain xall real

Range y y 1$ -


Solution


parabola will lie




left



0

1 0

1

x

x

x

1 2= +

+ =

= -

] g


1

y 0 1 2= +

=

] g

-1

-

4

3

2

1

5

-2

-4

-5

-

1 41-2 5-4

y

x









Solution



x x

x xx x

x x

0 6

3 23 0 2 0

3 2

2= + -

= + -

+ = - =

= - =

] ]g g


f 0 0 0 66

2= + -

= -

] ] ]g g g



x -intercepts


23 2

21- +

= -


-c m

f 21

21

21

6

41

21

6

641

2

- = - + - -

= - -

= -

c c cm m m


-

CONTINUED

You will learn more


-1

-3

4

3

2

1

5

-2

-4

-5

1 4-1-2 32-3-4

y

x










41



2

11 -





Solution


So f x 0=] g

0 4 54 5 0

0

x xx x

x x5 1

2

2

= - + +

=

+ =

- -

-] ]g g

x x

x x

5 0 1 0

5 1

- = + =

= = -


f 0 0 4 0 5

5

2= - + +

=

] ] ]g g g

-1

-3

4

2

5

-2

-4

-5

-

y

1 41-2 5-4

x

-2 4






21 5

2- +

=

f 2 2 4 2 59

= - + +

=

2] ] ]g g g










each function

(a) 2 y x x2= +

(b) 3 y x x2= - +

(c) f x x 12= -] g

(d) y x x 22= - -

(e) y x x9 82= +-

2 Sketch

(a) 2 y x2= +

(b) y x 12= - +

(c) f x x 42= -] g

(d) 2 y x x2= +

(e) y x x2= - -

(f) f x x 3= -2] ]g g

55 Exercises

-1

9

8

7

5

4

3

2

6

1

-2

-3

-4

-5

y

2 51 643-1-2-3-4

x




EXAMPLES


Solution









(g) f x x 1 2= +] ]g g

(h) y x x3 42= + -

(i) y x x2 5 32= - +

(j) f x x x3 22= - + -] g



(a) ndash y x x7 122= +

(b) f x x x42= +] g

(c) y x x2 82= - -

(d) y x x6 92= +-

(e) f t t 4 2= -] g


(a) y x 52= -

(b) f x x x6

2= -

] g (c) f x x x 22= - -] g

(d) y x2= -

(e) f x x 7 2= -] ]g g



(a) y x2= for x0 3

(b) y x 42= - + for x1 2 -

(c) f x x 12= -] g for x2 5 -

(d) y x x2 32= + - for x2 4 -

(e) y x x

22= - +- for

x0 4


function is

increasing(i)

decreasing(ii)

(a) y x2=

(b) y x2

= - (c) f x x 92

= -] g

(d) y x x42= - +

(e) f x x 5 2= +] ]g g


function



(a) y x 12= +

(b) f x x 32= -

] g

(c) y x2 2= -

(d) f x x x32= -] g

(e) f x x x2= +] g

(f) y x 42= -

(g) y x x2 32= - -

(h) y x x5 42= +-

(i) p x x 1 2= +] ]g g

(j) y x 2= -2] g




CONTINUED

eg When x 3= -

| | y 3 13 12

= - -

= -

=

x -3 -2 -1 0 1 2 3

y 2 1 0 -1 0 1 2


y

-2

4

3

2

1

5

-1

-3

-4

-5

1 4-1-2 32-3-4

x


| | y x

x x

x x1

1 0

1 0

when

when 1

$= - =

-


y x x1 0$= - ] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x - 1




y x 1= - - x 01] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y =- x - 1



-3

4

3

2

1

5

-2

-1

-4

-5

y y

3-1-2 421-3-4

x

y = - x - 1

y = x - 1


intercepts


So f x 0=

] g

| |

| |

x

x

x

0 1

1

1`

= -

=

=


( ) | |f 0 0 11

= -

= -





-3

4

3

2

1

5

-2

-1

-4

-5

y

4-2 5321-1-3-4

x



Domain all real x

Range y y 1$ -

2 Sketch | | y x 2= +

Solution


| | ( ) y x x xx x2 2 2 0

2 2 0whenwhen 1

$= + = + +

- + +


2 y x= + when x 2 0$+

x 2$ -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x + 2






y-intercept of 1-

CONTINUED




2 y x= - +] g when x 2 01+

ie y x 2= - - when x 21 -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2


-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2

y = x + 2



So 0f x =] g

0 | 2 |

0 2

2

x

x

x

= +

= +

- =


(0) | 0 2 |

2

f = +

=

There is only one

solution for the


Can you see why





-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

If you know how to

sketch the graph of




Investigation




are their equations





1 | |f x k x=] g

2 | |f x x k= +] g

3 | |f x x k= +] g







EXAMPLES

Solve

1 |2 1 | 3x - =

Solution



is x 1 2= -

2 |2 1 | 3 2x x= -+

Solution



3 | 1 | 2x 1+

Solution






and then check them







each function

(a) | | y x=

(b) | |f x x 7= +] g

(c) | |f x x 2= -] g

(d) 5 | | y x=

(e) | |f x x 3= - +] g

(f) | 6 | y x= +

(g) | |f x x3 2= -] g

(h) | 5 4 | y x= +

(i) | 7 1 | y x= -

(j) | |f x x2 9= +] g


plane

(a) | | y x=

(b) | |f x x 1= +] g

(c) | |f x x 3= -] g

(d) 2 | | y x=

(e) | |f x x= -] g

(f) | 1 | y x= +

(g) | |f x x 1= - -] g

(h) | 2 3 | y x= -

(i) | 4 2 | y x= +

(j) | |f x x3 1= +] g


each function

(a) | 1 | y x= -

(b) | |f x x 8= -] g

(c) | |f x x2 5= +] g

(d) 2 | | 3 y x= -

(e) | |f x x 3= - -] g


function is

increasing(i)

decreasing(ii)

(a) | 2 | y x= -

(b) | |f x x 2= +

] g

(c) | |f x x2 3= -] g

(d) 4 | | 1 y x= -

(e) | |f x x= -] g


of each function

(a) | | y x= for x2 2 -

(b) | |f x x 4= - -] g for

x4 3 -

(c) | |f x x 4= +] g for x7 2 -

(d) | 2 5 | y x= - for x3 3 -

(e) | |f x x= -] g for x1 1 -


function increasing

(a) | 3 | y x= +

(b) | |f x x 4= - +] g

(c) | |f x x 9= -] g

(d) | | y x 2 1= - -

(e) | |f x x 2= - +] g

56 Exercises




7 Solve graphically

(a) | | 3x =

(b) | |x 12

(c) | |x 2

(d) | 2 | 1x + =

(e) | 3 | 0x- =

(f) |2 3 | 1x - =

(g) | |x 1 41-

(h) | |x 1 3+

(i) | |x 2 22-

(j) | |x 3 1$-

(k) | |x2 3 5+

(l) | |x2 1 1$-

(m) |3 1 | 3x x- = +

(n) |3 2 | 4x x- = -

(o) | 1 | 1x x- = +

(p) | 3 | 2 2x x+ = + (q) |2 1 | 1x x+ = -

(r) |2 5 | 3x x- = -

(s) | 1 | 2x x- =

(t) |2 3 | 3x x- = +

The Hyperbola


or= =

EXAMPLE

Sketch1

y x=

Solution



x -3 -2 -121-

41- 0

41

21 1 2 3

y 3

1-

2

1- -1 -2 -4 mdash 4 2 1

2

1

3

1

Class Discussion



y is never 0 Why






but never touches)



Investigation




equations




1 y xk

=

21

y x k= +

31

y x k

=+


EXAMPLES


3=

-

] g


Solution



So x 3






be zero

CONTINUED







1

y 0 3

3=

-

= -

-3

4

3

2

1

5

-2

-1

-4

-5

y

-1-2 4 521-3-4

x

x = 3

y = 0

Asymptotes

3

2 Sketch y x2 4

1= -

+

Solution




x

x

2 4

2

-

-






( ) y

2 0 41

41

= -+

= -


a translation of3

y x

=





y

-2

x

-

1

4


a=

+


domain x xb



1 For each graph



exists


(a)2

y x=

(b)1

y x= -

(c) f xx 1

1=

+] g

(d) f xx 2

3=

-

] g

(e)3 6

1 y

x=

+

(f) f xx 3

2= -

-

] g

(g) f xx 1

4=

-

] g

(h)1

2 y

x= -

+

(i) f xx6 3

2=

-

] g

(j)2

6 y

x= -

+

2 Show that f x x2

=] g is an odd

function



(a) f xx2 5

1=

+

] g for x2 2 -

(b)3

1 y

x=

+ for x2 0 -

(c) f xx2 4

5=

-

] g for x3 1 -

57 Exercises




(d) f xx 4

3= -

-

] g for x3 3 -

(e)3 1

2 y

x= -

+ for x0 5



(a)3

y x= for y 1 3

(b)2

y x= - for y 221

- -

(c) f xx 1

1=

-

] g for y 171

- -

(d) f xx2 1

3= -

+

] g for

y 131

- -

(e)3 2

6 y

x=

- for y 1

21

6



Circle gate





Proof

r y

x

( x y)

y

x








c a b

r x y

2 2 2

2 2 2

`

= +

= +

EXAMPLE



Solution


y

x

-2

-2 2

2


one place

y

x

2

2

2

-2

The radius is 4

CONTINUED






Domain 2 2 x x -

Range 2 2 y y -


range y r y r -



Proof


shown

y

x

(a b)

x

y

r

( x y)

a

b x - a

y - b




ndash ndash

c a b

r x a y b

2 2 2

2 2 2

= +

= +] ^g h




EXAMPLES



Solution



y

x 9

9

-9

-9



Domain 9 9 x x -

Range 9 9 y y -



Solution


ndash

ndash ndash

x y

x y

1 2 4

1 2 2

2

2 2

+ + =

+ - =

2

2

] ^

] ]_

g h

g gi

So 1 2a b= = - and 2r =

CONTINUED







y

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

(1 -2)



Domain 1 3 x x -

Range 4 0 y y -


Solution


a b2 1= - = and r 3=

Substituting

ndash ndash

ndash

ndash

x a y b r

x y

x y

2 1 3

2 1 9

2 2 2

2 2 2

2 2

+ =

- - + =

+ + =

] ^]] ^

] ^

g hg g hg h


ndash

ndash

a b a ab b

x x x

x x

a b a ab b

y y y

y y

2

2 2 2 2

4 4

2

1 2 1 1

2 1

So

So

2 2 2

2 2 2

2

2 2 2

2 2 2

2

+ = + +

+ = + +

= + +

= - +

= - +

= - +

]] ] ]]^ ^ ]

gg g ggh h g


ndash

x x y y

x x y y

x x y y

x x y y

4 4 2 1 9

4 2 5 9

4 2 5 9

4 2 4 0

9 9

2

2

2

2

+ + + - + =

+ + - + =

+ + + =

+ + - - =

- -


in Chapter 2




Investigation


two functions





circle






for all values

y

x r

r

-r


Proof

ndash

x y r

y r x y r x

2 2 2

2 2 2

2 2

+ =

=

= -



semi-circles





y 0 for all values

y

x r

-

r

-r


EXAMPLES


1 f x x92

= -] g

Solution



y

x 3

3

-3

Domain 3 3 x x -

Range 0 3 y y




2 y x4 2= - -

Solution



y

x 2

-2

-2

Domain 2 2 x x -

Range 2 0 y y -




range

(a) 9x y 2 2+ =

(b) x y 16 02 2+ =-


(d) 1 9x y 2 2+ + =

] g

(e) ndashx y 2 1 12 2+ + =] ^g h



below the x -axis



range

(a) 25 y x2= - -

(b) 1 y x2= -

(c) 36 y x2= -

(d) 64 y x2= - -

(e) 7 y x2= - -



each circle

(a) 100x y 2 2+ =

(b) 5x y 2 2+ =


(d) ndashx y 5 6 492 2+ + =] ^g h

(e) ndashx y 3 812 2+ =^ h

58 Exercises






grouping symbols)



Centre(c) 1 5-







Centre(j) 4 7- -

^ h and radius 3

Other Graphs




EXAMPLES


Solution



eg When 0x =

y 3

1

c=

=

x

y

1

3

3

1

31

When1

1

= -

=

=

=

-

x 3- 2- 1- 0 1 2 3

y 271

91

31

1 3 9 27



You will meet these

graphs again in the

HSC Course





other point

When

x

y

1

3

3

1

=

=

=

y

x

1

2

1

3







Solution



x minus2 minus1 0 05 1 2 3 4

y minus03 0 03 05 06

y

x

1

2

1 2 3 4

-1




You learned about


stages of maths





range y y 02



range all real y

Cubic function


EXAMPLE



Solution



y minus25 minus6 1 2 3 10 29

y

x

1

1

-2 2 3 4

-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4



If you already

know the shape of


3






Domain and range


EXAMPLE


Solution




function are

f

f

1 1 2

1 21

4 4 2

64 2

66

3

3

- = - +

= - +

=

= +

= +

=

] ]

] ]

g g

g g


these points

y

x

(-1 1)

(4 66)

Range 1 66 y y










positive number

EXAMPLE


Solution


x 4So $

$


(or zero) So y 0$


59 Exercises


(a) 4 3 y x= +

(b) f x 4= -] g

(c) 3x =



(f) f x xx 12 2= - -] g

(g) 64x y 2 2+ =

(h) f t t 4

3=

-] g

(i) ( ) g 2

5zz

= +

(j) | |f x x=] g


(a) y x=

(b) 2 y x= -

(c) | |f x x2 3= -] g

(d) | | 2 y x= -

(e) f x x2 5= - +] g

(f) | | y x5= -

(g) 2 y x=

(h) y 5x= -

(i) f x xx 1

= +] g

(j)2

4 3 y x

x= -


(a) y x x 5 2= -] g


= +-

(d) g x x x164 2= -] g

(e) 49x y 2 2+ =

You may like to


by dividing by x




4 (a) Solve x1 02$-


f x x1 2= -] g


(a) 2 y x x2= - -

(b) g t t t 62= +] g



in each case


x3 3 -


x2 3 -


x2 1 - (d)


x1 5


0 4x


x3 3 -


x1 1 -


x2 3 -


x4 4 -

(j) y x x7 62= - + - in the

domain 0 7x


function y x 1

3=

+





x=] g


function

find its range(b)


plane

(a) f x x4=] g

(b) y x3= -

(c) y x 34= -

(d) 2 p x x3=] g

(e) 1 g x x3= +] g

(f) 100x y 2 2+ =

(g) 2 1 y x= +


y x 1= -









(b) f x x 22= -] g

(c)1

y x=

(d) f x x3=] g

(e) f x 3x=] g

13 (a) Solve x4 02$-


(i) 4 y x2= -

(ii) y x4

2= - -




DID YOU KNOW




a light



Limits






a a1

1

03

Z

=

=

3

3

-


lim a 0x=

x 3-



asymptote

EXAMPLES

1 Find lim x

x x5x 0

2+

Solution


which is undefined


( )

lim lim

lim

xx x

x

x x

x

5 5

5

5

x x

x

0

2

0 1

1

0

+=

+

= +

=

] g




2 Find limx

x

4

22

-

-

x 2

Solution


lim lim

lim

x

x

x x

x

x

4

2

2 2

2

21

41

2 1

1

-

-=

+ -

-

=+

=

x x2 2

x 2

^ _h i

3 Find limh

h x hx h2 72 2+ -

h 0

Solution

lim lim

lim

h

h x hx h

h

h hx x

hx x

x

2 7 2 7

2 7

7

2 2 2

2

2

+ -=

+ -

= + -

= -

h 0

h h0 0

^ h

Continuity





EXAMPLES

1 Find lim

x

x

1

12

-

-

x 1


11

y x

x2

=-

- Sketch the curve

Solution


xx2

-

- gives

00

CONTINUED




( )

lim lim

limx

x

x

x x

x11

1

1 1

1

2

x x

x

1

2

1

1

-

-=

-

+ -

= +

=

-

] ]g g

11

y xx2

=-





y xx

x

x x

x

11

1

1 1

1

2

=-

-

=

+

= +

-

-^ ^h h


2 Find limx

x x2

2x 2

2

+

+ -

-


x x2

22

= + -

+

Solution


x x2

22

+

+ - gives

00

lim lim

lim

xx x

x

x x

x

22

2

1 2

1

3

x x

x

2

2

2

2

+

+ -=

+

- +

=

= -

-

- -

-

^^ ^

^

hh h

h2 y

x

x xx

y x

x

x

x

22

2

2

1

1


=+

+ -= -

=+

= -

+ -^ ^h h


Remember that x 1




1 Find(a) lim x 52

+x 4

(b) lim t 7-t 3 -

(c) lim x x2 43+ -

x 2

(d) lim xx x32

+

x 0

(e) limh

h h

2

22

-

- -

h 2

(f) lim y

y

5

1253

-

-

y 5

(g) limx

x x

12 12

+

+ +

x 1-

(h) limx

x x4

2 82

+

+ -

x 4 -

(i) limc

c

4

22

-

-

c 2

(j) limx x

x 12

-

-

x 1

(k) lim h

h h h2 73 2+ -

h 0

(l) limh

hx hx h32 2- +

h 0

(m) limh

hx h x hx h2 3 53 2 2- + -

h 0

(n) lim x c x c 3 3

-

-

x c



discontinuous

(a) 3 y x2= -

(b)1

1 y

x=

+

(c) f x x 1= -] g

(d)4

1 y

x2=

+

(e)4

1

y x2=

-



(a)3

y xx x2

= +

(b)33

y x

x x2

=+

+

(c)1

5 4 y

xx x2

=+

+ +

510 Exercises




Regions

Class Investigation


them all







EXAMPLE


Solution



y

x =3

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4







region

EXAMPLE


Solution



included


shown

y

x

1

1

-2 2 3 4

-3

-4

-5

2

3

4

5

-1

-2

-3-4

-1 y = -1



y -intercept 1-






EXAMPLES


1 y x 2$ +

Solution






y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0



2 x y 2 3 61-

Solution


region


side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g








3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g






EXAMPLES


Solution




-1

-3

y

4

3

2

1

5

-2

-4

-5

1 4-1-2

x

32-3-4


2 Sketch x 1= -


Solution





-

-

4

3

2

5

-2

-4

-5

1 4-1-2-4

y

x







Domain all real x

Range y y b=

54 Exercises


each function

(a) y x 2= -

(b) f x x2 3= +] g (c) x y 2 1 0+ =-

(d) x y 3 0+ =-

(e) x y 3 6 2 0=- -


line

(a) x 4=

(b) x 3 0=-

(c) y 5=

(d) y 1 0+ =

(e) f x x2 1= -] g (f) y x 4= +

(g) f x x3 2= +] g

(h) x y 3+ =

(i) x y 1 0=- -

(j) x y 2 3 0+ =-


(a) x y 3 2 7 0+ =-

(b) y 2=

(c) x 4= - (d) x 2 0=-

(e) y 3 0=-


are even or odd

(a) y x2=

(b) y 3=

(c) x 4=

(d) y x= -

(e) y x=




meet




Applications






The pronumeral

a is called the


Quadratic Function










EXAMPLES



Solution



x

x

x

0 1

1

1

2

2

= -

=

=


0 1

1

y 2= -

= -



Domain xall real

Range y y 1$ -


Solution


parabola will lie




left



0

1 0

1

x

x

x

1 2= +

+ =

= -

] g


1

y 0 1 2= +

=

] g

-1

-

4

3

2

1

5

-2

-4

-5

-

1 41-2 5-4

y

x









Solution



x x

x xx x

x x

0 6

3 23 0 2 0

3 2

2= + -

= + -

+ = - =

= - =

] ]g g


f 0 0 0 66

2= + -

= -

] ] ]g g g



x -intercepts


23 2

21- +

= -


-c m

f 21

21

21

6

41

21

6

641

2

- = - + - -

= - -

= -

c c cm m m


-

CONTINUED

You will learn more


-1

-3

4

3

2

1

5

-2

-4

-5

1 4-1-2 32-3-4

y

x










41



2

11 -





Solution


So f x 0=] g

0 4 54 5 0

0

x xx x

x x5 1

2

2

= - + +

=

+ =

- -

-] ]g g

x x

x x

5 0 1 0

5 1

- = + =

= = -


f 0 0 4 0 5

5

2= - + +

=

] ] ]g g g

-1

-3

4

2

5

-2

-4

-5

-

y

1 41-2 5-4

x

-2 4






21 5

2- +

=

f 2 2 4 2 59

= - + +

=

2] ] ]g g g










each function

(a) 2 y x x2= +

(b) 3 y x x2= - +

(c) f x x 12= -] g

(d) y x x 22= - -

(e) y x x9 82= +-

2 Sketch

(a) 2 y x2= +

(b) y x 12= - +

(c) f x x 42= -] g

(d) 2 y x x2= +

(e) y x x2= - -

(f) f x x 3= -2] ]g g

55 Exercises

-1

9

8

7

5

4

3

2

6

1

-2

-3

-4

-5

y

2 51 643-1-2-3-4

x




EXAMPLES


Solution









(g) f x x 1 2= +] ]g g

(h) y x x3 42= + -

(i) y x x2 5 32= - +

(j) f x x x3 22= - + -] g



(a) ndash y x x7 122= +

(b) f x x x42= +] g

(c) y x x2 82= - -

(d) y x x6 92= +-

(e) f t t 4 2= -] g


(a) y x 52= -

(b) f x x x6

2= -

] g (c) f x x x 22= - -] g

(d) y x2= -

(e) f x x 7 2= -] ]g g



(a) y x2= for x0 3

(b) y x 42= - + for x1 2 -

(c) f x x 12= -] g for x2 5 -

(d) y x x2 32= + - for x2 4 -

(e) y x x

22= - +- for

x0 4


function is

increasing(i)

decreasing(ii)

(a) y x2=

(b) y x2

= - (c) f x x 92

= -] g

(d) y x x42= - +

(e) f x x 5 2= +] ]g g


function



(a) y x 12= +

(b) f x x 32= -

] g

(c) y x2 2= -

(d) f x x x32= -] g

(e) f x x x2= +] g

(f) y x 42= -

(g) y x x2 32= - -

(h) y x x5 42= +-

(i) p x x 1 2= +] ]g g

(j) y x 2= -2] g




CONTINUED

eg When x 3= -

| | y 3 13 12

= - -

= -

=

x -3 -2 -1 0 1 2 3

y 2 1 0 -1 0 1 2


y

-2

4

3

2

1

5

-1

-3

-4

-5

1 4-1-2 32-3-4

x


| | y x

x x

x x1

1 0

1 0

when

when 1

$= - =

-


y x x1 0$= - ] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x - 1




y x 1= - - x 01] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y =- x - 1



-3

4

3

2

1

5

-2

-1

-4

-5

y y

3-1-2 421-3-4

x

y = - x - 1

y = x - 1


intercepts


So f x 0=

] g

| |

| |

x

x

x

0 1

1

1`

= -

=

=


( ) | |f 0 0 11

= -

= -





-3

4

3

2

1

5

-2

-1

-4

-5

y

4-2 5321-1-3-4

x



Domain all real x

Range y y 1$ -

2 Sketch | | y x 2= +

Solution


| | ( ) y x x xx x2 2 2 0

2 2 0whenwhen 1

$= + = + +

- + +


2 y x= + when x 2 0$+

x 2$ -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x + 2






y-intercept of 1-

CONTINUED




2 y x= - +] g when x 2 01+

ie y x 2= - - when x 21 -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2


-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2

y = x + 2



So 0f x =] g

0 | 2 |

0 2

2

x

x

x

= +

= +

- =


(0) | 0 2 |

2

f = +

=

There is only one

solution for the


Can you see why





-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

If you know how to

sketch the graph of




Investigation




are their equations





1 | |f x k x=] g

2 | |f x x k= +] g

3 | |f x x k= +] g







EXAMPLES

Solve

1 |2 1 | 3x - =

Solution



is x 1 2= -

2 |2 1 | 3 2x x= -+

Solution



3 | 1 | 2x 1+

Solution






and then check them







each function

(a) | | y x=

(b) | |f x x 7= +] g

(c) | |f x x 2= -] g

(d) 5 | | y x=

(e) | |f x x 3= - +] g

(f) | 6 | y x= +

(g) | |f x x3 2= -] g

(h) | 5 4 | y x= +

(i) | 7 1 | y x= -

(j) | |f x x2 9= +] g


plane

(a) | | y x=

(b) | |f x x 1= +] g

(c) | |f x x 3= -] g

(d) 2 | | y x=

(e) | |f x x= -] g

(f) | 1 | y x= +

(g) | |f x x 1= - -] g

(h) | 2 3 | y x= -

(i) | 4 2 | y x= +

(j) | |f x x3 1= +] g


each function

(a) | 1 | y x= -

(b) | |f x x 8= -] g

(c) | |f x x2 5= +] g

(d) 2 | | 3 y x= -

(e) | |f x x 3= - -] g


function is

increasing(i)

decreasing(ii)

(a) | 2 | y x= -

(b) | |f x x 2= +

] g

(c) | |f x x2 3= -] g

(d) 4 | | 1 y x= -

(e) | |f x x= -] g


of each function

(a) | | y x= for x2 2 -

(b) | |f x x 4= - -] g for

x4 3 -

(c) | |f x x 4= +] g for x7 2 -

(d) | 2 5 | y x= - for x3 3 -

(e) | |f x x= -] g for x1 1 -


function increasing

(a) | 3 | y x= +

(b) | |f x x 4= - +] g

(c) | |f x x 9= -] g

(d) | | y x 2 1= - -

(e) | |f x x 2= - +] g

56 Exercises




7 Solve graphically

(a) | | 3x =

(b) | |x 12

(c) | |x 2

(d) | 2 | 1x + =

(e) | 3 | 0x- =

(f) |2 3 | 1x - =

(g) | |x 1 41-

(h) | |x 1 3+

(i) | |x 2 22-

(j) | |x 3 1$-

(k) | |x2 3 5+

(l) | |x2 1 1$-

(m) |3 1 | 3x x- = +

(n) |3 2 | 4x x- = -

(o) | 1 | 1x x- = +

(p) | 3 | 2 2x x+ = + (q) |2 1 | 1x x+ = -

(r) |2 5 | 3x x- = -

(s) | 1 | 2x x- =

(t) |2 3 | 3x x- = +

The Hyperbola


or= =

EXAMPLE

Sketch1

y x=

Solution



x -3 -2 -121-

41- 0

41

21 1 2 3

y 3

1-

2

1- -1 -2 -4 mdash 4 2 1

2

1

3

1

Class Discussion



y is never 0 Why






but never touches)



Investigation




equations




1 y xk

=

21

y x k= +

31

y x k

=+


EXAMPLES


3=

-

] g


Solution



So x 3






be zero

CONTINUED







1

y 0 3

3=

-

= -

-3

4

3

2

1

5

-2

-1

-4

-5

y

-1-2 4 521-3-4

x

x = 3

y = 0

Asymptotes

3

2 Sketch y x2 4

1= -

+

Solution




x

x

2 4

2

-

-






( ) y

2 0 41

41

= -+

= -


a translation of3

y x

=





y

-2

x

-

1

4


a=

+


domain x xb



1 For each graph



exists


(a)2

y x=

(b)1

y x= -

(c) f xx 1

1=

+] g

(d) f xx 2

3=

-

] g

(e)3 6

1 y

x=

+

(f) f xx 3

2= -

-

] g

(g) f xx 1

4=

-

] g

(h)1

2 y

x= -

+

(i) f xx6 3

2=

-

] g

(j)2

6 y

x= -

+

2 Show that f x x2

=] g is an odd

function



(a) f xx2 5

1=

+

] g for x2 2 -

(b)3

1 y

x=

+ for x2 0 -

(c) f xx2 4

5=

-

] g for x3 1 -

57 Exercises




(d) f xx 4

3= -

-

] g for x3 3 -

(e)3 1

2 y

x= -

+ for x0 5



(a)3

y x= for y 1 3

(b)2

y x= - for y 221

- -

(c) f xx 1

1=

-

] g for y 171

- -

(d) f xx2 1

3= -

+

] g for

y 131

- -

(e)3 2

6 y

x=

- for y 1

21

6



Circle gate





Proof

r y

x

( x y)

y

x








c a b

r x y

2 2 2

2 2 2

`

= +

= +

EXAMPLE



Solution


y

x

-2

-2 2

2


one place

y

x

2

2

2

-2

The radius is 4

CONTINUED






Domain 2 2 x x -

Range 2 2 y y -


range y r y r -



Proof


shown

y

x

(a b)

x

y

r

( x y)

a

b x - a

y - b




ndash ndash

c a b

r x a y b

2 2 2

2 2 2

= +

= +] ^g h




EXAMPLES



Solution



y

x 9

9

-9

-9



Domain 9 9 x x -

Range 9 9 y y -



Solution


ndash

ndash ndash

x y

x y

1 2 4

1 2 2

2

2 2

+ + =

+ - =

2

2

] ^

] ]_

g h

g gi

So 1 2a b= = - and 2r =

CONTINUED







y

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

(1 -2)



Domain 1 3 x x -

Range 4 0 y y -


Solution


a b2 1= - = and r 3=

Substituting

ndash ndash

ndash

ndash

x a y b r

x y

x y

2 1 3

2 1 9

2 2 2

2 2 2

2 2

+ =

- - + =

+ + =

] ^]] ^

] ^

g hg g hg h


ndash

ndash

a b a ab b

x x x

x x

a b a ab b

y y y

y y

2

2 2 2 2

4 4

2

1 2 1 1

2 1

So

So

2 2 2

2 2 2

2

2 2 2

2 2 2

2

+ = + +

+ = + +

= + +

= - +

= - +

= - +

]] ] ]]^ ^ ]

gg g ggh h g


ndash

x x y y

x x y y

x x y y

x x y y

4 4 2 1 9

4 2 5 9

4 2 5 9

4 2 4 0

9 9

2

2

2

2

+ + + - + =

+ + - + =

+ + + =

+ + - - =

- -


in Chapter 2




Investigation


two functions





circle






for all values

y

x r

r

-r


Proof

ndash

x y r

y r x y r x

2 2 2

2 2 2

2 2

+ =

=

= -



semi-circles





y 0 for all values

y

x r

-

r

-r


EXAMPLES


1 f x x92

= -] g

Solution



y

x 3

3

-3

Domain 3 3 x x -

Range 0 3 y y




2 y x4 2= - -

Solution



y

x 2

-2

-2

Domain 2 2 x x -

Range 2 0 y y -




range

(a) 9x y 2 2+ =

(b) x y 16 02 2+ =-


(d) 1 9x y 2 2+ + =

] g

(e) ndashx y 2 1 12 2+ + =] ^g h



below the x -axis



range

(a) 25 y x2= - -

(b) 1 y x2= -

(c) 36 y x2= -

(d) 64 y x2= - -

(e) 7 y x2= - -



each circle

(a) 100x y 2 2+ =

(b) 5x y 2 2+ =


(d) ndashx y 5 6 492 2+ + =] ^g h

(e) ndashx y 3 812 2+ =^ h

58 Exercises






grouping symbols)



Centre(c) 1 5-







Centre(j) 4 7- -

^ h and radius 3

Other Graphs




EXAMPLES


Solution



eg When 0x =

y 3

1

c=

=

x

y

1

3

3

1

31

When1

1

= -

=

=

=

-

x 3- 2- 1- 0 1 2 3

y 271

91

31

1 3 9 27



You will meet these

graphs again in the

HSC Course





other point

When

x

y

1

3

3

1

=

=

=

y

x

1

2

1

3







Solution



x minus2 minus1 0 05 1 2 3 4

y minus03 0 03 05 06

y

x

1

2

1 2 3 4

-1




You learned about


stages of maths





range y y 02



range all real y

Cubic function


EXAMPLE



Solution



y minus25 minus6 1 2 3 10 29

y

x

1

1

-2 2 3 4

-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4



If you already

know the shape of


3






Domain and range


EXAMPLE


Solution




function are

f

f

1 1 2

1 21

4 4 2

64 2

66

3

3

- = - +

= - +

=

= +

= +

=

] ]

] ]

g g

g g


these points

y

x

(-1 1)

(4 66)

Range 1 66 y y










positive number

EXAMPLE


Solution


x 4So $

$


(or zero) So y 0$


59 Exercises


(a) 4 3 y x= +

(b) f x 4= -] g

(c) 3x =



(f) f x xx 12 2= - -] g

(g) 64x y 2 2+ =

(h) f t t 4

3=

-] g

(i) ( ) g 2

5zz

= +

(j) | |f x x=] g


(a) y x=

(b) 2 y x= -

(c) | |f x x2 3= -] g

(d) | | 2 y x= -

(e) f x x2 5= - +] g

(f) | | y x5= -

(g) 2 y x=

(h) y 5x= -

(i) f x xx 1

= +] g

(j)2

4 3 y x

x= -


(a) y x x 5 2= -] g


= +-

(d) g x x x164 2= -] g

(e) 49x y 2 2+ =

You may like to


by dividing by x




4 (a) Solve x1 02$-


f x x1 2= -] g


(a) 2 y x x2= - -

(b) g t t t 62= +] g



in each case


x3 3 -


x2 3 -


x2 1 - (d)


x1 5


0 4x


x3 3 -


x1 1 -


x2 3 -


x4 4 -

(j) y x x7 62= - + - in the

domain 0 7x


function y x 1

3=

+





x=] g


function

find its range(b)


plane

(a) f x x4=] g

(b) y x3= -

(c) y x 34= -

(d) 2 p x x3=] g

(e) 1 g x x3= +] g

(f) 100x y 2 2+ =

(g) 2 1 y x= +


y x 1= -









(b) f x x 22= -] g

(c)1

y x=

(d) f x x3=] g

(e) f x 3x=] g

13 (a) Solve x4 02$-


(i) 4 y x2= -

(ii) y x4

2= - -




DID YOU KNOW




a light



Limits






a a1

1

03

Z

=

=

3

3

-


lim a 0x=

x 3-



asymptote

EXAMPLES

1 Find lim x

x x5x 0

2+

Solution


which is undefined


( )

lim lim

lim

xx x

x

x x

x

5 5

5

5

x x

x

0

2

0 1

1

0

+=

+

= +

=

] g




2 Find limx

x

4

22

-

-

x 2

Solution


lim lim

lim

x

x

x x

x

x

4

2

2 2

2

21

41

2 1

1

-

-=

+ -

-

=+

=

x x2 2

x 2

^ _h i

3 Find limh

h x hx h2 72 2+ -

h 0

Solution

lim lim

lim

h

h x hx h

h

h hx x

hx x

x

2 7 2 7

2 7

7

2 2 2

2

2

+ -=

+ -

= + -

= -

h 0

h h0 0

^ h

Continuity





EXAMPLES

1 Find lim

x

x

1

12

-

-

x 1


11

y x

x2

=-

- Sketch the curve

Solution


xx2

-

- gives

00

CONTINUED




( )

lim lim

limx

x

x

x x

x11

1

1 1

1

2

x x

x

1

2

1

1

-

-=

-

+ -

= +

=

-

] ]g g

11

y xx2

=-





y xx

x

x x

x

11

1

1 1

1

2

=-

-

=

+

= +

-

-^ ^h h


2 Find limx

x x2

2x 2

2

+

+ -

-


x x2

22

= + -

+

Solution


x x2

22

+

+ - gives

00

lim lim

lim

xx x

x

x x

x

22

2

1 2

1

3

x x

x

2

2

2

2

+

+ -=

+

- +

=

= -

-

- -

-

^^ ^

^

hh h

h2 y

x

x xx

y x

x

x

x

22

2

2

1

1


=+

+ -= -

=+

= -

+ -^ ^h h


Remember that x 1




1 Find(a) lim x 52

+x 4

(b) lim t 7-t 3 -

(c) lim x x2 43+ -

x 2

(d) lim xx x32

+

x 0

(e) limh

h h

2

22

-

- -

h 2

(f) lim y

y

5

1253

-

-

y 5

(g) limx

x x

12 12

+

+ +

x 1-

(h) limx

x x4

2 82

+

+ -

x 4 -

(i) limc

c

4

22

-

-

c 2

(j) limx x

x 12

-

-

x 1

(k) lim h

h h h2 73 2+ -

h 0

(l) limh

hx hx h32 2- +

h 0

(m) limh

hx h x hx h2 3 53 2 2- + -

h 0

(n) lim x c x c 3 3

-

-

x c



discontinuous

(a) 3 y x2= -

(b)1

1 y

x=

+

(c) f x x 1= -] g

(d)4

1 y

x2=

+

(e)4

1

y x2=

-



(a)3

y xx x2

= +

(b)33

y x

x x2

=+

+

(c)1

5 4 y

xx x2

=+

+ +

510 Exercises




Regions

Class Investigation


them all







EXAMPLE


Solution



y

x =3

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4







region

EXAMPLE


Solution



included


shown

y

x

1

1

-2 2 3 4

-3

-4

-5

2

3

4

5

-1

-2

-3-4

-1 y = -1



y -intercept 1-






EXAMPLES


1 y x 2$ +

Solution






y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0



2 x y 2 3 61-

Solution


region


side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g








3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g









Domain all real x

Range y y b=

54 Exercises


each function

(a) y x 2= -

(b) f x x2 3= +] g (c) x y 2 1 0+ =-

(d) x y 3 0+ =-

(e) x y 3 6 2 0=- -


line

(a) x 4=

(b) x 3 0=-

(c) y 5=

(d) y 1 0+ =

(e) f x x2 1= -] g (f) y x 4= +

(g) f x x3 2= +] g

(h) x y 3+ =

(i) x y 1 0=- -

(j) x y 2 3 0+ =-


(a) x y 3 2 7 0+ =-

(b) y 2=

(c) x 4= - (d) x 2 0=-

(e) y 3 0=-


are even or odd

(a) y x2=

(b) y 3=

(c) x 4=

(d) y x= -

(e) y x=




meet




Applications






The pronumeral

a is called the


Quadratic Function










EXAMPLES



Solution



x

x

x

0 1

1

1

2

2

= -

=

=


0 1

1

y 2= -

= -



Domain xall real

Range y y 1$ -


Solution


parabola will lie




left



0

1 0

1

x

x

x

1 2= +

+ =

= -

] g


1

y 0 1 2= +

=

] g

-1

-

4

3

2

1

5

-2

-4

-5

-

1 41-2 5-4

y

x









Solution



x x

x xx x

x x

0 6

3 23 0 2 0

3 2

2= + -

= + -

+ = - =

= - =

] ]g g


f 0 0 0 66

2= + -

= -

] ] ]g g g



x -intercepts


23 2

21- +

= -


-c m

f 21

21

21

6

41

21

6

641

2

- = - + - -

= - -

= -

c c cm m m


-

CONTINUED

You will learn more


-1

-3

4

3

2

1

5

-2

-4

-5

1 4-1-2 32-3-4

y

x










41



2

11 -





Solution


So f x 0=] g

0 4 54 5 0

0

x xx x

x x5 1

2

2

= - + +

=

+ =

- -

-] ]g g

x x

x x

5 0 1 0

5 1

- = + =

= = -


f 0 0 4 0 5

5

2= - + +

=

] ] ]g g g

-1

-3

4

2

5

-2

-4

-5

-

y

1 41-2 5-4

x

-2 4






21 5

2- +

=

f 2 2 4 2 59

= - + +

=

2] ] ]g g g










each function

(a) 2 y x x2= +

(b) 3 y x x2= - +

(c) f x x 12= -] g

(d) y x x 22= - -

(e) y x x9 82= +-

2 Sketch

(a) 2 y x2= +

(b) y x 12= - +

(c) f x x 42= -] g

(d) 2 y x x2= +

(e) y x x2= - -

(f) f x x 3= -2] ]g g

55 Exercises

-1

9

8

7

5

4

3

2

6

1

-2

-3

-4

-5

y

2 51 643-1-2-3-4

x




EXAMPLES


Solution









(g) f x x 1 2= +] ]g g

(h) y x x3 42= + -

(i) y x x2 5 32= - +

(j) f x x x3 22= - + -] g



(a) ndash y x x7 122= +

(b) f x x x42= +] g

(c) y x x2 82= - -

(d) y x x6 92= +-

(e) f t t 4 2= -] g


(a) y x 52= -

(b) f x x x6

2= -

] g (c) f x x x 22= - -] g

(d) y x2= -

(e) f x x 7 2= -] ]g g



(a) y x2= for x0 3

(b) y x 42= - + for x1 2 -

(c) f x x 12= -] g for x2 5 -

(d) y x x2 32= + - for x2 4 -

(e) y x x

22= - +- for

x0 4


function is

increasing(i)

decreasing(ii)

(a) y x2=

(b) y x2

= - (c) f x x 92

= -] g

(d) y x x42= - +

(e) f x x 5 2= +] ]g g


function



(a) y x 12= +

(b) f x x 32= -

] g

(c) y x2 2= -

(d) f x x x32= -] g

(e) f x x x2= +] g

(f) y x 42= -

(g) y x x2 32= - -

(h) y x x5 42= +-

(i) p x x 1 2= +] ]g g

(j) y x 2= -2] g




CONTINUED

eg When x 3= -

| | y 3 13 12

= - -

= -

=

x -3 -2 -1 0 1 2 3

y 2 1 0 -1 0 1 2


y

-2

4

3

2

1

5

-1

-3

-4

-5

1 4-1-2 32-3-4

x


| | y x

x x

x x1

1 0

1 0

when

when 1

$= - =

-


y x x1 0$= - ] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x - 1




y x 1= - - x 01] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y =- x - 1



-3

4

3

2

1

5

-2

-1

-4

-5

y y

3-1-2 421-3-4

x

y = - x - 1

y = x - 1


intercepts


So f x 0=

] g

| |

| |

x

x

x

0 1

1

1`

= -

=

=


( ) | |f 0 0 11

= -

= -





-3

4

3

2

1

5

-2

-1

-4

-5

y

4-2 5321-1-3-4

x



Domain all real x

Range y y 1$ -

2 Sketch | | y x 2= +

Solution


| | ( ) y x x xx x2 2 2 0

2 2 0whenwhen 1

$= + = + +

- + +


2 y x= + when x 2 0$+

x 2$ -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x + 2






y-intercept of 1-

CONTINUED




2 y x= - +] g when x 2 01+

ie y x 2= - - when x 21 -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2


-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2

y = x + 2



So 0f x =] g

0 | 2 |

0 2

2

x

x

x

= +

= +

- =


(0) | 0 2 |

2

f = +

=

There is only one

solution for the


Can you see why





-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

If you know how to

sketch the graph of




Investigation




are their equations





1 | |f x k x=] g

2 | |f x x k= +] g

3 | |f x x k= +] g







EXAMPLES

Solve

1 |2 1 | 3x - =

Solution



is x 1 2= -

2 |2 1 | 3 2x x= -+

Solution



3 | 1 | 2x 1+

Solution






and then check them







each function

(a) | | y x=

(b) | |f x x 7= +] g

(c) | |f x x 2= -] g

(d) 5 | | y x=

(e) | |f x x 3= - +] g

(f) | 6 | y x= +

(g) | |f x x3 2= -] g

(h) | 5 4 | y x= +

(i) | 7 1 | y x= -

(j) | |f x x2 9= +] g


plane

(a) | | y x=

(b) | |f x x 1= +] g

(c) | |f x x 3= -] g

(d) 2 | | y x=

(e) | |f x x= -] g

(f) | 1 | y x= +

(g) | |f x x 1= - -] g

(h) | 2 3 | y x= -

(i) | 4 2 | y x= +

(j) | |f x x3 1= +] g


each function

(a) | 1 | y x= -

(b) | |f x x 8= -] g

(c) | |f x x2 5= +] g

(d) 2 | | 3 y x= -

(e) | |f x x 3= - -] g


function is

increasing(i)

decreasing(ii)

(a) | 2 | y x= -

(b) | |f x x 2= +

] g

(c) | |f x x2 3= -] g

(d) 4 | | 1 y x= -

(e) | |f x x= -] g


of each function

(a) | | y x= for x2 2 -

(b) | |f x x 4= - -] g for

x4 3 -

(c) | |f x x 4= +] g for x7 2 -

(d) | 2 5 | y x= - for x3 3 -

(e) | |f x x= -] g for x1 1 -


function increasing

(a) | 3 | y x= +

(b) | |f x x 4= - +] g

(c) | |f x x 9= -] g

(d) | | y x 2 1= - -

(e) | |f x x 2= - +] g

56 Exercises




7 Solve graphically

(a) | | 3x =

(b) | |x 12

(c) | |x 2

(d) | 2 | 1x + =

(e) | 3 | 0x- =

(f) |2 3 | 1x - =

(g) | |x 1 41-

(h) | |x 1 3+

(i) | |x 2 22-

(j) | |x 3 1$-

(k) | |x2 3 5+

(l) | |x2 1 1$-

(m) |3 1 | 3x x- = +

(n) |3 2 | 4x x- = -

(o) | 1 | 1x x- = +

(p) | 3 | 2 2x x+ = + (q) |2 1 | 1x x+ = -

(r) |2 5 | 3x x- = -

(s) | 1 | 2x x- =

(t) |2 3 | 3x x- = +

The Hyperbola


or= =

EXAMPLE

Sketch1

y x=

Solution



x -3 -2 -121-

41- 0

41

21 1 2 3

y 3

1-

2

1- -1 -2 -4 mdash 4 2 1

2

1

3

1

Class Discussion



y is never 0 Why






but never touches)



Investigation




equations




1 y xk

=

21

y x k= +

31

y x k

=+


EXAMPLES


3=

-

] g


Solution



So x 3






be zero

CONTINUED







1

y 0 3

3=

-

= -

-3

4

3

2

1

5

-2

-1

-4

-5

y

-1-2 4 521-3-4

x

x = 3

y = 0

Asymptotes

3

2 Sketch y x2 4

1= -

+

Solution




x

x

2 4

2

-

-






( ) y

2 0 41

41

= -+

= -


a translation of3

y x

=





y

-2

x

-

1

4


a=

+


domain x xb



1 For each graph



exists


(a)2

y x=

(b)1

y x= -

(c) f xx 1

1=

+] g

(d) f xx 2

3=

-

] g

(e)3 6

1 y

x=

+

(f) f xx 3

2= -

-

] g

(g) f xx 1

4=

-

] g

(h)1

2 y

x= -

+

(i) f xx6 3

2=

-

] g

(j)2

6 y

x= -

+

2 Show that f x x2

=] g is an odd

function



(a) f xx2 5

1=

+

] g for x2 2 -

(b)3

1 y

x=

+ for x2 0 -

(c) f xx2 4

5=

-

] g for x3 1 -

57 Exercises




(d) f xx 4

3= -

-

] g for x3 3 -

(e)3 1

2 y

x= -

+ for x0 5



(a)3

y x= for y 1 3

(b)2

y x= - for y 221

- -

(c) f xx 1

1=

-

] g for y 171

- -

(d) f xx2 1

3= -

+

] g for

y 131

- -

(e)3 2

6 y

x=

- for y 1

21

6



Circle gate





Proof

r y

x

( x y)

y

x








c a b

r x y

2 2 2

2 2 2

`

= +

= +

EXAMPLE



Solution


y

x

-2

-2 2

2


one place

y

x

2

2

2

-2

The radius is 4

CONTINUED






Domain 2 2 x x -

Range 2 2 y y -


range y r y r -



Proof


shown

y

x

(a b)

x

y

r

( x y)

a

b x - a

y - b




ndash ndash

c a b

r x a y b

2 2 2

2 2 2

= +

= +] ^g h




EXAMPLES



Solution



y

x 9

9

-9

-9



Domain 9 9 x x -

Range 9 9 y y -



Solution


ndash

ndash ndash

x y

x y

1 2 4

1 2 2

2

2 2

+ + =

+ - =

2

2

] ^

] ]_

g h

g gi

So 1 2a b= = - and 2r =

CONTINUED







y

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

(1 -2)



Domain 1 3 x x -

Range 4 0 y y -


Solution


a b2 1= - = and r 3=

Substituting

ndash ndash

ndash

ndash

x a y b r

x y

x y

2 1 3

2 1 9

2 2 2

2 2 2

2 2

+ =

- - + =

+ + =

] ^]] ^

] ^

g hg g hg h


ndash

ndash

a b a ab b

x x x

x x

a b a ab b

y y y

y y

2

2 2 2 2

4 4

2

1 2 1 1

2 1

So

So

2 2 2

2 2 2

2

2 2 2

2 2 2

2

+ = + +

+ = + +

= + +

= - +

= - +

= - +

]] ] ]]^ ^ ]

gg g ggh h g


ndash

x x y y

x x y y

x x y y

x x y y

4 4 2 1 9

4 2 5 9

4 2 5 9

4 2 4 0

9 9

2

2

2

2

+ + + - + =

+ + - + =

+ + + =

+ + - - =

- -


in Chapter 2




Investigation


two functions





circle






for all values

y

x r

r

-r


Proof

ndash

x y r

y r x y r x

2 2 2

2 2 2

2 2

+ =

=

= -



semi-circles





y 0 for all values

y

x r

-

r

-r


EXAMPLES


1 f x x92

= -] g

Solution



y

x 3

3

-3

Domain 3 3 x x -

Range 0 3 y y




2 y x4 2= - -

Solution



y

x 2

-2

-2

Domain 2 2 x x -

Range 2 0 y y -




range

(a) 9x y 2 2+ =

(b) x y 16 02 2+ =-


(d) 1 9x y 2 2+ + =

] g

(e) ndashx y 2 1 12 2+ + =] ^g h



below the x -axis



range

(a) 25 y x2= - -

(b) 1 y x2= -

(c) 36 y x2= -

(d) 64 y x2= - -

(e) 7 y x2= - -



each circle

(a) 100x y 2 2+ =

(b) 5x y 2 2+ =


(d) ndashx y 5 6 492 2+ + =] ^g h

(e) ndashx y 3 812 2+ =^ h

58 Exercises






grouping symbols)



Centre(c) 1 5-







Centre(j) 4 7- -

^ h and radius 3

Other Graphs




EXAMPLES


Solution



eg When 0x =

y 3

1

c=

=

x

y

1

3

3

1

31

When1

1

= -

=

=

=

-

x 3- 2- 1- 0 1 2 3

y 271

91

31

1 3 9 27



You will meet these

graphs again in the

HSC Course





other point

When

x

y

1

3

3

1

=

=

=

y

x

1

2

1

3







Solution



x minus2 minus1 0 05 1 2 3 4

y minus03 0 03 05 06

y

x

1

2

1 2 3 4

-1




You learned about


stages of maths





range y y 02



range all real y

Cubic function


EXAMPLE



Solution



y minus25 minus6 1 2 3 10 29

y

x

1

1

-2 2 3 4

-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4



If you already

know the shape of


3






Domain and range


EXAMPLE


Solution




function are

f

f

1 1 2

1 21

4 4 2

64 2

66

3

3

- = - +

= - +

=

= +

= +

=

] ]

] ]

g g

g g


these points

y

x

(-1 1)

(4 66)

Range 1 66 y y










positive number

EXAMPLE


Solution


x 4So $

$


(or zero) So y 0$


59 Exercises


(a) 4 3 y x= +

(b) f x 4= -] g

(c) 3x =



(f) f x xx 12 2= - -] g

(g) 64x y 2 2+ =

(h) f t t 4

3=

-] g

(i) ( ) g 2

5zz

= +

(j) | |f x x=] g


(a) y x=

(b) 2 y x= -

(c) | |f x x2 3= -] g

(d) | | 2 y x= -

(e) f x x2 5= - +] g

(f) | | y x5= -

(g) 2 y x=

(h) y 5x= -

(i) f x xx 1

= +] g

(j)2

4 3 y x

x= -


(a) y x x 5 2= -] g


= +-

(d) g x x x164 2= -] g

(e) 49x y 2 2+ =

You may like to


by dividing by x




4 (a) Solve x1 02$-


f x x1 2= -] g


(a) 2 y x x2= - -

(b) g t t t 62= +] g



in each case


x3 3 -


x2 3 -


x2 1 - (d)


x1 5


0 4x


x3 3 -


x1 1 -


x2 3 -


x4 4 -

(j) y x x7 62= - + - in the

domain 0 7x


function y x 1

3=

+





x=] g


function

find its range(b)


plane

(a) f x x4=] g

(b) y x3= -

(c) y x 34= -

(d) 2 p x x3=] g

(e) 1 g x x3= +] g

(f) 100x y 2 2+ =

(g) 2 1 y x= +


y x 1= -









(b) f x x 22= -] g

(c)1

y x=

(d) f x x3=] g

(e) f x 3x=] g

13 (a) Solve x4 02$-


(i) 4 y x2= -

(ii) y x4

2= - -




DID YOU KNOW




a light



Limits






a a1

1

03

Z

=

=

3

3

-


lim a 0x=

x 3-



asymptote

EXAMPLES

1 Find lim x

x x5x 0

2+

Solution


which is undefined


( )

lim lim

lim

xx x

x

x x

x

5 5

5

5

x x

x

0

2

0 1

1

0

+=

+

= +

=

] g




2 Find limx

x

4

22

-

-

x 2

Solution


lim lim

lim

x

x

x x

x

x

4

2

2 2

2

21

41

2 1

1

-

-=

+ -

-

=+

=

x x2 2

x 2

^ _h i

3 Find limh

h x hx h2 72 2+ -

h 0

Solution

lim lim

lim

h

h x hx h

h

h hx x

hx x

x

2 7 2 7

2 7

7

2 2 2

2

2

+ -=

+ -

= + -

= -

h 0

h h0 0

^ h

Continuity





EXAMPLES

1 Find lim

x

x

1

12

-

-

x 1


11

y x

x2

=-

- Sketch the curve

Solution


xx2

-

- gives

00

CONTINUED




( )

lim lim

limx

x

x

x x

x11

1

1 1

1

2

x x

x

1

2

1

1

-

-=

-

+ -

= +

=

-

] ]g g

11

y xx2

=-





y xx

x

x x

x

11

1

1 1

1

2

=-

-

=

+

= +

-

-^ ^h h


2 Find limx

x x2

2x 2

2

+

+ -

-


x x2

22

= + -

+

Solution


x x2

22

+

+ - gives

00

lim lim

lim

xx x

x

x x

x

22

2

1 2

1

3

x x

x

2

2

2

2

+

+ -=

+

- +

=

= -

-

- -

-

^^ ^

^

hh h

h2 y

x

x xx

y x

x

x

x

22

2

2

1

1


=+

+ -= -

=+

= -

+ -^ ^h h


Remember that x 1




1 Find(a) lim x 52

+x 4

(b) lim t 7-t 3 -

(c) lim x x2 43+ -

x 2

(d) lim xx x32

+

x 0

(e) limh

h h

2

22

-

- -

h 2

(f) lim y

y

5

1253

-

-

y 5

(g) limx

x x

12 12

+

+ +

x 1-

(h) limx

x x4

2 82

+

+ -

x 4 -

(i) limc

c

4

22

-

-

c 2

(j) limx x

x 12

-

-

x 1

(k) lim h

h h h2 73 2+ -

h 0

(l) limh

hx hx h32 2- +

h 0

(m) limh

hx h x hx h2 3 53 2 2- + -

h 0

(n) lim x c x c 3 3

-

-

x c



discontinuous

(a) 3 y x2= -

(b)1

1 y

x=

+

(c) f x x 1= -] g

(d)4

1 y

x2=

+

(e)4

1

y x2=

-



(a)3

y xx x2

= +

(b)33

y x

x x2

=+

+

(c)1

5 4 y

xx x2

=+

+ +

510 Exercises




Regions

Class Investigation


them all







EXAMPLE


Solution



y

x =3

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4







region

EXAMPLE


Solution



included


shown

y

x

1

1

-2 2 3 4

-3

-4

-5

2

3

4

5

-1

-2

-3-4

-1 y = -1



y -intercept 1-






EXAMPLES


1 y x 2$ +

Solution






y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0



2 x y 2 3 61-

Solution


region


side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g








3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g






Applications






The pronumeral

a is called the


Quadratic Function










EXAMPLES



Solution



x

x

x

0 1

1

1

2

2

= -

=

=


0 1

1

y 2= -

= -



Domain xall real

Range y y 1$ -


Solution


parabola will lie




left



0

1 0

1

x

x

x

1 2= +

+ =

= -

] g


1

y 0 1 2= +

=

] g

-1

-

4

3

2

1

5

-2

-4

-5

-

1 41-2 5-4

y

x









Solution



x x

x xx x

x x

0 6

3 23 0 2 0

3 2

2= + -

= + -

+ = - =

= - =

] ]g g


f 0 0 0 66

2= + -

= -

] ] ]g g g



x -intercepts


23 2

21- +

= -


-c m

f 21

21

21

6

41

21

6

641

2

- = - + - -

= - -

= -

c c cm m m


-

CONTINUED

You will learn more


-1

-3

4

3

2

1

5

-2

-4

-5

1 4-1-2 32-3-4

y

x










41



2

11 -





Solution


So f x 0=] g

0 4 54 5 0

0

x xx x

x x5 1

2

2

= - + +

=

+ =

- -

-] ]g g

x x

x x

5 0 1 0

5 1

- = + =

= = -


f 0 0 4 0 5

5

2= - + +

=

] ] ]g g g

-1

-3

4

2

5

-2

-4

-5

-

y

1 41-2 5-4

x

-2 4






21 5

2- +

=

f 2 2 4 2 59

= - + +

=

2] ] ]g g g










each function

(a) 2 y x x2= +

(b) 3 y x x2= - +

(c) f x x 12= -] g

(d) y x x 22= - -

(e) y x x9 82= +-

2 Sketch

(a) 2 y x2= +

(b) y x 12= - +

(c) f x x 42= -] g

(d) 2 y x x2= +

(e) y x x2= - -

(f) f x x 3= -2] ]g g

55 Exercises

-1

9

8

7

5

4

3

2

6

1

-2

-3

-4

-5

y

2 51 643-1-2-3-4

x




EXAMPLES


Solution









(g) f x x 1 2= +] ]g g

(h) y x x3 42= + -

(i) y x x2 5 32= - +

(j) f x x x3 22= - + -] g



(a) ndash y x x7 122= +

(b) f x x x42= +] g

(c) y x x2 82= - -

(d) y x x6 92= +-

(e) f t t 4 2= -] g


(a) y x 52= -

(b) f x x x6

2= -

] g (c) f x x x 22= - -] g

(d) y x2= -

(e) f x x 7 2= -] ]g g



(a) y x2= for x0 3

(b) y x 42= - + for x1 2 -

(c) f x x 12= -] g for x2 5 -

(d) y x x2 32= + - for x2 4 -

(e) y x x

22= - +- for

x0 4


function is

increasing(i)

decreasing(ii)

(a) y x2=

(b) y x2

= - (c) f x x 92

= -] g

(d) y x x42= - +

(e) f x x 5 2= +] ]g g


function



(a) y x 12= +

(b) f x x 32= -

] g

(c) y x2 2= -

(d) f x x x32= -] g

(e) f x x x2= +] g

(f) y x 42= -

(g) y x x2 32= - -

(h) y x x5 42= +-

(i) p x x 1 2= +] ]g g

(j) y x 2= -2] g




CONTINUED

eg When x 3= -

| | y 3 13 12

= - -

= -

=

x -3 -2 -1 0 1 2 3

y 2 1 0 -1 0 1 2


y

-2

4

3

2

1

5

-1

-3

-4

-5

1 4-1-2 32-3-4

x


| | y x

x x

x x1

1 0

1 0

when

when 1

$= - =

-


y x x1 0$= - ] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x - 1




y x 1= - - x 01] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y =- x - 1



-3

4

3

2

1

5

-2

-1

-4

-5

y y

3-1-2 421-3-4

x

y = - x - 1

y = x - 1


intercepts


So f x 0=

] g

| |

| |

x

x

x

0 1

1

1`

= -

=

=


( ) | |f 0 0 11

= -

= -





-3

4

3

2

1

5

-2

-1

-4

-5

y

4-2 5321-1-3-4

x



Domain all real x

Range y y 1$ -

2 Sketch | | y x 2= +

Solution


| | ( ) y x x xx x2 2 2 0

2 2 0whenwhen 1

$= + = + +

- + +


2 y x= + when x 2 0$+

x 2$ -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x + 2






y-intercept of 1-

CONTINUED




2 y x= - +] g when x 2 01+

ie y x 2= - - when x 21 -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2


-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2

y = x + 2



So 0f x =] g

0 | 2 |

0 2

2

x

x

x

= +

= +

- =


(0) | 0 2 |

2

f = +

=

There is only one

solution for the


Can you see why





-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

If you know how to

sketch the graph of




Investigation




are their equations





1 | |f x k x=] g

2 | |f x x k= +] g

3 | |f x x k= +] g







EXAMPLES

Solve

1 |2 1 | 3x - =

Solution



is x 1 2= -

2 |2 1 | 3 2x x= -+

Solution



3 | 1 | 2x 1+

Solution






and then check them







each function

(a) | | y x=

(b) | |f x x 7= +] g

(c) | |f x x 2= -] g

(d) 5 | | y x=

(e) | |f x x 3= - +] g

(f) | 6 | y x= +

(g) | |f x x3 2= -] g

(h) | 5 4 | y x= +

(i) | 7 1 | y x= -

(j) | |f x x2 9= +] g


plane

(a) | | y x=

(b) | |f x x 1= +] g

(c) | |f x x 3= -] g

(d) 2 | | y x=

(e) | |f x x= -] g

(f) | 1 | y x= +

(g) | |f x x 1= - -] g

(h) | 2 3 | y x= -

(i) | 4 2 | y x= +

(j) | |f x x3 1= +] g


each function

(a) | 1 | y x= -

(b) | |f x x 8= -] g

(c) | |f x x2 5= +] g

(d) 2 | | 3 y x= -

(e) | |f x x 3= - -] g


function is

increasing(i)

decreasing(ii)

(a) | 2 | y x= -

(b) | |f x x 2= +

] g

(c) | |f x x2 3= -] g

(d) 4 | | 1 y x= -

(e) | |f x x= -] g


of each function

(a) | | y x= for x2 2 -

(b) | |f x x 4= - -] g for

x4 3 -

(c) | |f x x 4= +] g for x7 2 -

(d) | 2 5 | y x= - for x3 3 -

(e) | |f x x= -] g for x1 1 -


function increasing

(a) | 3 | y x= +

(b) | |f x x 4= - +] g

(c) | |f x x 9= -] g

(d) | | y x 2 1= - -

(e) | |f x x 2= - +] g

56 Exercises




7 Solve graphically

(a) | | 3x =

(b) | |x 12

(c) | |x 2

(d) | 2 | 1x + =

(e) | 3 | 0x- =

(f) |2 3 | 1x - =

(g) | |x 1 41-

(h) | |x 1 3+

(i) | |x 2 22-

(j) | |x 3 1$-

(k) | |x2 3 5+

(l) | |x2 1 1$-

(m) |3 1 | 3x x- = +

(n) |3 2 | 4x x- = -

(o) | 1 | 1x x- = +

(p) | 3 | 2 2x x+ = + (q) |2 1 | 1x x+ = -

(r) |2 5 | 3x x- = -

(s) | 1 | 2x x- =

(t) |2 3 | 3x x- = +

The Hyperbola


or= =

EXAMPLE

Sketch1

y x=

Solution



x -3 -2 -121-

41- 0

41

21 1 2 3

y 3

1-

2

1- -1 -2 -4 mdash 4 2 1

2

1

3

1

Class Discussion



y is never 0 Why






but never touches)



Investigation




equations




1 y xk

=

21

y x k= +

31

y x k

=+


EXAMPLES


3=

-

] g


Solution



So x 3






be zero

CONTINUED







1

y 0 3

3=

-

= -

-3

4

3

2

1

5

-2

-1

-4

-5

y

-1-2 4 521-3-4

x

x = 3

y = 0

Asymptotes

3

2 Sketch y x2 4

1= -

+

Solution




x

x

2 4

2

-

-






( ) y

2 0 41

41

= -+

= -


a translation of3

y x

=





y

-2

x

-

1

4


a=

+


domain x xb



1 For each graph



exists


(a)2

y x=

(b)1

y x= -

(c) f xx 1

1=

+] g

(d) f xx 2

3=

-

] g

(e)3 6

1 y

x=

+

(f) f xx 3

2= -

-

] g

(g) f xx 1

4=

-

] g

(h)1

2 y

x= -

+

(i) f xx6 3

2=

-

] g

(j)2

6 y

x= -

+

2 Show that f x x2

=] g is an odd

function



(a) f xx2 5

1=

+

] g for x2 2 -

(b)3

1 y

x=

+ for x2 0 -

(c) f xx2 4

5=

-

] g for x3 1 -

57 Exercises




(d) f xx 4

3= -

-

] g for x3 3 -

(e)3 1

2 y

x= -

+ for x0 5



(a)3

y x= for y 1 3

(b)2

y x= - for y 221

- -

(c) f xx 1

1=

-

] g for y 171

- -

(d) f xx2 1

3= -

+

] g for

y 131

- -

(e)3 2

6 y

x=

- for y 1

21

6



Circle gate





Proof

r y

x

( x y)

y

x








c a b

r x y

2 2 2

2 2 2

`

= +

= +

EXAMPLE



Solution


y

x

-2

-2 2

2


one place

y

x

2

2

2

-2

The radius is 4

CONTINUED






Domain 2 2 x x -

Range 2 2 y y -


range y r y r -



Proof


shown

y

x

(a b)

x

y

r

( x y)

a

b x - a

y - b




ndash ndash

c a b

r x a y b

2 2 2

2 2 2

= +

= +] ^g h




EXAMPLES



Solution



y

x 9

9

-9

-9



Domain 9 9 x x -

Range 9 9 y y -



Solution


ndash

ndash ndash

x y

x y

1 2 4

1 2 2

2

2 2

+ + =

+ - =

2

2

] ^

] ]_

g h

g gi

So 1 2a b= = - and 2r =

CONTINUED







y

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

(1 -2)



Domain 1 3 x x -

Range 4 0 y y -


Solution


a b2 1= - = and r 3=

Substituting

ndash ndash

ndash

ndash

x a y b r

x y

x y

2 1 3

2 1 9

2 2 2

2 2 2

2 2

+ =

- - + =

+ + =

] ^]] ^

] ^

g hg g hg h


ndash

ndash

a b a ab b

x x x

x x

a b a ab b

y y y

y y

2

2 2 2 2

4 4

2

1 2 1 1

2 1

So

So

2 2 2

2 2 2

2

2 2 2

2 2 2

2

+ = + +

+ = + +

= + +

= - +

= - +

= - +

]] ] ]]^ ^ ]

gg g ggh h g


ndash

x x y y

x x y y

x x y y

x x y y

4 4 2 1 9

4 2 5 9

4 2 5 9

4 2 4 0

9 9

2

2

2

2

+ + + - + =

+ + - + =

+ + + =

+ + - - =

- -


in Chapter 2




Investigation


two functions





circle






for all values

y

x r

r

-r


Proof

ndash

x y r

y r x y r x

2 2 2

2 2 2

2 2

+ =

=

= -



semi-circles





y 0 for all values

y

x r

-

r

-r


EXAMPLES


1 f x x92

= -] g

Solution



y

x 3

3

-3

Domain 3 3 x x -

Range 0 3 y y




2 y x4 2= - -

Solution



y

x 2

-2

-2

Domain 2 2 x x -

Range 2 0 y y -




range

(a) 9x y 2 2+ =

(b) x y 16 02 2+ =-


(d) 1 9x y 2 2+ + =

] g

(e) ndashx y 2 1 12 2+ + =] ^g h



below the x -axis



range

(a) 25 y x2= - -

(b) 1 y x2= -

(c) 36 y x2= -

(d) 64 y x2= - -

(e) 7 y x2= - -



each circle

(a) 100x y 2 2+ =

(b) 5x y 2 2+ =


(d) ndashx y 5 6 492 2+ + =] ^g h

(e) ndashx y 3 812 2+ =^ h

58 Exercises






grouping symbols)



Centre(c) 1 5-







Centre(j) 4 7- -

^ h and radius 3

Other Graphs




EXAMPLES


Solution



eg When 0x =

y 3

1

c=

=

x

y

1

3

3

1

31

When1

1

= -

=

=

=

-

x 3- 2- 1- 0 1 2 3

y 271

91

31

1 3 9 27



You will meet these

graphs again in the

HSC Course





other point

When

x

y

1

3

3

1

=

=

=

y

x

1

2

1

3







Solution



x minus2 minus1 0 05 1 2 3 4

y minus03 0 03 05 06

y

x

1

2

1 2 3 4

-1




You learned about


stages of maths





range y y 02



range all real y

Cubic function


EXAMPLE



Solution



y minus25 minus6 1 2 3 10 29

y

x

1

1

-2 2 3 4

-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4



If you already

know the shape of


3






Domain and range


EXAMPLE


Solution




function are

f

f

1 1 2

1 21

4 4 2

64 2

66

3

3

- = - +

= - +

=

= +

= +

=

] ]

] ]

g g

g g


these points

y

x

(-1 1)

(4 66)

Range 1 66 y y










positive number

EXAMPLE


Solution


x 4So $

$


(or zero) So y 0$


59 Exercises


(a) 4 3 y x= +

(b) f x 4= -] g

(c) 3x =



(f) f x xx 12 2= - -] g

(g) 64x y 2 2+ =

(h) f t t 4

3=

-] g

(i) ( ) g 2

5zz

= +

(j) | |f x x=] g


(a) y x=

(b) 2 y x= -

(c) | |f x x2 3= -] g

(d) | | 2 y x= -

(e) f x x2 5= - +] g

(f) | | y x5= -

(g) 2 y x=

(h) y 5x= -

(i) f x xx 1

= +] g

(j)2

4 3 y x

x= -


(a) y x x 5 2= -] g


= +-

(d) g x x x164 2= -] g

(e) 49x y 2 2+ =

You may like to


by dividing by x




4 (a) Solve x1 02$-


f x x1 2= -] g


(a) 2 y x x2= - -

(b) g t t t 62= +] g



in each case


x3 3 -


x2 3 -


x2 1 - (d)


x1 5


0 4x


x3 3 -


x1 1 -


x2 3 -


x4 4 -

(j) y x x7 62= - + - in the

domain 0 7x


function y x 1

3=

+





x=] g


function

find its range(b)


plane

(a) f x x4=] g

(b) y x3= -

(c) y x 34= -

(d) 2 p x x3=] g

(e) 1 g x x3= +] g

(f) 100x y 2 2+ =

(g) 2 1 y x= +


y x 1= -









(b) f x x 22= -] g

(c)1

y x=

(d) f x x3=] g

(e) f x 3x=] g

13 (a) Solve x4 02$-


(i) 4 y x2= -

(ii) y x4

2= - -




DID YOU KNOW




a light



Limits






a a1

1

03

Z

=

=

3

3

-


lim a 0x=

x 3-



asymptote

EXAMPLES

1 Find lim x

x x5x 0

2+

Solution


which is undefined


( )

lim lim

lim

xx x

x

x x

x

5 5

5

5

x x

x

0

2

0 1

1

0

+=

+

= +

=

] g




2 Find limx

x

4

22

-

-

x 2

Solution


lim lim

lim

x

x

x x

x

x

4

2

2 2

2

21

41

2 1

1

-

-=

+ -

-

=+

=

x x2 2

x 2

^ _h i

3 Find limh

h x hx h2 72 2+ -

h 0

Solution

lim lim

lim

h

h x hx h

h

h hx x

hx x

x

2 7 2 7

2 7

7

2 2 2

2

2

+ -=

+ -

= + -

= -

h 0

h h0 0

^ h

Continuity





EXAMPLES

1 Find lim

x

x

1

12

-

-

x 1


11

y x

x2

=-

- Sketch the curve

Solution


xx2

-

- gives

00

CONTINUED




( )

lim lim

limx

x

x

x x

x11

1

1 1

1

2

x x

x

1

2

1

1

-

-=

-

+ -

= +

=

-

] ]g g

11

y xx2

=-





y xx

x

x x

x

11

1

1 1

1

2

=-

-

=

+

= +

-

-^ ^h h


2 Find limx

x x2

2x 2

2

+

+ -

-


x x2

22

= + -

+

Solution


x x2

22

+

+ - gives

00

lim lim

lim

xx x

x

x x

x

22

2

1 2

1

3

x x

x

2

2

2

2

+

+ -=

+

- +

=

= -

-

- -

-

^^ ^

^

hh h

h2 y

x

x xx

y x

x

x

x

22

2

2

1

1


=+

+ -= -

=+

= -

+ -^ ^h h


Remember that x 1




1 Find(a) lim x 52

+x 4

(b) lim t 7-t 3 -

(c) lim x x2 43+ -

x 2

(d) lim xx x32

+

x 0

(e) limh

h h

2

22

-

- -

h 2

(f) lim y

y

5

1253

-

-

y 5

(g) limx

x x

12 12

+

+ +

x 1-

(h) limx

x x4

2 82

+

+ -

x 4 -

(i) limc

c

4

22

-

-

c 2

(j) limx x

x 12

-

-

x 1

(k) lim h

h h h2 73 2+ -

h 0

(l) limh

hx hx h32 2- +

h 0

(m) limh

hx h x hx h2 3 53 2 2- + -

h 0

(n) lim x c x c 3 3

-

-

x c



discontinuous

(a) 3 y x2= -

(b)1

1 y

x=

+

(c) f x x 1= -] g

(d)4

1 y

x2=

+

(e)4

1

y x2=

-



(a)3

y xx x2

= +

(b)33

y x

x x2

=+

+

(c)1

5 4 y

xx x2

=+

+ +

510 Exercises




Regions

Class Investigation


them all







EXAMPLE


Solution



y

x =3

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4







region

EXAMPLE


Solution



included


shown

y

x

1

1

-2 2 3 4

-3

-4

-5

2

3

4

5

-1

-2

-3-4

-1 y = -1



y -intercept 1-






EXAMPLES


1 y x 2$ +

Solution






y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0



2 x y 2 3 61-

Solution


region


side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g








3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g








EXAMPLES



Solution



x

x

x

0 1

1

1

2

2

= -

=

=


0 1

1

y 2= -

= -



Domain xall real

Range y y 1$ -


Solution


parabola will lie




left



0

1 0

1

x

x

x

1 2= +

+ =

= -

] g


1

y 0 1 2= +

=

] g

-1

-

4

3

2

1

5

-2

-4

-5

-

1 41-2 5-4

y

x









Solution



x x

x xx x

x x

0 6

3 23 0 2 0

3 2

2= + -

= + -

+ = - =

= - =

] ]g g


f 0 0 0 66

2= + -

= -

] ] ]g g g



x -intercepts


23 2

21- +

= -


-c m

f 21

21

21

6

41

21

6

641

2

- = - + - -

= - -

= -

c c cm m m


-

CONTINUED

You will learn more


-1

-3

4

3

2

1

5

-2

-4

-5

1 4-1-2 32-3-4

y

x










41



2

11 -





Solution


So f x 0=] g

0 4 54 5 0

0

x xx x

x x5 1

2

2

= - + +

=

+ =

- -

-] ]g g

x x

x x

5 0 1 0

5 1

- = + =

= = -


f 0 0 4 0 5

5

2= - + +

=

] ] ]g g g

-1

-3

4

2

5

-2

-4

-5

-

y

1 41-2 5-4

x

-2 4






21 5

2- +

=

f 2 2 4 2 59

= - + +

=

2] ] ]g g g










each function

(a) 2 y x x2= +

(b) 3 y x x2= - +

(c) f x x 12= -] g

(d) y x x 22= - -

(e) y x x9 82= +-

2 Sketch

(a) 2 y x2= +

(b) y x 12= - +

(c) f x x 42= -] g

(d) 2 y x x2= +

(e) y x x2= - -

(f) f x x 3= -2] ]g g

55 Exercises

-1

9

8

7

5

4

3

2

6

1

-2

-3

-4

-5

y

2 51 643-1-2-3-4

x




EXAMPLES


Solution









(g) f x x 1 2= +] ]g g

(h) y x x3 42= + -

(i) y x x2 5 32= - +

(j) f x x x3 22= - + -] g



(a) ndash y x x7 122= +

(b) f x x x42= +] g

(c) y x x2 82= - -

(d) y x x6 92= +-

(e) f t t 4 2= -] g


(a) y x 52= -

(b) f x x x6

2= -

] g (c) f x x x 22= - -] g

(d) y x2= -

(e) f x x 7 2= -] ]g g



(a) y x2= for x0 3

(b) y x 42= - + for x1 2 -

(c) f x x 12= -] g for x2 5 -

(d) y x x2 32= + - for x2 4 -

(e) y x x

22= - +- for

x0 4


function is

increasing(i)

decreasing(ii)

(a) y x2=

(b) y x2

= - (c) f x x 92

= -] g

(d) y x x42= - +

(e) f x x 5 2= +] ]g g


function



(a) y x 12= +

(b) f x x 32= -

] g

(c) y x2 2= -

(d) f x x x32= -] g

(e) f x x x2= +] g

(f) y x 42= -

(g) y x x2 32= - -

(h) y x x5 42= +-

(i) p x x 1 2= +] ]g g

(j) y x 2= -2] g




CONTINUED

eg When x 3= -

| | y 3 13 12

= - -

= -

=

x -3 -2 -1 0 1 2 3

y 2 1 0 -1 0 1 2


y

-2

4

3

2

1

5

-1

-3

-4

-5

1 4-1-2 32-3-4

x


| | y x

x x

x x1

1 0

1 0

when

when 1

$= - =

-


y x x1 0$= - ] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x - 1




y x 1= - - x 01] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y =- x - 1



-3

4

3

2

1

5

-2

-1

-4

-5

y y

3-1-2 421-3-4

x

y = - x - 1

y = x - 1


intercepts


So f x 0=

] g

| |

| |

x

x

x

0 1

1

1`

= -

=

=


( ) | |f 0 0 11

= -

= -





-3

4

3

2

1

5

-2

-1

-4

-5

y

4-2 5321-1-3-4

x



Domain all real x

Range y y 1$ -

2 Sketch | | y x 2= +

Solution


| | ( ) y x x xx x2 2 2 0

2 2 0whenwhen 1

$= + = + +

- + +


2 y x= + when x 2 0$+

x 2$ -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x + 2






y-intercept of 1-

CONTINUED




2 y x= - +] g when x 2 01+

ie y x 2= - - when x 21 -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2


-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2

y = x + 2



So 0f x =] g

0 | 2 |

0 2

2

x

x

x

= +

= +

- =


(0) | 0 2 |

2

f = +

=

There is only one

solution for the


Can you see why





-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

If you know how to

sketch the graph of




Investigation




are their equations





1 | |f x k x=] g

2 | |f x x k= +] g

3 | |f x x k= +] g







EXAMPLES

Solve

1 |2 1 | 3x - =

Solution



is x 1 2= -

2 |2 1 | 3 2x x= -+

Solution



3 | 1 | 2x 1+

Solution






and then check them







each function

(a) | | y x=

(b) | |f x x 7= +] g

(c) | |f x x 2= -] g

(d) 5 | | y x=

(e) | |f x x 3= - +] g

(f) | 6 | y x= +

(g) | |f x x3 2= -] g

(h) | 5 4 | y x= +

(i) | 7 1 | y x= -

(j) | |f x x2 9= +] g


plane

(a) | | y x=

(b) | |f x x 1= +] g

(c) | |f x x 3= -] g

(d) 2 | | y x=

(e) | |f x x= -] g

(f) | 1 | y x= +

(g) | |f x x 1= - -] g

(h) | 2 3 | y x= -

(i) | 4 2 | y x= +

(j) | |f x x3 1= +] g


each function

(a) | 1 | y x= -

(b) | |f x x 8= -] g

(c) | |f x x2 5= +] g

(d) 2 | | 3 y x= -

(e) | |f x x 3= - -] g


function is

increasing(i)

decreasing(ii)

(a) | 2 | y x= -

(b) | |f x x 2= +

] g

(c) | |f x x2 3= -] g

(d) 4 | | 1 y x= -

(e) | |f x x= -] g


of each function

(a) | | y x= for x2 2 -

(b) | |f x x 4= - -] g for

x4 3 -

(c) | |f x x 4= +] g for x7 2 -

(d) | 2 5 | y x= - for x3 3 -

(e) | |f x x= -] g for x1 1 -


function increasing

(a) | 3 | y x= +

(b) | |f x x 4= - +] g

(c) | |f x x 9= -] g

(d) | | y x 2 1= - -

(e) | |f x x 2= - +] g

56 Exercises




7 Solve graphically

(a) | | 3x =

(b) | |x 12

(c) | |x 2

(d) | 2 | 1x + =

(e) | 3 | 0x- =

(f) |2 3 | 1x - =

(g) | |x 1 41-

(h) | |x 1 3+

(i) | |x 2 22-

(j) | |x 3 1$-

(k) | |x2 3 5+

(l) | |x2 1 1$-

(m) |3 1 | 3x x- = +

(n) |3 2 | 4x x- = -

(o) | 1 | 1x x- = +

(p) | 3 | 2 2x x+ = + (q) |2 1 | 1x x+ = -

(r) |2 5 | 3x x- = -

(s) | 1 | 2x x- =

(t) |2 3 | 3x x- = +

The Hyperbola


or= =

EXAMPLE

Sketch1

y x=

Solution



x -3 -2 -121-

41- 0

41

21 1 2 3

y 3

1-

2

1- -1 -2 -4 mdash 4 2 1

2

1

3

1

Class Discussion



y is never 0 Why






but never touches)



Investigation




equations




1 y xk

=

21

y x k= +

31

y x k

=+


EXAMPLES


3=

-

] g


Solution



So x 3






be zero

CONTINUED







1

y 0 3

3=

-

= -

-3

4

3

2

1

5

-2

-1

-4

-5

y

-1-2 4 521-3-4

x

x = 3

y = 0

Asymptotes

3

2 Sketch y x2 4

1= -

+

Solution




x

x

2 4

2

-

-






( ) y

2 0 41

41

= -+

= -


a translation of3

y x

=





y

-2

x

-

1

4


a=

+


domain x xb



1 For each graph



exists


(a)2

y x=

(b)1

y x= -

(c) f xx 1

1=

+] g

(d) f xx 2

3=

-

] g

(e)3 6

1 y

x=

+

(f) f xx 3

2= -

-

] g

(g) f xx 1

4=

-

] g

(h)1

2 y

x= -

+

(i) f xx6 3

2=

-

] g

(j)2

6 y

x= -

+

2 Show that f x x2

=] g is an odd

function



(a) f xx2 5

1=

+

] g for x2 2 -

(b)3

1 y

x=

+ for x2 0 -

(c) f xx2 4

5=

-

] g for x3 1 -

57 Exercises




(d) f xx 4

3= -

-

] g for x3 3 -

(e)3 1

2 y

x= -

+ for x0 5



(a)3

y x= for y 1 3

(b)2

y x= - for y 221

- -

(c) f xx 1

1=

-

] g for y 171

- -

(d) f xx2 1

3= -

+

] g for

y 131

- -

(e)3 2

6 y

x=

- for y 1

21

6



Circle gate





Proof

r y

x

( x y)

y

x








c a b

r x y

2 2 2

2 2 2

`

= +

= +

EXAMPLE



Solution


y

x

-2

-2 2

2


one place

y

x

2

2

2

-2

The radius is 4

CONTINUED






Domain 2 2 x x -

Range 2 2 y y -


range y r y r -



Proof


shown

y

x

(a b)

x

y

r

( x y)

a

b x - a

y - b




ndash ndash

c a b

r x a y b

2 2 2

2 2 2

= +

= +] ^g h




EXAMPLES



Solution



y

x 9

9

-9

-9



Domain 9 9 x x -

Range 9 9 y y -



Solution


ndash

ndash ndash

x y

x y

1 2 4

1 2 2

2

2 2

+ + =

+ - =

2

2

] ^

] ]_

g h

g gi

So 1 2a b= = - and 2r =

CONTINUED







y

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

(1 -2)



Domain 1 3 x x -

Range 4 0 y y -


Solution


a b2 1= - = and r 3=

Substituting

ndash ndash

ndash

ndash

x a y b r

x y

x y

2 1 3

2 1 9

2 2 2

2 2 2

2 2

+ =

- - + =

+ + =

] ^]] ^

] ^

g hg g hg h


ndash

ndash

a b a ab b

x x x

x x

a b a ab b

y y y

y y

2

2 2 2 2

4 4

2

1 2 1 1

2 1

So

So

2 2 2

2 2 2

2

2 2 2

2 2 2

2

+ = + +

+ = + +

= + +

= - +

= - +

= - +

]] ] ]]^ ^ ]

gg g ggh h g


ndash

x x y y

x x y y

x x y y

x x y y

4 4 2 1 9

4 2 5 9

4 2 5 9

4 2 4 0

9 9

2

2

2

2

+ + + - + =

+ + - + =

+ + + =

+ + - - =

- -


in Chapter 2




Investigation


two functions





circle






for all values

y

x r

r

-r


Proof

ndash

x y r

y r x y r x

2 2 2

2 2 2

2 2

+ =

=

= -



semi-circles





y 0 for all values

y

x r

-

r

-r


EXAMPLES


1 f x x92

= -] g

Solution



y

x 3

3

-3

Domain 3 3 x x -

Range 0 3 y y




2 y x4 2= - -

Solution



y

x 2

-2

-2

Domain 2 2 x x -

Range 2 0 y y -




range

(a) 9x y 2 2+ =

(b) x y 16 02 2+ =-


(d) 1 9x y 2 2+ + =

] g

(e) ndashx y 2 1 12 2+ + =] ^g h



below the x -axis



range

(a) 25 y x2= - -

(b) 1 y x2= -

(c) 36 y x2= -

(d) 64 y x2= - -

(e) 7 y x2= - -



each circle

(a) 100x y 2 2+ =

(b) 5x y 2 2+ =


(d) ndashx y 5 6 492 2+ + =] ^g h

(e) ndashx y 3 812 2+ =^ h

58 Exercises






grouping symbols)



Centre(c) 1 5-







Centre(j) 4 7- -

^ h and radius 3

Other Graphs




EXAMPLES


Solution



eg When 0x =

y 3

1

c=

=

x

y

1

3

3

1

31

When1

1

= -

=

=

=

-

x 3- 2- 1- 0 1 2 3

y 271

91

31

1 3 9 27



You will meet these

graphs again in the

HSC Course





other point

When

x

y

1

3

3

1

=

=

=

y

x

1

2

1

3







Solution



x minus2 minus1 0 05 1 2 3 4

y minus03 0 03 05 06

y

x

1

2

1 2 3 4

-1




You learned about


stages of maths





range y y 02



range all real y

Cubic function


EXAMPLE



Solution



y minus25 minus6 1 2 3 10 29

y

x

1

1

-2 2 3 4

-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4



If you already

know the shape of


3






Domain and range


EXAMPLE


Solution




function are

f

f

1 1 2

1 21

4 4 2

64 2

66

3

3

- = - +

= - +

=

= +

= +

=

] ]

] ]

g g

g g


these points

y

x

(-1 1)

(4 66)

Range 1 66 y y










positive number

EXAMPLE


Solution


x 4So $

$


(or zero) So y 0$


59 Exercises


(a) 4 3 y x= +

(b) f x 4= -] g

(c) 3x =



(f) f x xx 12 2= - -] g

(g) 64x y 2 2+ =

(h) f t t 4

3=

-] g

(i) ( ) g 2

5zz

= +

(j) | |f x x=] g


(a) y x=

(b) 2 y x= -

(c) | |f x x2 3= -] g

(d) | | 2 y x= -

(e) f x x2 5= - +] g

(f) | | y x5= -

(g) 2 y x=

(h) y 5x= -

(i) f x xx 1

= +] g

(j)2

4 3 y x

x= -


(a) y x x 5 2= -] g


= +-

(d) g x x x164 2= -] g

(e) 49x y 2 2+ =

You may like to


by dividing by x




4 (a) Solve x1 02$-


f x x1 2= -] g


(a) 2 y x x2= - -

(b) g t t t 62= +] g



in each case


x3 3 -


x2 3 -


x2 1 - (d)


x1 5


0 4x


x3 3 -


x1 1 -


x2 3 -


x4 4 -

(j) y x x7 62= - + - in the

domain 0 7x


function y x 1

3=

+





x=] g


function

find its range(b)


plane

(a) f x x4=] g

(b) y x3= -

(c) y x 34= -

(d) 2 p x x3=] g

(e) 1 g x x3= +] g

(f) 100x y 2 2+ =

(g) 2 1 y x= +


y x 1= -









(b) f x x 22= -] g

(c)1

y x=

(d) f x x3=] g

(e) f x 3x=] g

13 (a) Solve x4 02$-


(i) 4 y x2= -

(ii) y x4

2= - -




DID YOU KNOW




a light



Limits






a a1

1

03

Z

=

=

3

3

-


lim a 0x=

x 3-



asymptote

EXAMPLES

1 Find lim x

x x5x 0

2+

Solution


which is undefined


( )

lim lim

lim

xx x

x

x x

x

5 5

5

5

x x

x

0

2

0 1

1

0

+=

+

= +

=

] g




2 Find limx

x

4

22

-

-

x 2

Solution


lim lim

lim

x

x

x x

x

x

4

2

2 2

2

21

41

2 1

1

-

-=

+ -

-

=+

=

x x2 2

x 2

^ _h i

3 Find limh

h x hx h2 72 2+ -

h 0

Solution

lim lim

lim

h

h x hx h

h

h hx x

hx x

x

2 7 2 7

2 7

7

2 2 2

2

2

+ -=

+ -

= + -

= -

h 0

h h0 0

^ h

Continuity





EXAMPLES

1 Find lim

x

x

1

12

-

-

x 1


11

y x

x2

=-

- Sketch the curve

Solution


xx2

-

- gives

00

CONTINUED




( )

lim lim

limx

x

x

x x

x11

1

1 1

1

2

x x

x

1

2

1

1

-

-=

-

+ -

= +

=

-

] ]g g

11

y xx2

=-





y xx

x

x x

x

11

1

1 1

1

2

=-

-

=

+

= +

-

-^ ^h h


2 Find limx

x x2

2x 2

2

+

+ -

-


x x2

22

= + -

+

Solution


x x2

22

+

+ - gives

00

lim lim

lim

xx x

x

x x

x

22

2

1 2

1

3

x x

x

2

2

2

2

+

+ -=

+

- +

=

= -

-

- -

-

^^ ^

^

hh h

h2 y

x

x xx

y x

x

x

x

22

2

2

1

1


=+

+ -= -

=+

= -

+ -^ ^h h


Remember that x 1




1 Find(a) lim x 52

+x 4

(b) lim t 7-t 3 -

(c) lim x x2 43+ -

x 2

(d) lim xx x32

+

x 0

(e) limh

h h

2

22

-

- -

h 2

(f) lim y

y

5

1253

-

-

y 5

(g) limx

x x

12 12

+

+ +

x 1-

(h) limx

x x4

2 82

+

+ -

x 4 -

(i) limc

c

4

22

-

-

c 2

(j) limx x

x 12

-

-

x 1

(k) lim h

h h h2 73 2+ -

h 0

(l) limh

hx hx h32 2- +

h 0

(m) limh

hx h x hx h2 3 53 2 2- + -

h 0

(n) lim x c x c 3 3

-

-

x c



discontinuous

(a) 3 y x2= -

(b)1

1 y

x=

+

(c) f x x 1= -] g

(d)4

1 y

x2=

+

(e)4

1

y x2=

-



(a)3

y xx x2

= +

(b)33

y x

x x2

=+

+

(c)1

5 4 y

xx x2

=+

+ +

510 Exercises




Regions

Class Investigation


them all







EXAMPLE


Solution



y

x =3

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4







region

EXAMPLE


Solution



included


shown

y

x

1

1

-2 2 3 4

-3

-4

-5

2

3

4

5

-1

-2

-3-4

-1 y = -1



y -intercept 1-






EXAMPLES


1 y x 2$ +

Solution






y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0



2 x y 2 3 61-

Solution


region


side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g








3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g






EXAMPLES



Solution



x

x

x

0 1

1

1

2

2

= -

=

=


0 1

1

y 2= -

= -



Domain xall real

Range y y 1$ -


Solution


parabola will lie




left



0

1 0

1

x

x

x

1 2= +

+ =

= -

] g


1

y 0 1 2= +

=

] g

-1

-

4

3

2

1

5

-2

-4

-5

-

1 41-2 5-4

y

x









Solution



x x

x xx x

x x

0 6

3 23 0 2 0

3 2

2= + -

= + -

+ = - =

= - =

] ]g g


f 0 0 0 66

2= + -

= -

] ] ]g g g



x -intercepts


23 2

21- +

= -


-c m

f 21

21

21

6

41

21

6

641

2

- = - + - -

= - -

= -

c c cm m m


-

CONTINUED

You will learn more


-1

-3

4

3

2

1

5

-2

-4

-5

1 4-1-2 32-3-4

y

x










41



2

11 -





Solution


So f x 0=] g

0 4 54 5 0

0

x xx x

x x5 1

2

2

= - + +

=

+ =

- -

-] ]g g

x x

x x

5 0 1 0

5 1

- = + =

= = -


f 0 0 4 0 5

5

2= - + +

=

] ] ]g g g

-1

-3

4

2

5

-2

-4

-5

-

y

1 41-2 5-4

x

-2 4






21 5

2- +

=

f 2 2 4 2 59

= - + +

=

2] ] ]g g g










each function

(a) 2 y x x2= +

(b) 3 y x x2= - +

(c) f x x 12= -] g

(d) y x x 22= - -

(e) y x x9 82= +-

2 Sketch

(a) 2 y x2= +

(b) y x 12= - +

(c) f x x 42= -] g

(d) 2 y x x2= +

(e) y x x2= - -

(f) f x x 3= -2] ]g g

55 Exercises

-1

9

8

7

5

4

3

2

6

1

-2

-3

-4

-5

y

2 51 643-1-2-3-4

x




EXAMPLES


Solution









(g) f x x 1 2= +] ]g g

(h) y x x3 42= + -

(i) y x x2 5 32= - +

(j) f x x x3 22= - + -] g



(a) ndash y x x7 122= +

(b) f x x x42= +] g

(c) y x x2 82= - -

(d) y x x6 92= +-

(e) f t t 4 2= -] g


(a) y x 52= -

(b) f x x x6

2= -

] g (c) f x x x 22= - -] g

(d) y x2= -

(e) f x x 7 2= -] ]g g



(a) y x2= for x0 3

(b) y x 42= - + for x1 2 -

(c) f x x 12= -] g for x2 5 -

(d) y x x2 32= + - for x2 4 -

(e) y x x

22= - +- for

x0 4


function is

increasing(i)

decreasing(ii)

(a) y x2=

(b) y x2

= - (c) f x x 92

= -] g

(d) y x x42= - +

(e) f x x 5 2= +] ]g g


function



(a) y x 12= +

(b) f x x 32= -

] g

(c) y x2 2= -

(d) f x x x32= -] g

(e) f x x x2= +] g

(f) y x 42= -

(g) y x x2 32= - -

(h) y x x5 42= +-

(i) p x x 1 2= +] ]g g

(j) y x 2= -2] g




CONTINUED

eg When x 3= -

| | y 3 13 12

= - -

= -

=

x -3 -2 -1 0 1 2 3

y 2 1 0 -1 0 1 2


y

-2

4

3

2

1

5

-1

-3

-4

-5

1 4-1-2 32-3-4

x


| | y x

x x

x x1

1 0

1 0

when

when 1

$= - =

-


y x x1 0$= - ] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x - 1




y x 1= - - x 01] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y =- x - 1



-3

4

3

2

1

5

-2

-1

-4

-5

y y

3-1-2 421-3-4

x

y = - x - 1

y = x - 1


intercepts


So f x 0=

] g

| |

| |

x

x

x

0 1

1

1`

= -

=

=


( ) | |f 0 0 11

= -

= -





-3

4

3

2

1

5

-2

-1

-4

-5

y

4-2 5321-1-3-4

x



Domain all real x

Range y y 1$ -

2 Sketch | | y x 2= +

Solution


| | ( ) y x x xx x2 2 2 0

2 2 0whenwhen 1

$= + = + +

- + +


2 y x= + when x 2 0$+

x 2$ -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x + 2






y-intercept of 1-

CONTINUED




2 y x= - +] g when x 2 01+

ie y x 2= - - when x 21 -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2


-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2

y = x + 2



So 0f x =] g

0 | 2 |

0 2

2

x

x

x

= +

= +

- =


(0) | 0 2 |

2

f = +

=

There is only one

solution for the


Can you see why





-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

If you know how to

sketch the graph of




Investigation




are their equations





1 | |f x k x=] g

2 | |f x x k= +] g

3 | |f x x k= +] g







EXAMPLES

Solve

1 |2 1 | 3x - =

Solution



is x 1 2= -

2 |2 1 | 3 2x x= -+

Solution



3 | 1 | 2x 1+

Solution






and then check them







each function

(a) | | y x=

(b) | |f x x 7= +] g

(c) | |f x x 2= -] g

(d) 5 | | y x=

(e) | |f x x 3= - +] g

(f) | 6 | y x= +

(g) | |f x x3 2= -] g

(h) | 5 4 | y x= +

(i) | 7 1 | y x= -

(j) | |f x x2 9= +] g


plane

(a) | | y x=

(b) | |f x x 1= +] g

(c) | |f x x 3= -] g

(d) 2 | | y x=

(e) | |f x x= -] g

(f) | 1 | y x= +

(g) | |f x x 1= - -] g

(h) | 2 3 | y x= -

(i) | 4 2 | y x= +

(j) | |f x x3 1= +] g


each function

(a) | 1 | y x= -

(b) | |f x x 8= -] g

(c) | |f x x2 5= +] g

(d) 2 | | 3 y x= -

(e) | |f x x 3= - -] g


function is

increasing(i)

decreasing(ii)

(a) | 2 | y x= -

(b) | |f x x 2= +

] g

(c) | |f x x2 3= -] g

(d) 4 | | 1 y x= -

(e) | |f x x= -] g


of each function

(a) | | y x= for x2 2 -

(b) | |f x x 4= - -] g for

x4 3 -

(c) | |f x x 4= +] g for x7 2 -

(d) | 2 5 | y x= - for x3 3 -

(e) | |f x x= -] g for x1 1 -


function increasing

(a) | 3 | y x= +

(b) | |f x x 4= - +] g

(c) | |f x x 9= -] g

(d) | | y x 2 1= - -

(e) | |f x x 2= - +] g

56 Exercises




7 Solve graphically

(a) | | 3x =

(b) | |x 12

(c) | |x 2

(d) | 2 | 1x + =

(e) | 3 | 0x- =

(f) |2 3 | 1x - =

(g) | |x 1 41-

(h) | |x 1 3+

(i) | |x 2 22-

(j) | |x 3 1$-

(k) | |x2 3 5+

(l) | |x2 1 1$-

(m) |3 1 | 3x x- = +

(n) |3 2 | 4x x- = -

(o) | 1 | 1x x- = +

(p) | 3 | 2 2x x+ = + (q) |2 1 | 1x x+ = -

(r) |2 5 | 3x x- = -

(s) | 1 | 2x x- =

(t) |2 3 | 3x x- = +

The Hyperbola


or= =

EXAMPLE

Sketch1

y x=

Solution



x -3 -2 -121-

41- 0

41

21 1 2 3

y 3

1-

2

1- -1 -2 -4 mdash 4 2 1

2

1

3

1

Class Discussion



y is never 0 Why






but never touches)



Investigation




equations




1 y xk

=

21

y x k= +

31

y x k

=+


EXAMPLES


3=

-

] g


Solution



So x 3






be zero

CONTINUED







1

y 0 3

3=

-

= -

-3

4

3

2

1

5

-2

-1

-4

-5

y

-1-2 4 521-3-4

x

x = 3

y = 0

Asymptotes

3

2 Sketch y x2 4

1= -

+

Solution




x

x

2 4

2

-

-






( ) y

2 0 41

41

= -+

= -


a translation of3

y x

=





y

-2

x

-

1

4


a=

+


domain x xb



1 For each graph



exists


(a)2

y x=

(b)1

y x= -

(c) f xx 1

1=

+] g

(d) f xx 2

3=

-

] g

(e)3 6

1 y

x=

+

(f) f xx 3

2= -

-

] g

(g) f xx 1

4=

-

] g

(h)1

2 y

x= -

+

(i) f xx6 3

2=

-

] g

(j)2

6 y

x= -

+

2 Show that f x x2

=] g is an odd

function



(a) f xx2 5

1=

+

] g for x2 2 -

(b)3

1 y

x=

+ for x2 0 -

(c) f xx2 4

5=

-

] g for x3 1 -

57 Exercises




(d) f xx 4

3= -

-

] g for x3 3 -

(e)3 1

2 y

x= -

+ for x0 5



(a)3

y x= for y 1 3

(b)2

y x= - for y 221

- -

(c) f xx 1

1=

-

] g for y 171

- -

(d) f xx2 1

3= -

+

] g for

y 131

- -

(e)3 2

6 y

x=

- for y 1

21

6



Circle gate





Proof

r y

x

( x y)

y

x








c a b

r x y

2 2 2

2 2 2

`

= +

= +

EXAMPLE



Solution


y

x

-2

-2 2

2


one place

y

x

2

2

2

-2

The radius is 4

CONTINUED






Domain 2 2 x x -

Range 2 2 y y -


range y r y r -



Proof


shown

y

x

(a b)

x

y

r

( x y)

a

b x - a

y - b




ndash ndash

c a b

r x a y b

2 2 2

2 2 2

= +

= +] ^g h




EXAMPLES



Solution



y

x 9

9

-9

-9



Domain 9 9 x x -

Range 9 9 y y -



Solution


ndash

ndash ndash

x y

x y

1 2 4

1 2 2

2

2 2

+ + =

+ - =

2

2

] ^

] ]_

g h

g gi

So 1 2a b= = - and 2r =

CONTINUED







y

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

(1 -2)



Domain 1 3 x x -

Range 4 0 y y -


Solution


a b2 1= - = and r 3=

Substituting

ndash ndash

ndash

ndash

x a y b r

x y

x y

2 1 3

2 1 9

2 2 2

2 2 2

2 2

+ =

- - + =

+ + =

] ^]] ^

] ^

g hg g hg h


ndash

ndash

a b a ab b

x x x

x x

a b a ab b

y y y

y y

2

2 2 2 2

4 4

2

1 2 1 1

2 1

So

So

2 2 2

2 2 2

2

2 2 2

2 2 2

2

+ = + +

+ = + +

= + +

= - +

= - +

= - +

]] ] ]]^ ^ ]

gg g ggh h g


ndash

x x y y

x x y y

x x y y

x x y y

4 4 2 1 9

4 2 5 9

4 2 5 9

4 2 4 0

9 9

2

2

2

2

+ + + - + =

+ + - + =

+ + + =

+ + - - =

- -


in Chapter 2




Investigation


two functions





circle






for all values

y

x r

r

-r


Proof

ndash

x y r

y r x y r x

2 2 2

2 2 2

2 2

+ =

=

= -



semi-circles





y 0 for all values

y

x r

-

r

-r


EXAMPLES


1 f x x92

= -] g

Solution



y

x 3

3

-3

Domain 3 3 x x -

Range 0 3 y y




2 y x4 2= - -

Solution



y

x 2

-2

-2

Domain 2 2 x x -

Range 2 0 y y -




range

(a) 9x y 2 2+ =

(b) x y 16 02 2+ =-


(d) 1 9x y 2 2+ + =

] g

(e) ndashx y 2 1 12 2+ + =] ^g h



below the x -axis



range

(a) 25 y x2= - -

(b) 1 y x2= -

(c) 36 y x2= -

(d) 64 y x2= - -

(e) 7 y x2= - -



each circle

(a) 100x y 2 2+ =

(b) 5x y 2 2+ =


(d) ndashx y 5 6 492 2+ + =] ^g h

(e) ndashx y 3 812 2+ =^ h

58 Exercises






grouping symbols)



Centre(c) 1 5-







Centre(j) 4 7- -

^ h and radius 3

Other Graphs




EXAMPLES


Solution



eg When 0x =

y 3

1

c=

=

x

y

1

3

3

1

31

When1

1

= -

=

=

=

-

x 3- 2- 1- 0 1 2 3

y 271

91

31

1 3 9 27



You will meet these

graphs again in the

HSC Course





other point

When

x

y

1

3

3

1

=

=

=

y

x

1

2

1

3







Solution



x minus2 minus1 0 05 1 2 3 4

y minus03 0 03 05 06

y

x

1

2

1 2 3 4

-1




You learned about


stages of maths





range y y 02



range all real y

Cubic function


EXAMPLE



Solution



y minus25 minus6 1 2 3 10 29

y

x

1

1

-2 2 3 4

-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4



If you already

know the shape of


3






Domain and range


EXAMPLE


Solution




function are

f

f

1 1 2

1 21

4 4 2

64 2

66

3

3

- = - +

= - +

=

= +

= +

=

] ]

] ]

g g

g g


these points

y

x

(-1 1)

(4 66)

Range 1 66 y y










positive number

EXAMPLE


Solution


x 4So $

$


(or zero) So y 0$


59 Exercises


(a) 4 3 y x= +

(b) f x 4= -] g

(c) 3x =



(f) f x xx 12 2= - -] g

(g) 64x y 2 2+ =

(h) f t t 4

3=

-] g

(i) ( ) g 2

5zz

= +

(j) | |f x x=] g


(a) y x=

(b) 2 y x= -

(c) | |f x x2 3= -] g

(d) | | 2 y x= -

(e) f x x2 5= - +] g

(f) | | y x5= -

(g) 2 y x=

(h) y 5x= -

(i) f x xx 1

= +] g

(j)2

4 3 y x

x= -


(a) y x x 5 2= -] g


= +-

(d) g x x x164 2= -] g

(e) 49x y 2 2+ =

You may like to


by dividing by x




4 (a) Solve x1 02$-


f x x1 2= -] g


(a) 2 y x x2= - -

(b) g t t t 62= +] g



in each case


x3 3 -


x2 3 -


x2 1 - (d)


x1 5


0 4x


x3 3 -


x1 1 -


x2 3 -


x4 4 -

(j) y x x7 62= - + - in the

domain 0 7x


function y x 1

3=

+





x=] g


function

find its range(b)


plane

(a) f x x4=] g

(b) y x3= -

(c) y x 34= -

(d) 2 p x x3=] g

(e) 1 g x x3= +] g

(f) 100x y 2 2+ =

(g) 2 1 y x= +


y x 1= -









(b) f x x 22= -] g

(c)1

y x=

(d) f x x3=] g

(e) f x 3x=] g

13 (a) Solve x4 02$-


(i) 4 y x2= -

(ii) y x4

2= - -




DID YOU KNOW




a light



Limits






a a1

1

03

Z

=

=

3

3

-


lim a 0x=

x 3-



asymptote

EXAMPLES

1 Find lim x

x x5x 0

2+

Solution


which is undefined


( )

lim lim

lim

xx x

x

x x

x

5 5

5

5

x x

x

0

2

0 1

1

0

+=

+

= +

=

] g




2 Find limx

x

4

22

-

-

x 2

Solution


lim lim

lim

x

x

x x

x

x

4

2

2 2

2

21

41

2 1

1

-

-=

+ -

-

=+

=

x x2 2

x 2

^ _h i

3 Find limh

h x hx h2 72 2+ -

h 0

Solution

lim lim

lim

h

h x hx h

h

h hx x

hx x

x

2 7 2 7

2 7

7

2 2 2

2

2

+ -=

+ -

= + -

= -

h 0

h h0 0

^ h

Continuity





EXAMPLES

1 Find lim

x

x

1

12

-

-

x 1


11

y x

x2

=-

- Sketch the curve

Solution


xx2

-

- gives

00

CONTINUED




( )

lim lim

limx

x

x

x x

x11

1

1 1

1

2

x x

x

1

2

1

1

-

-=

-

+ -

= +

=

-

] ]g g

11

y xx2

=-





y xx

x

x x

x

11

1

1 1

1

2

=-

-

=

+

= +

-

-^ ^h h


2 Find limx

x x2

2x 2

2

+

+ -

-


x x2

22

= + -

+

Solution


x x2

22

+

+ - gives

00

lim lim

lim

xx x

x

x x

x

22

2

1 2

1

3

x x

x

2

2

2

2

+

+ -=

+

- +

=

= -

-

- -

-

^^ ^

^

hh h

h2 y

x

x xx

y x

x

x

x

22

2

2

1

1


=+

+ -= -

=+

= -

+ -^ ^h h


Remember that x 1




1 Find(a) lim x 52

+x 4

(b) lim t 7-t 3 -

(c) lim x x2 43+ -

x 2

(d) lim xx x32

+

x 0

(e) limh

h h

2

22

-

- -

h 2

(f) lim y

y

5

1253

-

-

y 5

(g) limx

x x

12 12

+

+ +

x 1-

(h) limx

x x4

2 82

+

+ -

x 4 -

(i) limc

c

4

22

-

-

c 2

(j) limx x

x 12

-

-

x 1

(k) lim h

h h h2 73 2+ -

h 0

(l) limh

hx hx h32 2- +

h 0

(m) limh

hx h x hx h2 3 53 2 2- + -

h 0

(n) lim x c x c 3 3

-

-

x c



discontinuous

(a) 3 y x2= -

(b)1

1 y

x=

+

(c) f x x 1= -] g

(d)4

1 y

x2=

+

(e)4

1

y x2=

-



(a)3

y xx x2

= +

(b)33

y x

x x2

=+

+

(c)1

5 4 y

xx x2

=+

+ +

510 Exercises




Regions

Class Investigation


them all







EXAMPLE


Solution



y

x =3

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4







region

EXAMPLE


Solution



included


shown

y

x

1

1

-2 2 3 4

-3

-4

-5

2

3

4

5

-1

-2

-3-4

-1 y = -1



y -intercept 1-






EXAMPLES


1 y x 2$ +

Solution






y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0



2 x y 2 3 61-

Solution


region


side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g








3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g











Solution



x x

x xx x

x x

0 6

3 23 0 2 0

3 2

2= + -

= + -

+ = - =

= - =

] ]g g


f 0 0 0 66

2= + -

= -

] ] ]g g g



x -intercepts


23 2

21- +

= -


-c m

f 21

21

21

6

41

21

6

641

2

- = - + - -

= - -

= -

c c cm m m


-

CONTINUED

You will learn more


-1

-3

4

3

2

1

5

-2

-4

-5

1 4-1-2 32-3-4

y

x










41



2

11 -





Solution


So f x 0=] g

0 4 54 5 0

0

x xx x

x x5 1

2

2

= - + +

=

+ =

- -

-] ]g g

x x

x x

5 0 1 0

5 1

- = + =

= = -


f 0 0 4 0 5

5

2= - + +

=

] ] ]g g g

-1

-3

4

2

5

-2

-4

-5

-

y

1 41-2 5-4

x

-2 4






21 5

2- +

=

f 2 2 4 2 59

= - + +

=

2] ] ]g g g










each function

(a) 2 y x x2= +

(b) 3 y x x2= - +

(c) f x x 12= -] g

(d) y x x 22= - -

(e) y x x9 82= +-

2 Sketch

(a) 2 y x2= +

(b) y x 12= - +

(c) f x x 42= -] g

(d) 2 y x x2= +

(e) y x x2= - -

(f) f x x 3= -2] ]g g

55 Exercises

-1

9

8

7

5

4

3

2

6

1

-2

-3

-4

-5

y

2 51 643-1-2-3-4

x




EXAMPLES


Solution









(g) f x x 1 2= +] ]g g

(h) y x x3 42= + -

(i) y x x2 5 32= - +

(j) f x x x3 22= - + -] g



(a) ndash y x x7 122= +

(b) f x x x42= +] g

(c) y x x2 82= - -

(d) y x x6 92= +-

(e) f t t 4 2= -] g


(a) y x 52= -

(b) f x x x6

2= -

] g (c) f x x x 22= - -] g

(d) y x2= -

(e) f x x 7 2= -] ]g g



(a) y x2= for x0 3

(b) y x 42= - + for x1 2 -

(c) f x x 12= -] g for x2 5 -

(d) y x x2 32= + - for x2 4 -

(e) y x x

22= - +- for

x0 4


function is

increasing(i)

decreasing(ii)

(a) y x2=

(b) y x2

= - (c) f x x 92

= -] g

(d) y x x42= - +

(e) f x x 5 2= +] ]g g


function



(a) y x 12= +

(b) f x x 32= -

] g

(c) y x2 2= -

(d) f x x x32= -] g

(e) f x x x2= +] g

(f) y x 42= -

(g) y x x2 32= - -

(h) y x x5 42= +-

(i) p x x 1 2= +] ]g g

(j) y x 2= -2] g




CONTINUED

eg When x 3= -

| | y 3 13 12

= - -

= -

=

x -3 -2 -1 0 1 2 3

y 2 1 0 -1 0 1 2


y

-2

4

3

2

1

5

-1

-3

-4

-5

1 4-1-2 32-3-4

x


| | y x

x x

x x1

1 0

1 0

when

when 1

$= - =

-


y x x1 0$= - ] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x - 1




y x 1= - - x 01] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y =- x - 1



-3

4

3

2

1

5

-2

-1

-4

-5

y y

3-1-2 421-3-4

x

y = - x - 1

y = x - 1


intercepts


So f x 0=

] g

| |

| |

x

x

x

0 1

1

1`

= -

=

=


( ) | |f 0 0 11

= -

= -





-3

4

3

2

1

5

-2

-1

-4

-5

y

4-2 5321-1-3-4

x



Domain all real x

Range y y 1$ -

2 Sketch | | y x 2= +

Solution


| | ( ) y x x xx x2 2 2 0

2 2 0whenwhen 1

$= + = + +

- + +


2 y x= + when x 2 0$+

x 2$ -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x + 2






y-intercept of 1-

CONTINUED




2 y x= - +] g when x 2 01+

ie y x 2= - - when x 21 -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2


-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2

y = x + 2



So 0f x =] g

0 | 2 |

0 2

2

x

x

x

= +

= +

- =


(0) | 0 2 |

2

f = +

=

There is only one

solution for the


Can you see why





-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

If you know how to

sketch the graph of




Investigation




are their equations





1 | |f x k x=] g

2 | |f x x k= +] g

3 | |f x x k= +] g







EXAMPLES

Solve

1 |2 1 | 3x - =

Solution



is x 1 2= -

2 |2 1 | 3 2x x= -+

Solution



3 | 1 | 2x 1+

Solution






and then check them







each function

(a) | | y x=

(b) | |f x x 7= +] g

(c) | |f x x 2= -] g

(d) 5 | | y x=

(e) | |f x x 3= - +] g

(f) | 6 | y x= +

(g) | |f x x3 2= -] g

(h) | 5 4 | y x= +

(i) | 7 1 | y x= -

(j) | |f x x2 9= +] g


plane

(a) | | y x=

(b) | |f x x 1= +] g

(c) | |f x x 3= -] g

(d) 2 | | y x=

(e) | |f x x= -] g

(f) | 1 | y x= +

(g) | |f x x 1= - -] g

(h) | 2 3 | y x= -

(i) | 4 2 | y x= +

(j) | |f x x3 1= +] g


each function

(a) | 1 | y x= -

(b) | |f x x 8= -] g

(c) | |f x x2 5= +] g

(d) 2 | | 3 y x= -

(e) | |f x x 3= - -] g


function is

increasing(i)

decreasing(ii)

(a) | 2 | y x= -

(b) | |f x x 2= +

] g

(c) | |f x x2 3= -] g

(d) 4 | | 1 y x= -

(e) | |f x x= -] g


of each function

(a) | | y x= for x2 2 -

(b) | |f x x 4= - -] g for

x4 3 -

(c) | |f x x 4= +] g for x7 2 -

(d) | 2 5 | y x= - for x3 3 -

(e) | |f x x= -] g for x1 1 -


function increasing

(a) | 3 | y x= +

(b) | |f x x 4= - +] g

(c) | |f x x 9= -] g

(d) | | y x 2 1= - -

(e) | |f x x 2= - +] g

56 Exercises




7 Solve graphically

(a) | | 3x =

(b) | |x 12

(c) | |x 2

(d) | 2 | 1x + =

(e) | 3 | 0x- =

(f) |2 3 | 1x - =

(g) | |x 1 41-

(h) | |x 1 3+

(i) | |x 2 22-

(j) | |x 3 1$-

(k) | |x2 3 5+

(l) | |x2 1 1$-

(m) |3 1 | 3x x- = +

(n) |3 2 | 4x x- = -

(o) | 1 | 1x x- = +

(p) | 3 | 2 2x x+ = + (q) |2 1 | 1x x+ = -

(r) |2 5 | 3x x- = -

(s) | 1 | 2x x- =

(t) |2 3 | 3x x- = +

The Hyperbola


or= =

EXAMPLE

Sketch1

y x=

Solution



x -3 -2 -121-

41- 0

41

21 1 2 3

y 3

1-

2

1- -1 -2 -4 mdash 4 2 1

2

1

3

1

Class Discussion



y is never 0 Why






but never touches)



Investigation




equations




1 y xk

=

21

y x k= +

31

y x k

=+


EXAMPLES


3=

-

] g


Solution



So x 3






be zero

CONTINUED







1

y 0 3

3=

-

= -

-3

4

3

2

1

5

-2

-1

-4

-5

y

-1-2 4 521-3-4

x

x = 3

y = 0

Asymptotes

3

2 Sketch y x2 4

1= -

+

Solution




x

x

2 4

2

-

-






( ) y

2 0 41

41

= -+

= -


a translation of3

y x

=





y

-2

x

-

1

4


a=

+


domain x xb



1 For each graph



exists


(a)2

y x=

(b)1

y x= -

(c) f xx 1

1=

+] g

(d) f xx 2

3=

-

] g

(e)3 6

1 y

x=

+

(f) f xx 3

2= -

-

] g

(g) f xx 1

4=

-

] g

(h)1

2 y

x= -

+

(i) f xx6 3

2=

-

] g

(j)2

6 y

x= -

+

2 Show that f x x2

=] g is an odd

function



(a) f xx2 5

1=

+

] g for x2 2 -

(b)3

1 y

x=

+ for x2 0 -

(c) f xx2 4

5=

-

] g for x3 1 -

57 Exercises




(d) f xx 4

3= -

-

] g for x3 3 -

(e)3 1

2 y

x= -

+ for x0 5



(a)3

y x= for y 1 3

(b)2

y x= - for y 221

- -

(c) f xx 1

1=

-

] g for y 171

- -

(d) f xx2 1

3= -

+

] g for

y 131

- -

(e)3 2

6 y

x=

- for y 1

21

6



Circle gate





Proof

r y

x

( x y)

y

x








c a b

r x y

2 2 2

2 2 2

`

= +

= +

EXAMPLE



Solution


y

x

-2

-2 2

2


one place

y

x

2

2

2

-2

The radius is 4

CONTINUED






Domain 2 2 x x -

Range 2 2 y y -


range y r y r -



Proof


shown

y

x

(a b)

x

y

r

( x y)

a

b x - a

y - b




ndash ndash

c a b

r x a y b

2 2 2

2 2 2

= +

= +] ^g h




EXAMPLES



Solution



y

x 9

9

-9

-9



Domain 9 9 x x -

Range 9 9 y y -



Solution


ndash

ndash ndash

x y

x y

1 2 4

1 2 2

2

2 2

+ + =

+ - =

2

2

] ^

] ]_

g h

g gi

So 1 2a b= = - and 2r =

CONTINUED







y

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

(1 -2)



Domain 1 3 x x -

Range 4 0 y y -


Solution


a b2 1= - = and r 3=

Substituting

ndash ndash

ndash

ndash

x a y b r

x y

x y

2 1 3

2 1 9

2 2 2

2 2 2

2 2

+ =

- - + =

+ + =

] ^]] ^

] ^

g hg g hg h


ndash

ndash

a b a ab b

x x x

x x

a b a ab b

y y y

y y

2

2 2 2 2

4 4

2

1 2 1 1

2 1

So

So

2 2 2

2 2 2

2

2 2 2

2 2 2

2

+ = + +

+ = + +

= + +

= - +

= - +

= - +

]] ] ]]^ ^ ]

gg g ggh h g


ndash

x x y y

x x y y

x x y y

x x y y

4 4 2 1 9

4 2 5 9

4 2 5 9

4 2 4 0

9 9

2

2

2

2

+ + + - + =

+ + - + =

+ + + =

+ + - - =

- -


in Chapter 2




Investigation


two functions





circle






for all values

y

x r

r

-r


Proof

ndash

x y r

y r x y r x

2 2 2

2 2 2

2 2

+ =

=

= -



semi-circles





y 0 for all values

y

x r

-

r

-r


EXAMPLES


1 f x x92

= -] g

Solution



y

x 3

3

-3

Domain 3 3 x x -

Range 0 3 y y




2 y x4 2= - -

Solution



y

x 2

-2

-2

Domain 2 2 x x -

Range 2 0 y y -




range

(a) 9x y 2 2+ =

(b) x y 16 02 2+ =-


(d) 1 9x y 2 2+ + =

] g

(e) ndashx y 2 1 12 2+ + =] ^g h



below the x -axis



range

(a) 25 y x2= - -

(b) 1 y x2= -

(c) 36 y x2= -

(d) 64 y x2= - -

(e) 7 y x2= - -



each circle

(a) 100x y 2 2+ =

(b) 5x y 2 2+ =


(d) ndashx y 5 6 492 2+ + =] ^g h

(e) ndashx y 3 812 2+ =^ h

58 Exercises






grouping symbols)



Centre(c) 1 5-







Centre(j) 4 7- -

^ h and radius 3

Other Graphs




EXAMPLES


Solution



eg When 0x =

y 3

1

c=

=

x

y

1

3

3

1

31

When1

1

= -

=

=

=

-

x 3- 2- 1- 0 1 2 3

y 271

91

31

1 3 9 27



You will meet these

graphs again in the

HSC Course





other point

When

x

y

1

3

3

1

=

=

=

y

x

1

2

1

3







Solution



x minus2 minus1 0 05 1 2 3 4

y minus03 0 03 05 06

y

x

1

2

1 2 3 4

-1




You learned about


stages of maths





range y y 02



range all real y

Cubic function


EXAMPLE



Solution



y minus25 minus6 1 2 3 10 29

y

x

1

1

-2 2 3 4

-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4



If you already

know the shape of


3






Domain and range


EXAMPLE


Solution




function are

f

f

1 1 2

1 21

4 4 2

64 2

66

3

3

- = - +

= - +

=

= +

= +

=

] ]

] ]

g g

g g


these points

y

x

(-1 1)

(4 66)

Range 1 66 y y










positive number

EXAMPLE


Solution


x 4So $

$


(or zero) So y 0$


59 Exercises


(a) 4 3 y x= +

(b) f x 4= -] g

(c) 3x =



(f) f x xx 12 2= - -] g

(g) 64x y 2 2+ =

(h) f t t 4

3=

-] g

(i) ( ) g 2

5zz

= +

(j) | |f x x=] g


(a) y x=

(b) 2 y x= -

(c) | |f x x2 3= -] g

(d) | | 2 y x= -

(e) f x x2 5= - +] g

(f) | | y x5= -

(g) 2 y x=

(h) y 5x= -

(i) f x xx 1

= +] g

(j)2

4 3 y x

x= -


(a) y x x 5 2= -] g


= +-

(d) g x x x164 2= -] g

(e) 49x y 2 2+ =

You may like to


by dividing by x




4 (a) Solve x1 02$-


f x x1 2= -] g


(a) 2 y x x2= - -

(b) g t t t 62= +] g



in each case


x3 3 -


x2 3 -


x2 1 - (d)


x1 5


0 4x


x3 3 -


x1 1 -


x2 3 -


x4 4 -

(j) y x x7 62= - + - in the

domain 0 7x


function y x 1

3=

+





x=] g


function

find its range(b)


plane

(a) f x x4=] g

(b) y x3= -

(c) y x 34= -

(d) 2 p x x3=] g

(e) 1 g x x3= +] g

(f) 100x y 2 2+ =

(g) 2 1 y x= +


y x 1= -









(b) f x x 22= -] g

(c)1

y x=

(d) f x x3=] g

(e) f x 3x=] g

13 (a) Solve x4 02$-


(i) 4 y x2= -

(ii) y x4

2= - -




DID YOU KNOW




a light



Limits






a a1

1

03

Z

=

=

3

3

-


lim a 0x=

x 3-



asymptote

EXAMPLES

1 Find lim x

x x5x 0

2+

Solution


which is undefined


( )

lim lim

lim

xx x

x

x x

x

5 5

5

5

x x

x

0

2

0 1

1

0

+=

+

= +

=

] g




2 Find limx

x

4

22

-

-

x 2

Solution


lim lim

lim

x

x

x x

x

x

4

2

2 2

2

21

41

2 1

1

-

-=

+ -

-

=+

=

x x2 2

x 2

^ _h i

3 Find limh

h x hx h2 72 2+ -

h 0

Solution

lim lim

lim

h

h x hx h

h

h hx x

hx x

x

2 7 2 7

2 7

7

2 2 2

2

2

+ -=

+ -

= + -

= -

h 0

h h0 0

^ h

Continuity





EXAMPLES

1 Find lim

x

x

1

12

-

-

x 1


11

y x

x2

=-

- Sketch the curve

Solution


xx2

-

- gives

00

CONTINUED




( )

lim lim

limx

x

x

x x

x11

1

1 1

1

2

x x

x

1

2

1

1

-

-=

-

+ -

= +

=

-

] ]g g

11

y xx2

=-





y xx

x

x x

x

11

1

1 1

1

2

=-

-

=

+

= +

-

-^ ^h h


2 Find limx

x x2

2x 2

2

+

+ -

-


x x2

22

= + -

+

Solution


x x2

22

+

+ - gives

00

lim lim

lim

xx x

x

x x

x

22

2

1 2

1

3

x x

x

2

2

2

2

+

+ -=

+

- +

=

= -

-

- -

-

^^ ^

^

hh h

h2 y

x

x xx

y x

x

x

x

22

2

2

1

1


=+

+ -= -

=+

= -

+ -^ ^h h


Remember that x 1




1 Find(a) lim x 52

+x 4

(b) lim t 7-t 3 -

(c) lim x x2 43+ -

x 2

(d) lim xx x32

+

x 0

(e) limh

h h

2

22

-

- -

h 2

(f) lim y

y

5

1253

-

-

y 5

(g) limx

x x

12 12

+

+ +

x 1-

(h) limx

x x4

2 82

+

+ -

x 4 -

(i) limc

c

4

22

-

-

c 2

(j) limx x

x 12

-

-

x 1

(k) lim h

h h h2 73 2+ -

h 0

(l) limh

hx hx h32 2- +

h 0

(m) limh

hx h x hx h2 3 53 2 2- + -

h 0

(n) lim x c x c 3 3

-

-

x c



discontinuous

(a) 3 y x2= -

(b)1

1 y

x=

+

(c) f x x 1= -] g

(d)4

1 y

x2=

+

(e)4

1

y x2=

-



(a)3

y xx x2

= +

(b)33

y x

x x2

=+

+

(c)1

5 4 y

xx x2

=+

+ +

510 Exercises




Regions

Class Investigation


them all







EXAMPLE


Solution



y

x =3

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4







region

EXAMPLE


Solution



included


shown

y

x

1

1

-2 2 3 4

-3

-4

-5

2

3

4

5

-1

-2

-3-4

-1 y = -1



y -intercept 1-






EXAMPLES


1 y x 2$ +

Solution






y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0



2 x y 2 3 61-

Solution


region


side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g








3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g












41



2

11 -





Solution


So f x 0=] g

0 4 54 5 0

0

x xx x

x x5 1

2

2

= - + +

=

+ =

- -

-] ]g g

x x

x x

5 0 1 0

5 1

- = + =

= = -


f 0 0 4 0 5

5

2= - + +

=

] ] ]g g g

-1

-3

4

2

5

-2

-4

-5

-

y

1 41-2 5-4

x

-2 4






21 5

2- +

=

f 2 2 4 2 59

= - + +

=

2] ] ]g g g










each function

(a) 2 y x x2= +

(b) 3 y x x2= - +

(c) f x x 12= -] g

(d) y x x 22= - -

(e) y x x9 82= +-

2 Sketch

(a) 2 y x2= +

(b) y x 12= - +

(c) f x x 42= -] g

(d) 2 y x x2= +

(e) y x x2= - -

(f) f x x 3= -2] ]g g

55 Exercises

-1

9

8

7

5

4

3

2

6

1

-2

-3

-4

-5

y

2 51 643-1-2-3-4

x




EXAMPLES


Solution









(g) f x x 1 2= +] ]g g

(h) y x x3 42= + -

(i) y x x2 5 32= - +

(j) f x x x3 22= - + -] g



(a) ndash y x x7 122= +

(b) f x x x42= +] g

(c) y x x2 82= - -

(d) y x x6 92= +-

(e) f t t 4 2= -] g


(a) y x 52= -

(b) f x x x6

2= -

] g (c) f x x x 22= - -] g

(d) y x2= -

(e) f x x 7 2= -] ]g g



(a) y x2= for x0 3

(b) y x 42= - + for x1 2 -

(c) f x x 12= -] g for x2 5 -

(d) y x x2 32= + - for x2 4 -

(e) y x x

22= - +- for

x0 4


function is

increasing(i)

decreasing(ii)

(a) y x2=

(b) y x2

= - (c) f x x 92

= -] g

(d) y x x42= - +

(e) f x x 5 2= +] ]g g


function



(a) y x 12= +

(b) f x x 32= -

] g

(c) y x2 2= -

(d) f x x x32= -] g

(e) f x x x2= +] g

(f) y x 42= -

(g) y x x2 32= - -

(h) y x x5 42= +-

(i) p x x 1 2= +] ]g g

(j) y x 2= -2] g




CONTINUED

eg When x 3= -

| | y 3 13 12

= - -

= -

=

x -3 -2 -1 0 1 2 3

y 2 1 0 -1 0 1 2


y

-2

4

3

2

1

5

-1

-3

-4

-5

1 4-1-2 32-3-4

x


| | y x

x x

x x1

1 0

1 0

when

when 1

$= - =

-


y x x1 0$= - ] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x - 1




y x 1= - - x 01] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y =- x - 1



-3

4

3

2

1

5

-2

-1

-4

-5

y y

3-1-2 421-3-4

x

y = - x - 1

y = x - 1


intercepts


So f x 0=

] g

| |

| |

x

x

x

0 1

1

1`

= -

=

=


( ) | |f 0 0 11

= -

= -





-3

4

3

2

1

5

-2

-1

-4

-5

y

4-2 5321-1-3-4

x



Domain all real x

Range y y 1$ -

2 Sketch | | y x 2= +

Solution


| | ( ) y x x xx x2 2 2 0

2 2 0whenwhen 1

$= + = + +

- + +


2 y x= + when x 2 0$+

x 2$ -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x + 2






y-intercept of 1-

CONTINUED




2 y x= - +] g when x 2 01+

ie y x 2= - - when x 21 -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2


-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2

y = x + 2



So 0f x =] g

0 | 2 |

0 2

2

x

x

x

= +

= +

- =


(0) | 0 2 |

2

f = +

=

There is only one

solution for the


Can you see why





-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

If you know how to

sketch the graph of




Investigation




are their equations





1 | |f x k x=] g

2 | |f x x k= +] g

3 | |f x x k= +] g







EXAMPLES

Solve

1 |2 1 | 3x - =

Solution



is x 1 2= -

2 |2 1 | 3 2x x= -+

Solution



3 | 1 | 2x 1+

Solution






and then check them







each function

(a) | | y x=

(b) | |f x x 7= +] g

(c) | |f x x 2= -] g

(d) 5 | | y x=

(e) | |f x x 3= - +] g

(f) | 6 | y x= +

(g) | |f x x3 2= -] g

(h) | 5 4 | y x= +

(i) | 7 1 | y x= -

(j) | |f x x2 9= +] g


plane

(a) | | y x=

(b) | |f x x 1= +] g

(c) | |f x x 3= -] g

(d) 2 | | y x=

(e) | |f x x= -] g

(f) | 1 | y x= +

(g) | |f x x 1= - -] g

(h) | 2 3 | y x= -

(i) | 4 2 | y x= +

(j) | |f x x3 1= +] g


each function

(a) | 1 | y x= -

(b) | |f x x 8= -] g

(c) | |f x x2 5= +] g

(d) 2 | | 3 y x= -

(e) | |f x x 3= - -] g


function is

increasing(i)

decreasing(ii)

(a) | 2 | y x= -

(b) | |f x x 2= +

] g

(c) | |f x x2 3= -] g

(d) 4 | | 1 y x= -

(e) | |f x x= -] g


of each function

(a) | | y x= for x2 2 -

(b) | |f x x 4= - -] g for

x4 3 -

(c) | |f x x 4= +] g for x7 2 -

(d) | 2 5 | y x= - for x3 3 -

(e) | |f x x= -] g for x1 1 -


function increasing

(a) | 3 | y x= +

(b) | |f x x 4= - +] g

(c) | |f x x 9= -] g

(d) | | y x 2 1= - -

(e) | |f x x 2= - +] g

56 Exercises




7 Solve graphically

(a) | | 3x =

(b) | |x 12

(c) | |x 2

(d) | 2 | 1x + =

(e) | 3 | 0x- =

(f) |2 3 | 1x - =

(g) | |x 1 41-

(h) | |x 1 3+

(i) | |x 2 22-

(j) | |x 3 1$-

(k) | |x2 3 5+

(l) | |x2 1 1$-

(m) |3 1 | 3x x- = +

(n) |3 2 | 4x x- = -

(o) | 1 | 1x x- = +

(p) | 3 | 2 2x x+ = + (q) |2 1 | 1x x+ = -

(r) |2 5 | 3x x- = -

(s) | 1 | 2x x- =

(t) |2 3 | 3x x- = +

The Hyperbola


or= =

EXAMPLE

Sketch1

y x=

Solution



x -3 -2 -121-

41- 0

41

21 1 2 3

y 3

1-

2

1- -1 -2 -4 mdash 4 2 1

2

1

3

1

Class Discussion



y is never 0 Why






but never touches)



Investigation




equations




1 y xk

=

21

y x k= +

31

y x k

=+


EXAMPLES


3=

-

] g


Solution



So x 3






be zero

CONTINUED







1

y 0 3

3=

-

= -

-3

4

3

2

1

5

-2

-1

-4

-5

y

-1-2 4 521-3-4

x

x = 3

y = 0

Asymptotes

3

2 Sketch y x2 4

1= -

+

Solution




x

x

2 4

2

-

-






( ) y

2 0 41

41

= -+

= -


a translation of3

y x

=





y

-2

x

-

1

4


a=

+


domain x xb



1 For each graph



exists


(a)2

y x=

(b)1

y x= -

(c) f xx 1

1=

+] g

(d) f xx 2

3=

-

] g

(e)3 6

1 y

x=

+

(f) f xx 3

2= -

-

] g

(g) f xx 1

4=

-

] g

(h)1

2 y

x= -

+

(i) f xx6 3

2=

-

] g

(j)2

6 y

x= -

+

2 Show that f x x2

=] g is an odd

function



(a) f xx2 5

1=

+

] g for x2 2 -

(b)3

1 y

x=

+ for x2 0 -

(c) f xx2 4

5=

-

] g for x3 1 -

57 Exercises




(d) f xx 4

3= -

-

] g for x3 3 -

(e)3 1

2 y

x= -

+ for x0 5



(a)3

y x= for y 1 3

(b)2

y x= - for y 221

- -

(c) f xx 1

1=

-

] g for y 171

- -

(d) f xx2 1

3= -

+

] g for

y 131

- -

(e)3 2

6 y

x=

- for y 1

21

6



Circle gate





Proof

r y

x

( x y)

y

x








c a b

r x y

2 2 2

2 2 2

`

= +

= +

EXAMPLE



Solution


y

x

-2

-2 2

2


one place

y

x

2

2

2

-2

The radius is 4

CONTINUED






Domain 2 2 x x -

Range 2 2 y y -


range y r y r -



Proof


shown

y

x

(a b)

x

y

r

( x y)

a

b x - a

y - b




ndash ndash

c a b

r x a y b

2 2 2

2 2 2

= +

= +] ^g h




EXAMPLES



Solution



y

x 9

9

-9

-9



Domain 9 9 x x -

Range 9 9 y y -



Solution


ndash

ndash ndash

x y

x y

1 2 4

1 2 2

2

2 2

+ + =

+ - =

2

2

] ^

] ]_

g h

g gi

So 1 2a b= = - and 2r =

CONTINUED







y

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

(1 -2)



Domain 1 3 x x -

Range 4 0 y y -


Solution


a b2 1= - = and r 3=

Substituting

ndash ndash

ndash

ndash

x a y b r

x y

x y

2 1 3

2 1 9

2 2 2

2 2 2

2 2

+ =

- - + =

+ + =

] ^]] ^

] ^

g hg g hg h


ndash

ndash

a b a ab b

x x x

x x

a b a ab b

y y y

y y

2

2 2 2 2

4 4

2

1 2 1 1

2 1

So

So

2 2 2

2 2 2

2

2 2 2

2 2 2

2

+ = + +

+ = + +

= + +

= - +

= - +

= - +

]] ] ]]^ ^ ]

gg g ggh h g


ndash

x x y y

x x y y

x x y y

x x y y

4 4 2 1 9

4 2 5 9

4 2 5 9

4 2 4 0

9 9

2

2

2

2

+ + + - + =

+ + - + =

+ + + =

+ + - - =

- -


in Chapter 2




Investigation


two functions





circle






for all values

y

x r

r

-r


Proof

ndash

x y r

y r x y r x

2 2 2

2 2 2

2 2

+ =

=

= -



semi-circles





y 0 for all values

y

x r

-

r

-r


EXAMPLES


1 f x x92

= -] g

Solution



y

x 3

3

-3

Domain 3 3 x x -

Range 0 3 y y




2 y x4 2= - -

Solution



y

x 2

-2

-2

Domain 2 2 x x -

Range 2 0 y y -




range

(a) 9x y 2 2+ =

(b) x y 16 02 2+ =-


(d) 1 9x y 2 2+ + =

] g

(e) ndashx y 2 1 12 2+ + =] ^g h



below the x -axis



range

(a) 25 y x2= - -

(b) 1 y x2= -

(c) 36 y x2= -

(d) 64 y x2= - -

(e) 7 y x2= - -



each circle

(a) 100x y 2 2+ =

(b) 5x y 2 2+ =


(d) ndashx y 5 6 492 2+ + =] ^g h

(e) ndashx y 3 812 2+ =^ h

58 Exercises






grouping symbols)



Centre(c) 1 5-







Centre(j) 4 7- -

^ h and radius 3

Other Graphs




EXAMPLES


Solution



eg When 0x =

y 3

1

c=

=

x

y

1

3

3

1

31

When1

1

= -

=

=

=

-

x 3- 2- 1- 0 1 2 3

y 271

91

31

1 3 9 27



You will meet these

graphs again in the

HSC Course





other point

When

x

y

1

3

3

1

=

=

=

y

x

1

2

1

3







Solution



x minus2 minus1 0 05 1 2 3 4

y minus03 0 03 05 06

y

x

1

2

1 2 3 4

-1




You learned about


stages of maths





range y y 02



range all real y

Cubic function


EXAMPLE



Solution



y minus25 minus6 1 2 3 10 29

y

x

1

1

-2 2 3 4

-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4



If you already

know the shape of


3






Domain and range


EXAMPLE


Solution




function are

f

f

1 1 2

1 21

4 4 2

64 2

66

3

3

- = - +

= - +

=

= +

= +

=

] ]

] ]

g g

g g


these points

y

x

(-1 1)

(4 66)

Range 1 66 y y










positive number

EXAMPLE


Solution


x 4So $

$


(or zero) So y 0$


59 Exercises


(a) 4 3 y x= +

(b) f x 4= -] g

(c) 3x =



(f) f x xx 12 2= - -] g

(g) 64x y 2 2+ =

(h) f t t 4

3=

-] g

(i) ( ) g 2

5zz

= +

(j) | |f x x=] g


(a) y x=

(b) 2 y x= -

(c) | |f x x2 3= -] g

(d) | | 2 y x= -

(e) f x x2 5= - +] g

(f) | | y x5= -

(g) 2 y x=

(h) y 5x= -

(i) f x xx 1

= +] g

(j)2

4 3 y x

x= -


(a) y x x 5 2= -] g


= +-

(d) g x x x164 2= -] g

(e) 49x y 2 2+ =

You may like to


by dividing by x




4 (a) Solve x1 02$-


f x x1 2= -] g


(a) 2 y x x2= - -

(b) g t t t 62= +] g



in each case


x3 3 -


x2 3 -


x2 1 - (d)


x1 5


0 4x


x3 3 -


x1 1 -


x2 3 -


x4 4 -

(j) y x x7 62= - + - in the

domain 0 7x


function y x 1

3=

+





x=] g


function

find its range(b)


plane

(a) f x x4=] g

(b) y x3= -

(c) y x 34= -

(d) 2 p x x3=] g

(e) 1 g x x3= +] g

(f) 100x y 2 2+ =

(g) 2 1 y x= +


y x 1= -









(b) f x x 22= -] g

(c)1

y x=

(d) f x x3=] g

(e) f x 3x=] g

13 (a) Solve x4 02$-


(i) 4 y x2= -

(ii) y x4

2= - -




DID YOU KNOW




a light



Limits






a a1

1

03

Z

=

=

3

3

-


lim a 0x=

x 3-



asymptote

EXAMPLES

1 Find lim x

x x5x 0

2+

Solution


which is undefined


( )

lim lim

lim

xx x

x

x x

x

5 5

5

5

x x

x

0

2

0 1

1

0

+=

+

= +

=

] g




2 Find limx

x

4

22

-

-

x 2

Solution


lim lim

lim

x

x

x x

x

x

4

2

2 2

2

21

41

2 1

1

-

-=

+ -

-

=+

=

x x2 2

x 2

^ _h i

3 Find limh

h x hx h2 72 2+ -

h 0

Solution

lim lim

lim

h

h x hx h

h

h hx x

hx x

x

2 7 2 7

2 7

7

2 2 2

2

2

+ -=

+ -

= + -

= -

h 0

h h0 0

^ h

Continuity





EXAMPLES

1 Find lim

x

x

1

12

-

-

x 1


11

y x

x2

=-

- Sketch the curve

Solution


xx2

-

- gives

00

CONTINUED




( )

lim lim

limx

x

x

x x

x11

1

1 1

1

2

x x

x

1

2

1

1

-

-=

-

+ -

= +

=

-

] ]g g

11

y xx2

=-





y xx

x

x x

x

11

1

1 1

1

2

=-

-

=

+

= +

-

-^ ^h h


2 Find limx

x x2

2x 2

2

+

+ -

-


x x2

22

= + -

+

Solution


x x2

22

+

+ - gives

00

lim lim

lim

xx x

x

x x

x

22

2

1 2

1

3

x x

x

2

2

2

2

+

+ -=

+

- +

=

= -

-

- -

-

^^ ^

^

hh h

h2 y

x

x xx

y x

x

x

x

22

2

2

1

1


=+

+ -= -

=+

= -

+ -^ ^h h


Remember that x 1




1 Find(a) lim x 52

+x 4

(b) lim t 7-t 3 -

(c) lim x x2 43+ -

x 2

(d) lim xx x32

+

x 0

(e) limh

h h

2

22

-

- -

h 2

(f) lim y

y

5

1253

-

-

y 5

(g) limx

x x

12 12

+

+ +

x 1-

(h) limx

x x4

2 82

+

+ -

x 4 -

(i) limc

c

4

22

-

-

c 2

(j) limx x

x 12

-

-

x 1

(k) lim h

h h h2 73 2+ -

h 0

(l) limh

hx hx h32 2- +

h 0

(m) limh

hx h x hx h2 3 53 2 2- + -

h 0

(n) lim x c x c 3 3

-

-

x c



discontinuous

(a) 3 y x2= -

(b)1

1 y

x=

+

(c) f x x 1= -] g

(d)4

1 y

x2=

+

(e)4

1

y x2=

-



(a)3

y xx x2

= +

(b)33

y x

x x2

=+

+

(c)1

5 4 y

xx x2

=+

+ +

510 Exercises




Regions

Class Investigation


them all







EXAMPLE


Solution



y

x =3

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4







region

EXAMPLE


Solution



included


shown

y

x

1

1

-2 2 3 4

-3

-4

-5

2

3

4

5

-1

-2

-3-4

-1 y = -1



y -intercept 1-






EXAMPLES


1 y x 2$ +

Solution






y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0



2 x y 2 3 61-

Solution


region


side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g








3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g








21 5

2- +

=

f 2 2 4 2 59

= - + +

=

2] ] ]g g g










each function

(a) 2 y x x2= +

(b) 3 y x x2= - +

(c) f x x 12= -] g

(d) y x x 22= - -

(e) y x x9 82= +-

2 Sketch

(a) 2 y x2= +

(b) y x 12= - +

(c) f x x 42= -] g

(d) 2 y x x2= +

(e) y x x2= - -

(f) f x x 3= -2] ]g g

55 Exercises

-1

9

8

7

5

4

3

2

6

1

-2

-3

-4

-5

y

2 51 643-1-2-3-4

x




EXAMPLES


Solution









(g) f x x 1 2= +] ]g g

(h) y x x3 42= + -

(i) y x x2 5 32= - +

(j) f x x x3 22= - + -] g



(a) ndash y x x7 122= +

(b) f x x x42= +] g

(c) y x x2 82= - -

(d) y x x6 92= +-

(e) f t t 4 2= -] g


(a) y x 52= -

(b) f x x x6

2= -

] g (c) f x x x 22= - -] g

(d) y x2= -

(e) f x x 7 2= -] ]g g



(a) y x2= for x0 3

(b) y x 42= - + for x1 2 -

(c) f x x 12= -] g for x2 5 -

(d) y x x2 32= + - for x2 4 -

(e) y x x

22= - +- for

x0 4


function is

increasing(i)

decreasing(ii)

(a) y x2=

(b) y x2

= - (c) f x x 92

= -] g

(d) y x x42= - +

(e) f x x 5 2= +] ]g g


function



(a) y x 12= +

(b) f x x 32= -

] g

(c) y x2 2= -

(d) f x x x32= -] g

(e) f x x x2= +] g

(f) y x 42= -

(g) y x x2 32= - -

(h) y x x5 42= +-

(i) p x x 1 2= +] ]g g

(j) y x 2= -2] g




CONTINUED

eg When x 3= -

| | y 3 13 12

= - -

= -

=

x -3 -2 -1 0 1 2 3

y 2 1 0 -1 0 1 2


y

-2

4

3

2

1

5

-1

-3

-4

-5

1 4-1-2 32-3-4

x


| | y x

x x

x x1

1 0

1 0

when

when 1

$= - =

-


y x x1 0$= - ] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x - 1




y x 1= - - x 01] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y =- x - 1



-3

4

3

2

1

5

-2

-1

-4

-5

y y

3-1-2 421-3-4

x

y = - x - 1

y = x - 1


intercepts


So f x 0=

] g

| |

| |

x

x

x

0 1

1

1`

= -

=

=


( ) | |f 0 0 11

= -

= -





-3

4

3

2

1

5

-2

-1

-4

-5

y

4-2 5321-1-3-4

x



Domain all real x

Range y y 1$ -

2 Sketch | | y x 2= +

Solution


| | ( ) y x x xx x2 2 2 0

2 2 0whenwhen 1

$= + = + +

- + +


2 y x= + when x 2 0$+

x 2$ -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x + 2






y-intercept of 1-

CONTINUED




2 y x= - +] g when x 2 01+

ie y x 2= - - when x 21 -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2


-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2

y = x + 2



So 0f x =] g

0 | 2 |

0 2

2

x

x

x

= +

= +

- =


(0) | 0 2 |

2

f = +

=

There is only one

solution for the


Can you see why





-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

If you know how to

sketch the graph of




Investigation




are their equations





1 | |f x k x=] g

2 | |f x x k= +] g

3 | |f x x k= +] g







EXAMPLES

Solve

1 |2 1 | 3x - =

Solution



is x 1 2= -

2 |2 1 | 3 2x x= -+

Solution



3 | 1 | 2x 1+

Solution






and then check them







each function

(a) | | y x=

(b) | |f x x 7= +] g

(c) | |f x x 2= -] g

(d) 5 | | y x=

(e) | |f x x 3= - +] g

(f) | 6 | y x= +

(g) | |f x x3 2= -] g

(h) | 5 4 | y x= +

(i) | 7 1 | y x= -

(j) | |f x x2 9= +] g


plane

(a) | | y x=

(b) | |f x x 1= +] g

(c) | |f x x 3= -] g

(d) 2 | | y x=

(e) | |f x x= -] g

(f) | 1 | y x= +

(g) | |f x x 1= - -] g

(h) | 2 3 | y x= -

(i) | 4 2 | y x= +

(j) | |f x x3 1= +] g


each function

(a) | 1 | y x= -

(b) | |f x x 8= -] g

(c) | |f x x2 5= +] g

(d) 2 | | 3 y x= -

(e) | |f x x 3= - -] g


function is

increasing(i)

decreasing(ii)

(a) | 2 | y x= -

(b) | |f x x 2= +

] g

(c) | |f x x2 3= -] g

(d) 4 | | 1 y x= -

(e) | |f x x= -] g


of each function

(a) | | y x= for x2 2 -

(b) | |f x x 4= - -] g for

x4 3 -

(c) | |f x x 4= +] g for x7 2 -

(d) | 2 5 | y x= - for x3 3 -

(e) | |f x x= -] g for x1 1 -


function increasing

(a) | 3 | y x= +

(b) | |f x x 4= - +] g

(c) | |f x x 9= -] g

(d) | | y x 2 1= - -

(e) | |f x x 2= - +] g

56 Exercises




7 Solve graphically

(a) | | 3x =

(b) | |x 12

(c) | |x 2

(d) | 2 | 1x + =

(e) | 3 | 0x- =

(f) |2 3 | 1x - =

(g) | |x 1 41-

(h) | |x 1 3+

(i) | |x 2 22-

(j) | |x 3 1$-

(k) | |x2 3 5+

(l) | |x2 1 1$-

(m) |3 1 | 3x x- = +

(n) |3 2 | 4x x- = -

(o) | 1 | 1x x- = +

(p) | 3 | 2 2x x+ = + (q) |2 1 | 1x x+ = -

(r) |2 5 | 3x x- = -

(s) | 1 | 2x x- =

(t) |2 3 | 3x x- = +

The Hyperbola


or= =

EXAMPLE

Sketch1

y x=

Solution



x -3 -2 -121-

41- 0

41

21 1 2 3

y 3

1-

2

1- -1 -2 -4 mdash 4 2 1

2

1

3

1

Class Discussion



y is never 0 Why






but never touches)



Investigation




equations




1 y xk

=

21

y x k= +

31

y x k

=+


EXAMPLES


3=

-

] g


Solution



So x 3






be zero

CONTINUED







1

y 0 3

3=

-

= -

-3

4

3

2

1

5

-2

-1

-4

-5

y

-1-2 4 521-3-4

x

x = 3

y = 0

Asymptotes

3

2 Sketch y x2 4

1= -

+

Solution




x

x

2 4

2

-

-






( ) y

2 0 41

41

= -+

= -


a translation of3

y x

=





y

-2

x

-

1

4


a=

+


domain x xb



1 For each graph



exists


(a)2

y x=

(b)1

y x= -

(c) f xx 1

1=

+] g

(d) f xx 2

3=

-

] g

(e)3 6

1 y

x=

+

(f) f xx 3

2= -

-

] g

(g) f xx 1

4=

-

] g

(h)1

2 y

x= -

+

(i) f xx6 3

2=

-

] g

(j)2

6 y

x= -

+

2 Show that f x x2

=] g is an odd

function



(a) f xx2 5

1=

+

] g for x2 2 -

(b)3

1 y

x=

+ for x2 0 -

(c) f xx2 4

5=

-

] g for x3 1 -

57 Exercises




(d) f xx 4

3= -

-

] g for x3 3 -

(e)3 1

2 y

x= -

+ for x0 5



(a)3

y x= for y 1 3

(b)2

y x= - for y 221

- -

(c) f xx 1

1=

-

] g for y 171

- -

(d) f xx2 1

3= -

+

] g for

y 131

- -

(e)3 2

6 y

x=

- for y 1

21

6



Circle gate





Proof

r y

x

( x y)

y

x








c a b

r x y

2 2 2

2 2 2

`

= +

= +

EXAMPLE



Solution


y

x

-2

-2 2

2


one place

y

x

2

2

2

-2

The radius is 4

CONTINUED






Domain 2 2 x x -

Range 2 2 y y -


range y r y r -



Proof


shown

y

x

(a b)

x

y

r

( x y)

a

b x - a

y - b




ndash ndash

c a b

r x a y b

2 2 2

2 2 2

= +

= +] ^g h




EXAMPLES



Solution



y

x 9

9

-9

-9



Domain 9 9 x x -

Range 9 9 y y -



Solution


ndash

ndash ndash

x y

x y

1 2 4

1 2 2

2

2 2

+ + =

+ - =

2

2

] ^

] ]_

g h

g gi

So 1 2a b= = - and 2r =

CONTINUED







y

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

(1 -2)



Domain 1 3 x x -

Range 4 0 y y -


Solution


a b2 1= - = and r 3=

Substituting

ndash ndash

ndash

ndash

x a y b r

x y

x y

2 1 3

2 1 9

2 2 2

2 2 2

2 2

+ =

- - + =

+ + =

] ^]] ^

] ^

g hg g hg h


ndash

ndash

a b a ab b

x x x

x x

a b a ab b

y y y

y y

2

2 2 2 2

4 4

2

1 2 1 1

2 1

So

So

2 2 2

2 2 2

2

2 2 2

2 2 2

2

+ = + +

+ = + +

= + +

= - +

= - +

= - +

]] ] ]]^ ^ ]

gg g ggh h g


ndash

x x y y

x x y y

x x y y

x x y y

4 4 2 1 9

4 2 5 9

4 2 5 9

4 2 4 0

9 9

2

2

2

2

+ + + - + =

+ + - + =

+ + + =

+ + - - =

- -


in Chapter 2




Investigation


two functions





circle






for all values

y

x r

r

-r


Proof

ndash

x y r

y r x y r x

2 2 2

2 2 2

2 2

+ =

=

= -



semi-circles





y 0 for all values

y

x r

-

r

-r


EXAMPLES


1 f x x92

= -] g

Solution



y

x 3

3

-3

Domain 3 3 x x -

Range 0 3 y y




2 y x4 2= - -

Solution



y

x 2

-2

-2

Domain 2 2 x x -

Range 2 0 y y -




range

(a) 9x y 2 2+ =

(b) x y 16 02 2+ =-


(d) 1 9x y 2 2+ + =

] g

(e) ndashx y 2 1 12 2+ + =] ^g h



below the x -axis



range

(a) 25 y x2= - -

(b) 1 y x2= -

(c) 36 y x2= -

(d) 64 y x2= - -

(e) 7 y x2= - -



each circle

(a) 100x y 2 2+ =

(b) 5x y 2 2+ =


(d) ndashx y 5 6 492 2+ + =] ^g h

(e) ndashx y 3 812 2+ =^ h

58 Exercises






grouping symbols)



Centre(c) 1 5-







Centre(j) 4 7- -

^ h and radius 3

Other Graphs




EXAMPLES


Solution



eg When 0x =

y 3

1

c=

=

x

y

1

3

3

1

31

When1

1

= -

=

=

=

-

x 3- 2- 1- 0 1 2 3

y 271

91

31

1 3 9 27



You will meet these

graphs again in the

HSC Course





other point

When

x

y

1

3

3

1

=

=

=

y

x

1

2

1

3







Solution



x minus2 minus1 0 05 1 2 3 4

y minus03 0 03 05 06

y

x

1

2

1 2 3 4

-1




You learned about


stages of maths





range y y 02



range all real y

Cubic function


EXAMPLE



Solution



y minus25 minus6 1 2 3 10 29

y

x

1

1

-2 2 3 4

-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4



If you already

know the shape of


3






Domain and range


EXAMPLE


Solution




function are

f

f

1 1 2

1 21

4 4 2

64 2

66

3

3

- = - +

= - +

=

= +

= +

=

] ]

] ]

g g

g g


these points

y

x

(-1 1)

(4 66)

Range 1 66 y y










positive number

EXAMPLE


Solution


x 4So $

$


(or zero) So y 0$


59 Exercises


(a) 4 3 y x= +

(b) f x 4= -] g

(c) 3x =



(f) f x xx 12 2= - -] g

(g) 64x y 2 2+ =

(h) f t t 4

3=

-] g

(i) ( ) g 2

5zz

= +

(j) | |f x x=] g


(a) y x=

(b) 2 y x= -

(c) | |f x x2 3= -] g

(d) | | 2 y x= -

(e) f x x2 5= - +] g

(f) | | y x5= -

(g) 2 y x=

(h) y 5x= -

(i) f x xx 1

= +] g

(j)2

4 3 y x

x= -


(a) y x x 5 2= -] g


= +-

(d) g x x x164 2= -] g

(e) 49x y 2 2+ =

You may like to


by dividing by x




4 (a) Solve x1 02$-


f x x1 2= -] g


(a) 2 y x x2= - -

(b) g t t t 62= +] g



in each case


x3 3 -


x2 3 -


x2 1 - (d)


x1 5


0 4x


x3 3 -


x1 1 -


x2 3 -


x4 4 -

(j) y x x7 62= - + - in the

domain 0 7x


function y x 1

3=

+





x=] g


function

find its range(b)


plane

(a) f x x4=] g

(b) y x3= -

(c) y x 34= -

(d) 2 p x x3=] g

(e) 1 g x x3= +] g

(f) 100x y 2 2+ =

(g) 2 1 y x= +


y x 1= -









(b) f x x 22= -] g

(c)1

y x=

(d) f x x3=] g

(e) f x 3x=] g

13 (a) Solve x4 02$-


(i) 4 y x2= -

(ii) y x4

2= - -




DID YOU KNOW




a light



Limits






a a1

1

03

Z

=

=

3

3

-


lim a 0x=

x 3-



asymptote

EXAMPLES

1 Find lim x

x x5x 0

2+

Solution


which is undefined


( )

lim lim

lim

xx x

x

x x

x

5 5

5

5

x x

x

0

2

0 1

1

0

+=

+

= +

=

] g




2 Find limx

x

4

22

-

-

x 2

Solution


lim lim

lim

x

x

x x

x

x

4

2

2 2

2

21

41

2 1

1

-

-=

+ -

-

=+

=

x x2 2

x 2

^ _h i

3 Find limh

h x hx h2 72 2+ -

h 0

Solution

lim lim

lim

h

h x hx h

h

h hx x

hx x

x

2 7 2 7

2 7

7

2 2 2

2

2

+ -=

+ -

= + -

= -

h 0

h h0 0

^ h

Continuity





EXAMPLES

1 Find lim

x

x

1

12

-

-

x 1


11

y x

x2

=-

- Sketch the curve

Solution


xx2

-

- gives

00

CONTINUED




( )

lim lim

limx

x

x

x x

x11

1

1 1

1

2

x x

x

1

2

1

1

-

-=

-

+ -

= +

=

-

] ]g g

11

y xx2

=-





y xx

x

x x

x

11

1

1 1

1

2

=-

-

=

+

= +

-

-^ ^h h


2 Find limx

x x2

2x 2

2

+

+ -

-


x x2

22

= + -

+

Solution


x x2

22

+

+ - gives

00

lim lim

lim

xx x

x

x x

x

22

2

1 2

1

3

x x

x

2

2

2

2

+

+ -=

+

- +

=

= -

-

- -

-

^^ ^

^

hh h

h2 y

x

x xx

y x

x

x

x

22

2

2

1

1


=+

+ -= -

=+

= -

+ -^ ^h h


Remember that x 1




1 Find(a) lim x 52

+x 4

(b) lim t 7-t 3 -

(c) lim x x2 43+ -

x 2

(d) lim xx x32

+

x 0

(e) limh

h h

2

22

-

- -

h 2

(f) lim y

y

5

1253

-

-

y 5

(g) limx

x x

12 12

+

+ +

x 1-

(h) limx

x x4

2 82

+

+ -

x 4 -

(i) limc

c

4

22

-

-

c 2

(j) limx x

x 12

-

-

x 1

(k) lim h

h h h2 73 2+ -

h 0

(l) limh

hx hx h32 2- +

h 0

(m) limh

hx h x hx h2 3 53 2 2- + -

h 0

(n) lim x c x c 3 3

-

-

x c



discontinuous

(a) 3 y x2= -

(b)1

1 y

x=

+

(c) f x x 1= -] g

(d)4

1 y

x2=

+

(e)4

1

y x2=

-



(a)3

y xx x2

= +

(b)33

y x

x x2

=+

+

(c)1

5 4 y

xx x2

=+

+ +

510 Exercises




Regions

Class Investigation


them all







EXAMPLE


Solution



y

x =3

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4







region

EXAMPLE


Solution



included


shown

y

x

1

1

-2 2 3 4

-3

-4

-5

2

3

4

5

-1

-2

-3-4

-1 y = -1



y -intercept 1-






EXAMPLES


1 y x 2$ +

Solution






y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0



2 x y 2 3 61-

Solution


region


side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g








3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g






EXAMPLES


Solution









(g) f x x 1 2= +] ]g g

(h) y x x3 42= + -

(i) y x x2 5 32= - +

(j) f x x x3 22= - + -] g



(a) ndash y x x7 122= +

(b) f x x x42= +] g

(c) y x x2 82= - -

(d) y x x6 92= +-

(e) f t t 4 2= -] g


(a) y x 52= -

(b) f x x x6

2= -

] g (c) f x x x 22= - -] g

(d) y x2= -

(e) f x x 7 2= -] ]g g



(a) y x2= for x0 3

(b) y x 42= - + for x1 2 -

(c) f x x 12= -] g for x2 5 -

(d) y x x2 32= + - for x2 4 -

(e) y x x

22= - +- for

x0 4


function is

increasing(i)

decreasing(ii)

(a) y x2=

(b) y x2

= - (c) f x x 92

= -] g

(d) y x x42= - +

(e) f x x 5 2= +] ]g g


function



(a) y x 12= +

(b) f x x 32= -

] g

(c) y x2 2= -

(d) f x x x32= -] g

(e) f x x x2= +] g

(f) y x 42= -

(g) y x x2 32= - -

(h) y x x5 42= +-

(i) p x x 1 2= +] ]g g

(j) y x 2= -2] g




CONTINUED

eg When x 3= -

| | y 3 13 12

= - -

= -

=

x -3 -2 -1 0 1 2 3

y 2 1 0 -1 0 1 2


y

-2

4

3

2

1

5

-1

-3

-4

-5

1 4-1-2 32-3-4

x


| | y x

x x

x x1

1 0

1 0

when

when 1

$= - =

-


y x x1 0$= - ] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x - 1




y x 1= - - x 01] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y =- x - 1



-3

4

3

2

1

5

-2

-1

-4

-5

y y

3-1-2 421-3-4

x

y = - x - 1

y = x - 1


intercepts


So f x 0=

] g

| |

| |

x

x

x

0 1

1

1`

= -

=

=


( ) | |f 0 0 11

= -

= -





-3

4

3

2

1

5

-2

-1

-4

-5

y

4-2 5321-1-3-4

x



Domain all real x

Range y y 1$ -

2 Sketch | | y x 2= +

Solution


| | ( ) y x x xx x2 2 2 0

2 2 0whenwhen 1

$= + = + +

- + +


2 y x= + when x 2 0$+

x 2$ -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x + 2






y-intercept of 1-

CONTINUED




2 y x= - +] g when x 2 01+

ie y x 2= - - when x 21 -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2


-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2

y = x + 2



So 0f x =] g

0 | 2 |

0 2

2

x

x

x

= +

= +

- =


(0) | 0 2 |

2

f = +

=

There is only one

solution for the


Can you see why





-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

If you know how to

sketch the graph of




Investigation




are their equations





1 | |f x k x=] g

2 | |f x x k= +] g

3 | |f x x k= +] g







EXAMPLES

Solve

1 |2 1 | 3x - =

Solution



is x 1 2= -

2 |2 1 | 3 2x x= -+

Solution



3 | 1 | 2x 1+

Solution






and then check them







each function

(a) | | y x=

(b) | |f x x 7= +] g

(c) | |f x x 2= -] g

(d) 5 | | y x=

(e) | |f x x 3= - +] g

(f) | 6 | y x= +

(g) | |f x x3 2= -] g

(h) | 5 4 | y x= +

(i) | 7 1 | y x= -

(j) | |f x x2 9= +] g


plane

(a) | | y x=

(b) | |f x x 1= +] g

(c) | |f x x 3= -] g

(d) 2 | | y x=

(e) | |f x x= -] g

(f) | 1 | y x= +

(g) | |f x x 1= - -] g

(h) | 2 3 | y x= -

(i) | 4 2 | y x= +

(j) | |f x x3 1= +] g


each function

(a) | 1 | y x= -

(b) | |f x x 8= -] g

(c) | |f x x2 5= +] g

(d) 2 | | 3 y x= -

(e) | |f x x 3= - -] g


function is

increasing(i)

decreasing(ii)

(a) | 2 | y x= -

(b) | |f x x 2= +

] g

(c) | |f x x2 3= -] g

(d) 4 | | 1 y x= -

(e) | |f x x= -] g


of each function

(a) | | y x= for x2 2 -

(b) | |f x x 4= - -] g for

x4 3 -

(c) | |f x x 4= +] g for x7 2 -

(d) | 2 5 | y x= - for x3 3 -

(e) | |f x x= -] g for x1 1 -


function increasing

(a) | 3 | y x= +

(b) | |f x x 4= - +] g

(c) | |f x x 9= -] g

(d) | | y x 2 1= - -

(e) | |f x x 2= - +] g

56 Exercises




7 Solve graphically

(a) | | 3x =

(b) | |x 12

(c) | |x 2

(d) | 2 | 1x + =

(e) | 3 | 0x- =

(f) |2 3 | 1x - =

(g) | |x 1 41-

(h) | |x 1 3+

(i) | |x 2 22-

(j) | |x 3 1$-

(k) | |x2 3 5+

(l) | |x2 1 1$-

(m) |3 1 | 3x x- = +

(n) |3 2 | 4x x- = -

(o) | 1 | 1x x- = +

(p) | 3 | 2 2x x+ = + (q) |2 1 | 1x x+ = -

(r) |2 5 | 3x x- = -

(s) | 1 | 2x x- =

(t) |2 3 | 3x x- = +

The Hyperbola


or= =

EXAMPLE

Sketch1

y x=

Solution



x -3 -2 -121-

41- 0

41

21 1 2 3

y 3

1-

2

1- -1 -2 -4 mdash 4 2 1

2

1

3

1

Class Discussion



y is never 0 Why






but never touches)



Investigation




equations




1 y xk

=

21

y x k= +

31

y x k

=+


EXAMPLES


3=

-

] g


Solution



So x 3






be zero

CONTINUED







1

y 0 3

3=

-

= -

-3

4

3

2

1

5

-2

-1

-4

-5

y

-1-2 4 521-3-4

x

x = 3

y = 0

Asymptotes

3

2 Sketch y x2 4

1= -

+

Solution




x

x

2 4

2

-

-






( ) y

2 0 41

41

= -+

= -


a translation of3

y x

=





y

-2

x

-

1

4


a=

+


domain x xb



1 For each graph



exists


(a)2

y x=

(b)1

y x= -

(c) f xx 1

1=

+] g

(d) f xx 2

3=

-

] g

(e)3 6

1 y

x=

+

(f) f xx 3

2= -

-

] g

(g) f xx 1

4=

-

] g

(h)1

2 y

x= -

+

(i) f xx6 3

2=

-

] g

(j)2

6 y

x= -

+

2 Show that f x x2

=] g is an odd

function



(a) f xx2 5

1=

+

] g for x2 2 -

(b)3

1 y

x=

+ for x2 0 -

(c) f xx2 4

5=

-

] g for x3 1 -

57 Exercises




(d) f xx 4

3= -

-

] g for x3 3 -

(e)3 1

2 y

x= -

+ for x0 5



(a)3

y x= for y 1 3

(b)2

y x= - for y 221

- -

(c) f xx 1

1=

-

] g for y 171

- -

(d) f xx2 1

3= -

+

] g for

y 131

- -

(e)3 2

6 y

x=

- for y 1

21

6



Circle gate





Proof

r y

x

( x y)

y

x








c a b

r x y

2 2 2

2 2 2

`

= +

= +

EXAMPLE



Solution


y

x

-2

-2 2

2


one place

y

x

2

2

2

-2

The radius is 4

CONTINUED






Domain 2 2 x x -

Range 2 2 y y -


range y r y r -



Proof


shown

y

x

(a b)

x

y

r

( x y)

a

b x - a

y - b




ndash ndash

c a b

r x a y b

2 2 2

2 2 2

= +

= +] ^g h




EXAMPLES



Solution



y

x 9

9

-9

-9



Domain 9 9 x x -

Range 9 9 y y -



Solution


ndash

ndash ndash

x y

x y

1 2 4

1 2 2

2

2 2

+ + =

+ - =

2

2

] ^

] ]_

g h

g gi

So 1 2a b= = - and 2r =

CONTINUED







y

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

(1 -2)



Domain 1 3 x x -

Range 4 0 y y -


Solution


a b2 1= - = and r 3=

Substituting

ndash ndash

ndash

ndash

x a y b r

x y

x y

2 1 3

2 1 9

2 2 2

2 2 2

2 2

+ =

- - + =

+ + =

] ^]] ^

] ^

g hg g hg h


ndash

ndash

a b a ab b

x x x

x x

a b a ab b

y y y

y y

2

2 2 2 2

4 4

2

1 2 1 1

2 1

So

So

2 2 2

2 2 2

2

2 2 2

2 2 2

2

+ = + +

+ = + +

= + +

= - +

= - +

= - +

]] ] ]]^ ^ ]

gg g ggh h g


ndash

x x y y

x x y y

x x y y

x x y y

4 4 2 1 9

4 2 5 9

4 2 5 9

4 2 4 0

9 9

2

2

2

2

+ + + - + =

+ + - + =

+ + + =

+ + - - =

- -


in Chapter 2




Investigation


two functions





circle






for all values

y

x r

r

-r


Proof

ndash

x y r

y r x y r x

2 2 2

2 2 2

2 2

+ =

=

= -



semi-circles





y 0 for all values

y

x r

-

r

-r


EXAMPLES


1 f x x92

= -] g

Solution



y

x 3

3

-3

Domain 3 3 x x -

Range 0 3 y y




2 y x4 2= - -

Solution



y

x 2

-2

-2

Domain 2 2 x x -

Range 2 0 y y -




range

(a) 9x y 2 2+ =

(b) x y 16 02 2+ =-


(d) 1 9x y 2 2+ + =

] g

(e) ndashx y 2 1 12 2+ + =] ^g h



below the x -axis



range

(a) 25 y x2= - -

(b) 1 y x2= -

(c) 36 y x2= -

(d) 64 y x2= - -

(e) 7 y x2= - -



each circle

(a) 100x y 2 2+ =

(b) 5x y 2 2+ =


(d) ndashx y 5 6 492 2+ + =] ^g h

(e) ndashx y 3 812 2+ =^ h

58 Exercises






grouping symbols)



Centre(c) 1 5-







Centre(j) 4 7- -

^ h and radius 3

Other Graphs




EXAMPLES


Solution



eg When 0x =

y 3

1

c=

=

x

y

1

3

3

1

31

When1

1

= -

=

=

=

-

x 3- 2- 1- 0 1 2 3

y 271

91

31

1 3 9 27



You will meet these

graphs again in the

HSC Course





other point

When

x

y

1

3

3

1

=

=

=

y

x

1

2

1

3







Solution



x minus2 minus1 0 05 1 2 3 4

y minus03 0 03 05 06

y

x

1

2

1 2 3 4

-1




You learned about


stages of maths





range y y 02



range all real y

Cubic function


EXAMPLE



Solution



y minus25 minus6 1 2 3 10 29

y

x

1

1

-2 2 3 4

-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4



If you already

know the shape of


3






Domain and range


EXAMPLE


Solution




function are

f

f

1 1 2

1 21

4 4 2

64 2

66

3

3

- = - +

= - +

=

= +

= +

=

] ]

] ]

g g

g g


these points

y

x

(-1 1)

(4 66)

Range 1 66 y y










positive number

EXAMPLE


Solution


x 4So $

$


(or zero) So y 0$


59 Exercises


(a) 4 3 y x= +

(b) f x 4= -] g

(c) 3x =



(f) f x xx 12 2= - -] g

(g) 64x y 2 2+ =

(h) f t t 4

3=

-] g

(i) ( ) g 2

5zz

= +

(j) | |f x x=] g


(a) y x=

(b) 2 y x= -

(c) | |f x x2 3= -] g

(d) | | 2 y x= -

(e) f x x2 5= - +] g

(f) | | y x5= -

(g) 2 y x=

(h) y 5x= -

(i) f x xx 1

= +] g

(j)2

4 3 y x

x= -


(a) y x x 5 2= -] g


= +-

(d) g x x x164 2= -] g

(e) 49x y 2 2+ =

You may like to


by dividing by x




4 (a) Solve x1 02$-


f x x1 2= -] g


(a) 2 y x x2= - -

(b) g t t t 62= +] g



in each case


x3 3 -


x2 3 -


x2 1 - (d)


x1 5


0 4x


x3 3 -


x1 1 -


x2 3 -


x4 4 -

(j) y x x7 62= - + - in the

domain 0 7x


function y x 1

3=

+





x=] g


function

find its range(b)


plane

(a) f x x4=] g

(b) y x3= -

(c) y x 34= -

(d) 2 p x x3=] g

(e) 1 g x x3= +] g

(f) 100x y 2 2+ =

(g) 2 1 y x= +


y x 1= -









(b) f x x 22= -] g

(c)1

y x=

(d) f x x3=] g

(e) f x 3x=] g

13 (a) Solve x4 02$-


(i) 4 y x2= -

(ii) y x4

2= - -




DID YOU KNOW




a light



Limits






a a1

1

03

Z

=

=

3

3

-


lim a 0x=

x 3-



asymptote

EXAMPLES

1 Find lim x

x x5x 0

2+

Solution


which is undefined


( )

lim lim

lim

xx x

x

x x

x

5 5

5

5

x x

x

0

2

0 1

1

0

+=

+

= +

=

] g




2 Find limx

x

4

22

-

-

x 2

Solution


lim lim

lim

x

x

x x

x

x

4

2

2 2

2

21

41

2 1

1

-

-=

+ -

-

=+

=

x x2 2

x 2

^ _h i

3 Find limh

h x hx h2 72 2+ -

h 0

Solution

lim lim

lim

h

h x hx h

h

h hx x

hx x

x

2 7 2 7

2 7

7

2 2 2

2

2

+ -=

+ -

= + -

= -

h 0

h h0 0

^ h

Continuity





EXAMPLES

1 Find lim

x

x

1

12

-

-

x 1


11

y x

x2

=-

- Sketch the curve

Solution


xx2

-

- gives

00

CONTINUED




( )

lim lim

limx

x

x

x x

x11

1

1 1

1

2

x x

x

1

2

1

1

-

-=

-

+ -

= +

=

-

] ]g g

11

y xx2

=-





y xx

x

x x

x

11

1

1 1

1

2

=-

-

=

+

= +

-

-^ ^h h


2 Find limx

x x2

2x 2

2

+

+ -

-


x x2

22

= + -

+

Solution


x x2

22

+

+ - gives

00

lim lim

lim

xx x

x

x x

x

22

2

1 2

1

3

x x

x

2

2

2

2

+

+ -=

+

- +

=

= -

-

- -

-

^^ ^

^

hh h

h2 y

x

x xx

y x

x

x

x

22

2

2

1

1


=+

+ -= -

=+

= -

+ -^ ^h h


Remember that x 1




1 Find(a) lim x 52

+x 4

(b) lim t 7-t 3 -

(c) lim x x2 43+ -

x 2

(d) lim xx x32

+

x 0

(e) limh

h h

2

22

-

- -

h 2

(f) lim y

y

5

1253

-

-

y 5

(g) limx

x x

12 12

+

+ +

x 1-

(h) limx

x x4

2 82

+

+ -

x 4 -

(i) limc

c

4

22

-

-

c 2

(j) limx x

x 12

-

-

x 1

(k) lim h

h h h2 73 2+ -

h 0

(l) limh

hx hx h32 2- +

h 0

(m) limh

hx h x hx h2 3 53 2 2- + -

h 0

(n) lim x c x c 3 3

-

-

x c



discontinuous

(a) 3 y x2= -

(b)1

1 y

x=

+

(c) f x x 1= -] g

(d)4

1 y

x2=

+

(e)4

1

y x2=

-



(a)3

y xx x2

= +

(b)33

y x

x x2

=+

+

(c)1

5 4 y

xx x2

=+

+ +

510 Exercises




Regions

Class Investigation


them all







EXAMPLE


Solution



y

x =3

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4







region

EXAMPLE


Solution



included


shown

y

x

1

1

-2 2 3 4

-3

-4

-5

2

3

4

5

-1

-2

-3-4

-1 y = -1



y -intercept 1-






EXAMPLES


1 y x 2$ +

Solution






y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0



2 x y 2 3 61-

Solution


region


side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g








3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g






CONTINUED

eg When x 3= -

| | y 3 13 12

= - -

= -

=

x -3 -2 -1 0 1 2 3

y 2 1 0 -1 0 1 2


y

-2

4

3

2

1

5

-1

-3

-4

-5

1 4-1-2 32-3-4

x


| | y x

x x

x x1

1 0

1 0

when

when 1

$= - =

-


y x x1 0$= - ] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x - 1




y x 1= - - x 01] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y =- x - 1



-3

4

3

2

1

5

-2

-1

-4

-5

y y

3-1-2 421-3-4

x

y = - x - 1

y = x - 1


intercepts


So f x 0=

] g

| |

| |

x

x

x

0 1

1

1`

= -

=

=


( ) | |f 0 0 11

= -

= -





-3

4

3

2

1

5

-2

-1

-4

-5

y

4-2 5321-1-3-4

x



Domain all real x

Range y y 1$ -

2 Sketch | | y x 2= +

Solution


| | ( ) y x x xx x2 2 2 0

2 2 0whenwhen 1

$= + = + +

- + +


2 y x= + when x 2 0$+

x 2$ -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x + 2






y-intercept of 1-

CONTINUED




2 y x= - +] g when x 2 01+

ie y x 2= - - when x 21 -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2


-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2

y = x + 2



So 0f x =] g

0 | 2 |

0 2

2

x

x

x

= +

= +

- =


(0) | 0 2 |

2

f = +

=

There is only one

solution for the


Can you see why





-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

If you know how to

sketch the graph of




Investigation




are their equations





1 | |f x k x=] g

2 | |f x x k= +] g

3 | |f x x k= +] g







EXAMPLES

Solve

1 |2 1 | 3x - =

Solution



is x 1 2= -

2 |2 1 | 3 2x x= -+

Solution



3 | 1 | 2x 1+

Solution






and then check them







each function

(a) | | y x=

(b) | |f x x 7= +] g

(c) | |f x x 2= -] g

(d) 5 | | y x=

(e) | |f x x 3= - +] g

(f) | 6 | y x= +

(g) | |f x x3 2= -] g

(h) | 5 4 | y x= +

(i) | 7 1 | y x= -

(j) | |f x x2 9= +] g


plane

(a) | | y x=

(b) | |f x x 1= +] g

(c) | |f x x 3= -] g

(d) 2 | | y x=

(e) | |f x x= -] g

(f) | 1 | y x= +

(g) | |f x x 1= - -] g

(h) | 2 3 | y x= -

(i) | 4 2 | y x= +

(j) | |f x x3 1= +] g


each function

(a) | 1 | y x= -

(b) | |f x x 8= -] g

(c) | |f x x2 5= +] g

(d) 2 | | 3 y x= -

(e) | |f x x 3= - -] g


function is

increasing(i)

decreasing(ii)

(a) | 2 | y x= -

(b) | |f x x 2= +

] g

(c) | |f x x2 3= -] g

(d) 4 | | 1 y x= -

(e) | |f x x= -] g


of each function

(a) | | y x= for x2 2 -

(b) | |f x x 4= - -] g for

x4 3 -

(c) | |f x x 4= +] g for x7 2 -

(d) | 2 5 | y x= - for x3 3 -

(e) | |f x x= -] g for x1 1 -


function increasing

(a) | 3 | y x= +

(b) | |f x x 4= - +] g

(c) | |f x x 9= -] g

(d) | | y x 2 1= - -

(e) | |f x x 2= - +] g

56 Exercises




7 Solve graphically

(a) | | 3x =

(b) | |x 12

(c) | |x 2

(d) | 2 | 1x + =

(e) | 3 | 0x- =

(f) |2 3 | 1x - =

(g) | |x 1 41-

(h) | |x 1 3+

(i) | |x 2 22-

(j) | |x 3 1$-

(k) | |x2 3 5+

(l) | |x2 1 1$-

(m) |3 1 | 3x x- = +

(n) |3 2 | 4x x- = -

(o) | 1 | 1x x- = +

(p) | 3 | 2 2x x+ = + (q) |2 1 | 1x x+ = -

(r) |2 5 | 3x x- = -

(s) | 1 | 2x x- =

(t) |2 3 | 3x x- = +

The Hyperbola


or= =

EXAMPLE

Sketch1

y x=

Solution



x -3 -2 -121-

41- 0

41

21 1 2 3

y 3

1-

2

1- -1 -2 -4 mdash 4 2 1

2

1

3

1

Class Discussion



y is never 0 Why






but never touches)



Investigation




equations




1 y xk

=

21

y x k= +

31

y x k

=+


EXAMPLES


3=

-

] g


Solution



So x 3






be zero

CONTINUED







1

y 0 3

3=

-

= -

-3

4

3

2

1

5

-2

-1

-4

-5

y

-1-2 4 521-3-4

x

x = 3

y = 0

Asymptotes

3

2 Sketch y x2 4

1= -

+

Solution




x

x

2 4

2

-

-






( ) y

2 0 41

41

= -+

= -


a translation of3

y x

=





y

-2

x

-

1

4


a=

+


domain x xb



1 For each graph



exists


(a)2

y x=

(b)1

y x= -

(c) f xx 1

1=

+] g

(d) f xx 2

3=

-

] g

(e)3 6

1 y

x=

+

(f) f xx 3

2= -

-

] g

(g) f xx 1

4=

-

] g

(h)1

2 y

x= -

+

(i) f xx6 3

2=

-

] g

(j)2

6 y

x= -

+

2 Show that f x x2

=] g is an odd

function



(a) f xx2 5

1=

+

] g for x2 2 -

(b)3

1 y

x=

+ for x2 0 -

(c) f xx2 4

5=

-

] g for x3 1 -

57 Exercises




(d) f xx 4

3= -

-

] g for x3 3 -

(e)3 1

2 y

x= -

+ for x0 5



(a)3

y x= for y 1 3

(b)2

y x= - for y 221

- -

(c) f xx 1

1=

-

] g for y 171

- -

(d) f xx2 1

3= -

+

] g for

y 131

- -

(e)3 2

6 y

x=

- for y 1

21

6



Circle gate





Proof

r y

x

( x y)

y

x








c a b

r x y

2 2 2

2 2 2

`

= +

= +

EXAMPLE



Solution


y

x

-2

-2 2

2


one place

y

x

2

2

2

-2

The radius is 4

CONTINUED






Domain 2 2 x x -

Range 2 2 y y -


range y r y r -



Proof


shown

y

x

(a b)

x

y

r

( x y)

a

b x - a

y - b




ndash ndash

c a b

r x a y b

2 2 2

2 2 2

= +

= +] ^g h




EXAMPLES



Solution



y

x 9

9

-9

-9



Domain 9 9 x x -

Range 9 9 y y -



Solution


ndash

ndash ndash

x y

x y

1 2 4

1 2 2

2

2 2

+ + =

+ - =

2

2

] ^

] ]_

g h

g gi

So 1 2a b= = - and 2r =

CONTINUED







y

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

(1 -2)



Domain 1 3 x x -

Range 4 0 y y -


Solution


a b2 1= - = and r 3=

Substituting

ndash ndash

ndash

ndash

x a y b r

x y

x y

2 1 3

2 1 9

2 2 2

2 2 2

2 2

+ =

- - + =

+ + =

] ^]] ^

] ^

g hg g hg h


ndash

ndash

a b a ab b

x x x

x x

a b a ab b

y y y

y y

2

2 2 2 2

4 4

2

1 2 1 1

2 1

So

So

2 2 2

2 2 2

2

2 2 2

2 2 2

2

+ = + +

+ = + +

= + +

= - +

= - +

= - +

]] ] ]]^ ^ ]

gg g ggh h g


ndash

x x y y

x x y y

x x y y

x x y y

4 4 2 1 9

4 2 5 9

4 2 5 9

4 2 4 0

9 9

2

2

2

2

+ + + - + =

+ + - + =

+ + + =

+ + - - =

- -


in Chapter 2




Investigation


two functions





circle






for all values

y

x r

r

-r


Proof

ndash

x y r

y r x y r x

2 2 2

2 2 2

2 2

+ =

=

= -



semi-circles





y 0 for all values

y

x r

-

r

-r


EXAMPLES


1 f x x92

= -] g

Solution



y

x 3

3

-3

Domain 3 3 x x -

Range 0 3 y y




2 y x4 2= - -

Solution



y

x 2

-2

-2

Domain 2 2 x x -

Range 2 0 y y -




range

(a) 9x y 2 2+ =

(b) x y 16 02 2+ =-


(d) 1 9x y 2 2+ + =

] g

(e) ndashx y 2 1 12 2+ + =] ^g h



below the x -axis



range

(a) 25 y x2= - -

(b) 1 y x2= -

(c) 36 y x2= -

(d) 64 y x2= - -

(e) 7 y x2= - -



each circle

(a) 100x y 2 2+ =

(b) 5x y 2 2+ =


(d) ndashx y 5 6 492 2+ + =] ^g h

(e) ndashx y 3 812 2+ =^ h

58 Exercises






grouping symbols)



Centre(c) 1 5-







Centre(j) 4 7- -

^ h and radius 3

Other Graphs




EXAMPLES


Solution



eg When 0x =

y 3

1

c=

=

x

y

1

3

3

1

31

When1

1

= -

=

=

=

-

x 3- 2- 1- 0 1 2 3

y 271

91

31

1 3 9 27



You will meet these

graphs again in the

HSC Course





other point

When

x

y

1

3

3

1

=

=

=

y

x

1

2

1

3







Solution



x minus2 minus1 0 05 1 2 3 4

y minus03 0 03 05 06

y

x

1

2

1 2 3 4

-1




You learned about


stages of maths





range y y 02



range all real y

Cubic function


EXAMPLE



Solution



y minus25 minus6 1 2 3 10 29

y

x

1

1

-2 2 3 4

-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4



If you already

know the shape of


3






Domain and range


EXAMPLE


Solution




function are

f

f

1 1 2

1 21

4 4 2

64 2

66

3

3

- = - +

= - +

=

= +

= +

=

] ]

] ]

g g

g g


these points

y

x

(-1 1)

(4 66)

Range 1 66 y y










positive number

EXAMPLE


Solution


x 4So $

$


(or zero) So y 0$


59 Exercises


(a) 4 3 y x= +

(b) f x 4= -] g

(c) 3x =



(f) f x xx 12 2= - -] g

(g) 64x y 2 2+ =

(h) f t t 4

3=

-] g

(i) ( ) g 2

5zz

= +

(j) | |f x x=] g


(a) y x=

(b) 2 y x= -

(c) | |f x x2 3= -] g

(d) | | 2 y x= -

(e) f x x2 5= - +] g

(f) | | y x5= -

(g) 2 y x=

(h) y 5x= -

(i) f x xx 1

= +] g

(j)2

4 3 y x

x= -


(a) y x x 5 2= -] g


= +-

(d) g x x x164 2= -] g

(e) 49x y 2 2+ =

You may like to


by dividing by x




4 (a) Solve x1 02$-


f x x1 2= -] g


(a) 2 y x x2= - -

(b) g t t t 62= +] g



in each case


x3 3 -


x2 3 -


x2 1 - (d)


x1 5


0 4x


x3 3 -


x1 1 -


x2 3 -


x4 4 -

(j) y x x7 62= - + - in the

domain 0 7x


function y x 1

3=

+





x=] g


function

find its range(b)


plane

(a) f x x4=] g

(b) y x3= -

(c) y x 34= -

(d) 2 p x x3=] g

(e) 1 g x x3= +] g

(f) 100x y 2 2+ =

(g) 2 1 y x= +


y x 1= -









(b) f x x 22= -] g

(c)1

y x=

(d) f x x3=] g

(e) f x 3x=] g

13 (a) Solve x4 02$-


(i) 4 y x2= -

(ii) y x4

2= - -




DID YOU KNOW




a light



Limits






a a1

1

03

Z

=

=

3

3

-


lim a 0x=

x 3-



asymptote

EXAMPLES

1 Find lim x

x x5x 0

2+

Solution


which is undefined


( )

lim lim

lim

xx x

x

x x

x

5 5

5

5

x x

x

0

2

0 1

1

0

+=

+

= +

=

] g




2 Find limx

x

4

22

-

-

x 2

Solution


lim lim

lim

x

x

x x

x

x

4

2

2 2

2

21

41

2 1

1

-

-=

+ -

-

=+

=

x x2 2

x 2

^ _h i

3 Find limh

h x hx h2 72 2+ -

h 0

Solution

lim lim

lim

h

h x hx h

h

h hx x

hx x

x

2 7 2 7

2 7

7

2 2 2

2

2

+ -=

+ -

= + -

= -

h 0

h h0 0

^ h

Continuity





EXAMPLES

1 Find lim

x

x

1

12

-

-

x 1


11

y x

x2

=-

- Sketch the curve

Solution


xx2

-

- gives

00

CONTINUED




( )

lim lim

limx

x

x

x x

x11

1

1 1

1

2

x x

x

1

2

1

1

-

-=

-

+ -

= +

=

-

] ]g g

11

y xx2

=-





y xx

x

x x

x

11

1

1 1

1

2

=-

-

=

+

= +

-

-^ ^h h


2 Find limx

x x2

2x 2

2

+

+ -

-


x x2

22

= + -

+

Solution


x x2

22

+

+ - gives

00

lim lim

lim

xx x

x

x x

x

22

2

1 2

1

3

x x

x

2

2

2

2

+

+ -=

+

- +

=

= -

-

- -

-

^^ ^

^

hh h

h2 y

x

x xx

y x

x

x

x

22

2

2

1

1


=+

+ -= -

=+

= -

+ -^ ^h h


Remember that x 1




1 Find(a) lim x 52

+x 4

(b) lim t 7-t 3 -

(c) lim x x2 43+ -

x 2

(d) lim xx x32

+

x 0

(e) limh

h h

2

22

-

- -

h 2

(f) lim y

y

5

1253

-

-

y 5

(g) limx

x x

12 12

+

+ +

x 1-

(h) limx

x x4

2 82

+

+ -

x 4 -

(i) limc

c

4

22

-

-

c 2

(j) limx x

x 12

-

-

x 1

(k) lim h

h h h2 73 2+ -

h 0

(l) limh

hx hx h32 2- +

h 0

(m) limh

hx h x hx h2 3 53 2 2- + -

h 0

(n) lim x c x c 3 3

-

-

x c



discontinuous

(a) 3 y x2= -

(b)1

1 y

x=

+

(c) f x x 1= -] g

(d)4

1 y

x2=

+

(e)4

1

y x2=

-



(a)3

y xx x2

= +

(b)33

y x

x x2

=+

+

(c)1

5 4 y

xx x2

=+

+ +

510 Exercises




Regions

Class Investigation


them all







EXAMPLE


Solution



y

x =3

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4







region

EXAMPLE


Solution



included


shown

y

x

1

1

-2 2 3 4

-3

-4

-5

2

3

4

5

-1

-2

-3-4

-1 y = -1



y -intercept 1-






EXAMPLES


1 y x 2$ +

Solution






y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0



2 x y 2 3 61-

Solution


region


side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g








3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g






y x 1= - - x 01] g

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y =- x - 1



-3

4

3

2

1

5

-2

-1

-4

-5

y y

3-1-2 421-3-4

x

y = - x - 1

y = x - 1


intercepts


So f x 0=

] g

| |

| |

x

x

x

0 1

1

1`

= -

=

=


( ) | |f 0 0 11

= -

= -





-3

4

3

2

1

5

-2

-1

-4

-5

y

4-2 5321-1-3-4

x



Domain all real x

Range y y 1$ -

2 Sketch | | y x 2= +

Solution


| | ( ) y x x xx x2 2 2 0

2 2 0whenwhen 1

$= + = + +

- + +


2 y x= + when x 2 0$+

x 2$ -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x + 2






y-intercept of 1-

CONTINUED




2 y x= - +] g when x 2 01+

ie y x 2= - - when x 21 -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2


-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2

y = x + 2



So 0f x =] g

0 | 2 |

0 2

2

x

x

x

= +

= +

- =


(0) | 0 2 |

2

f = +

=

There is only one

solution for the


Can you see why





-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

If you know how to

sketch the graph of




Investigation




are their equations





1 | |f x k x=] g

2 | |f x x k= +] g

3 | |f x x k= +] g







EXAMPLES

Solve

1 |2 1 | 3x - =

Solution



is x 1 2= -

2 |2 1 | 3 2x x= -+

Solution



3 | 1 | 2x 1+

Solution






and then check them







each function

(a) | | y x=

(b) | |f x x 7= +] g

(c) | |f x x 2= -] g

(d) 5 | | y x=

(e) | |f x x 3= - +] g

(f) | 6 | y x= +

(g) | |f x x3 2= -] g

(h) | 5 4 | y x= +

(i) | 7 1 | y x= -

(j) | |f x x2 9= +] g


plane

(a) | | y x=

(b) | |f x x 1= +] g

(c) | |f x x 3= -] g

(d) 2 | | y x=

(e) | |f x x= -] g

(f) | 1 | y x= +

(g) | |f x x 1= - -] g

(h) | 2 3 | y x= -

(i) | 4 2 | y x= +

(j) | |f x x3 1= +] g


each function

(a) | 1 | y x= -

(b) | |f x x 8= -] g

(c) | |f x x2 5= +] g

(d) 2 | | 3 y x= -

(e) | |f x x 3= - -] g


function is

increasing(i)

decreasing(ii)

(a) | 2 | y x= -

(b) | |f x x 2= +

] g

(c) | |f x x2 3= -] g

(d) 4 | | 1 y x= -

(e) | |f x x= -] g


of each function

(a) | | y x= for x2 2 -

(b) | |f x x 4= - -] g for

x4 3 -

(c) | |f x x 4= +] g for x7 2 -

(d) | 2 5 | y x= - for x3 3 -

(e) | |f x x= -] g for x1 1 -


function increasing

(a) | 3 | y x= +

(b) | |f x x 4= - +] g

(c) | |f x x 9= -] g

(d) | | y x 2 1= - -

(e) | |f x x 2= - +] g

56 Exercises




7 Solve graphically

(a) | | 3x =

(b) | |x 12

(c) | |x 2

(d) | 2 | 1x + =

(e) | 3 | 0x- =

(f) |2 3 | 1x - =

(g) | |x 1 41-

(h) | |x 1 3+

(i) | |x 2 22-

(j) | |x 3 1$-

(k) | |x2 3 5+

(l) | |x2 1 1$-

(m) |3 1 | 3x x- = +

(n) |3 2 | 4x x- = -

(o) | 1 | 1x x- = +

(p) | 3 | 2 2x x+ = + (q) |2 1 | 1x x+ = -

(r) |2 5 | 3x x- = -

(s) | 1 | 2x x- =

(t) |2 3 | 3x x- = +

The Hyperbola


or= =

EXAMPLE

Sketch1

y x=

Solution



x -3 -2 -121-

41- 0

41

21 1 2 3

y 3

1-

2

1- -1 -2 -4 mdash 4 2 1

2

1

3

1

Class Discussion



y is never 0 Why






but never touches)



Investigation




equations




1 y xk

=

21

y x k= +

31

y x k

=+


EXAMPLES


3=

-

] g


Solution



So x 3






be zero

CONTINUED







1

y 0 3

3=

-

= -

-3

4

3

2

1

5

-2

-1

-4

-5

y

-1-2 4 521-3-4

x

x = 3

y = 0

Asymptotes

3

2 Sketch y x2 4

1= -

+

Solution




x

x

2 4

2

-

-






( ) y

2 0 41

41

= -+

= -


a translation of3

y x

=





y

-2

x

-

1

4


a=

+


domain x xb



1 For each graph



exists


(a)2

y x=

(b)1

y x= -

(c) f xx 1

1=

+] g

(d) f xx 2

3=

-

] g

(e)3 6

1 y

x=

+

(f) f xx 3

2= -

-

] g

(g) f xx 1

4=

-

] g

(h)1

2 y

x= -

+

(i) f xx6 3

2=

-

] g

(j)2

6 y

x= -

+

2 Show that f x x2

=] g is an odd

function



(a) f xx2 5

1=

+

] g for x2 2 -

(b)3

1 y

x=

+ for x2 0 -

(c) f xx2 4

5=

-

] g for x3 1 -

57 Exercises




(d) f xx 4

3= -

-

] g for x3 3 -

(e)3 1

2 y

x= -

+ for x0 5



(a)3

y x= for y 1 3

(b)2

y x= - for y 221

- -

(c) f xx 1

1=

-

] g for y 171

- -

(d) f xx2 1

3= -

+

] g for

y 131

- -

(e)3 2

6 y

x=

- for y 1

21

6



Circle gate





Proof

r y

x

( x y)

y

x








c a b

r x y

2 2 2

2 2 2

`

= +

= +

EXAMPLE



Solution


y

x

-2

-2 2

2


one place

y

x

2

2

2

-2

The radius is 4

CONTINUED






Domain 2 2 x x -

Range 2 2 y y -


range y r y r -



Proof


shown

y

x

(a b)

x

y

r

( x y)

a

b x - a

y - b




ndash ndash

c a b

r x a y b

2 2 2

2 2 2

= +

= +] ^g h




EXAMPLES



Solution



y

x 9

9

-9

-9



Domain 9 9 x x -

Range 9 9 y y -



Solution


ndash

ndash ndash

x y

x y

1 2 4

1 2 2

2

2 2

+ + =

+ - =

2

2

] ^

] ]_

g h

g gi

So 1 2a b= = - and 2r =

CONTINUED







y

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

(1 -2)



Domain 1 3 x x -

Range 4 0 y y -


Solution


a b2 1= - = and r 3=

Substituting

ndash ndash

ndash

ndash

x a y b r

x y

x y

2 1 3

2 1 9

2 2 2

2 2 2

2 2

+ =

- - + =

+ + =

] ^]] ^

] ^

g hg g hg h


ndash

ndash

a b a ab b

x x x

x x

a b a ab b

y y y

y y

2

2 2 2 2

4 4

2

1 2 1 1

2 1

So

So

2 2 2

2 2 2

2

2 2 2

2 2 2

2

+ = + +

+ = + +

= + +

= - +

= - +

= - +

]] ] ]]^ ^ ]

gg g ggh h g


ndash

x x y y

x x y y

x x y y

x x y y

4 4 2 1 9

4 2 5 9

4 2 5 9

4 2 4 0

9 9

2

2

2

2

+ + + - + =

+ + - + =

+ + + =

+ + - - =

- -


in Chapter 2




Investigation


two functions





circle






for all values

y

x r

r

-r


Proof

ndash

x y r

y r x y r x

2 2 2

2 2 2

2 2

+ =

=

= -



semi-circles





y 0 for all values

y

x r

-

r

-r


EXAMPLES


1 f x x92

= -] g

Solution



y

x 3

3

-3

Domain 3 3 x x -

Range 0 3 y y




2 y x4 2= - -

Solution



y

x 2

-2

-2

Domain 2 2 x x -

Range 2 0 y y -




range

(a) 9x y 2 2+ =

(b) x y 16 02 2+ =-


(d) 1 9x y 2 2+ + =

] g

(e) ndashx y 2 1 12 2+ + =] ^g h



below the x -axis



range

(a) 25 y x2= - -

(b) 1 y x2= -

(c) 36 y x2= -

(d) 64 y x2= - -

(e) 7 y x2= - -



each circle

(a) 100x y 2 2+ =

(b) 5x y 2 2+ =


(d) ndashx y 5 6 492 2+ + =] ^g h

(e) ndashx y 3 812 2+ =^ h

58 Exercises






grouping symbols)



Centre(c) 1 5-







Centre(j) 4 7- -

^ h and radius 3

Other Graphs




EXAMPLES


Solution



eg When 0x =

y 3

1

c=

=

x

y

1

3

3

1

31

When1

1

= -

=

=

=

-

x 3- 2- 1- 0 1 2 3

y 271

91

31

1 3 9 27



You will meet these

graphs again in the

HSC Course





other point

When

x

y

1

3

3

1

=

=

=

y

x

1

2

1

3







Solution



x minus2 minus1 0 05 1 2 3 4

y minus03 0 03 05 06

y

x

1

2

1 2 3 4

-1




You learned about


stages of maths





range y y 02



range all real y

Cubic function


EXAMPLE



Solution



y minus25 minus6 1 2 3 10 29

y

x

1

1

-2 2 3 4

-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4



If you already

know the shape of


3






Domain and range


EXAMPLE


Solution




function are

f

f

1 1 2

1 21

4 4 2

64 2

66

3

3

- = - +

= - +

=

= +

= +

=

] ]

] ]

g g

g g


these points

y

x

(-1 1)

(4 66)

Range 1 66 y y










positive number

EXAMPLE


Solution


x 4So $

$


(or zero) So y 0$


59 Exercises


(a) 4 3 y x= +

(b) f x 4= -] g

(c) 3x =



(f) f x xx 12 2= - -] g

(g) 64x y 2 2+ =

(h) f t t 4

3=

-] g

(i) ( ) g 2

5zz

= +

(j) | |f x x=] g


(a) y x=

(b) 2 y x= -

(c) | |f x x2 3= -] g

(d) | | 2 y x= -

(e) f x x2 5= - +] g

(f) | | y x5= -

(g) 2 y x=

(h) y 5x= -

(i) f x xx 1

= +] g

(j)2

4 3 y x

x= -


(a) y x x 5 2= -] g


= +-

(d) g x x x164 2= -] g

(e) 49x y 2 2+ =

You may like to


by dividing by x




4 (a) Solve x1 02$-


f x x1 2= -] g


(a) 2 y x x2= - -

(b) g t t t 62= +] g



in each case


x3 3 -


x2 3 -


x2 1 - (d)


x1 5


0 4x


x3 3 -


x1 1 -


x2 3 -


x4 4 -

(j) y x x7 62= - + - in the

domain 0 7x


function y x 1

3=

+





x=] g


function

find its range(b)


plane

(a) f x x4=] g

(b) y x3= -

(c) y x 34= -

(d) 2 p x x3=] g

(e) 1 g x x3= +] g

(f) 100x y 2 2+ =

(g) 2 1 y x= +


y x 1= -









(b) f x x 22= -] g

(c)1

y x=

(d) f x x3=] g

(e) f x 3x=] g

13 (a) Solve x4 02$-


(i) 4 y x2= -

(ii) y x4

2= - -




DID YOU KNOW




a light



Limits






a a1

1

03

Z

=

=

3

3

-


lim a 0x=

x 3-



asymptote

EXAMPLES

1 Find lim x

x x5x 0

2+

Solution


which is undefined


( )

lim lim

lim

xx x

x

x x

x

5 5

5

5

x x

x

0

2

0 1

1

0

+=

+

= +

=

] g




2 Find limx

x

4

22

-

-

x 2

Solution


lim lim

lim

x

x

x x

x

x

4

2

2 2

2

21

41

2 1

1

-

-=

+ -

-

=+

=

x x2 2

x 2

^ _h i

3 Find limh

h x hx h2 72 2+ -

h 0

Solution

lim lim

lim

h

h x hx h

h

h hx x

hx x

x

2 7 2 7

2 7

7

2 2 2

2

2

+ -=

+ -

= + -

= -

h 0

h h0 0

^ h

Continuity





EXAMPLES

1 Find lim

x

x

1

12

-

-

x 1


11

y x

x2

=-

- Sketch the curve

Solution


xx2

-

- gives

00

CONTINUED




( )

lim lim

limx

x

x

x x

x11

1

1 1

1

2

x x

x

1

2

1

1

-

-=

-

+ -

= +

=

-

] ]g g

11

y xx2

=-





y xx

x

x x

x

11

1

1 1

1

2

=-

-

=

+

= +

-

-^ ^h h


2 Find limx

x x2

2x 2

2

+

+ -

-


x x2

22

= + -

+

Solution


x x2

22

+

+ - gives

00

lim lim

lim

xx x

x

x x

x

22

2

1 2

1

3

x x

x

2

2

2

2

+

+ -=

+

- +

=

= -

-

- -

-

^^ ^

^

hh h

h2 y

x

x xx

y x

x

x

x

22

2

2

1

1


=+

+ -= -

=+

= -

+ -^ ^h h


Remember that x 1




1 Find(a) lim x 52

+x 4

(b) lim t 7-t 3 -

(c) lim x x2 43+ -

x 2

(d) lim xx x32

+

x 0

(e) limh

h h

2

22

-

- -

h 2

(f) lim y

y

5

1253

-

-

y 5

(g) limx

x x

12 12

+

+ +

x 1-

(h) limx

x x4

2 82

+

+ -

x 4 -

(i) limc

c

4

22

-

-

c 2

(j) limx x

x 12

-

-

x 1

(k) lim h

h h h2 73 2+ -

h 0

(l) limh

hx hx h32 2- +

h 0

(m) limh

hx h x hx h2 3 53 2 2- + -

h 0

(n) lim x c x c 3 3

-

-

x c



discontinuous

(a) 3 y x2= -

(b)1

1 y

x=

+

(c) f x x 1= -] g

(d)4

1 y

x2=

+

(e)4

1

y x2=

-



(a)3

y xx x2

= +

(b)33

y x

x x2

=+

+

(c)1

5 4 y

xx x2

=+

+ +

510 Exercises




Regions

Class Investigation


them all







EXAMPLE


Solution



y

x =3

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4







region

EXAMPLE


Solution



included


shown

y

x

1

1

-2 2 3 4

-3

-4

-5

2

3

4

5

-1

-2

-3-4

-1 y = -1



y -intercept 1-






EXAMPLES


1 y x 2$ +

Solution






y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0



2 x y 2 3 61-

Solution


region


side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g








3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g







-3

4

3

2

1

5

-2

-1

-4

-5

y

4-2 5321-1-3-4

x



Domain all real x

Range y y 1$ -

2 Sketch | | y x 2= +

Solution


| | ( ) y x x xx x2 2 2 0

2 2 0whenwhen 1

$= + = + +

- + +


2 y x= + when x 2 0$+

x 2$ -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = x + 2






y-intercept of 1-

CONTINUED




2 y x= - +] g when x 2 01+

ie y x 2= - - when x 21 -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2


-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2

y = x + 2



So 0f x =] g

0 | 2 |

0 2

2

x

x

x

= +

= +

- =


(0) | 0 2 |

2

f = +

=

There is only one

solution for the


Can you see why





-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

If you know how to

sketch the graph of




Investigation




are their equations





1 | |f x k x=] g

2 | |f x x k= +] g

3 | |f x x k= +] g







EXAMPLES

Solve

1 |2 1 | 3x - =

Solution



is x 1 2= -

2 |2 1 | 3 2x x= -+

Solution



3 | 1 | 2x 1+

Solution






and then check them







each function

(a) | | y x=

(b) | |f x x 7= +] g

(c) | |f x x 2= -] g

(d) 5 | | y x=

(e) | |f x x 3= - +] g

(f) | 6 | y x= +

(g) | |f x x3 2= -] g

(h) | 5 4 | y x= +

(i) | 7 1 | y x= -

(j) | |f x x2 9= +] g


plane

(a) | | y x=

(b) | |f x x 1= +] g

(c) | |f x x 3= -] g

(d) 2 | | y x=

(e) | |f x x= -] g

(f) | 1 | y x= +

(g) | |f x x 1= - -] g

(h) | 2 3 | y x= -

(i) | 4 2 | y x= +

(j) | |f x x3 1= +] g


each function

(a) | 1 | y x= -

(b) | |f x x 8= -] g

(c) | |f x x2 5= +] g

(d) 2 | | 3 y x= -

(e) | |f x x 3= - -] g


function is

increasing(i)

decreasing(ii)

(a) | 2 | y x= -

(b) | |f x x 2= +

] g

(c) | |f x x2 3= -] g

(d) 4 | | 1 y x= -

(e) | |f x x= -] g


of each function

(a) | | y x= for x2 2 -

(b) | |f x x 4= - -] g for

x4 3 -

(c) | |f x x 4= +] g for x7 2 -

(d) | 2 5 | y x= - for x3 3 -

(e) | |f x x= -] g for x1 1 -


function increasing

(a) | 3 | y x= +

(b) | |f x x 4= - +] g

(c) | |f x x 9= -] g

(d) | | y x 2 1= - -

(e) | |f x x 2= - +] g

56 Exercises




7 Solve graphically

(a) | | 3x =

(b) | |x 12

(c) | |x 2

(d) | 2 | 1x + =

(e) | 3 | 0x- =

(f) |2 3 | 1x - =

(g) | |x 1 41-

(h) | |x 1 3+

(i) | |x 2 22-

(j) | |x 3 1$-

(k) | |x2 3 5+

(l) | |x2 1 1$-

(m) |3 1 | 3x x- = +

(n) |3 2 | 4x x- = -

(o) | 1 | 1x x- = +

(p) | 3 | 2 2x x+ = + (q) |2 1 | 1x x+ = -

(r) |2 5 | 3x x- = -

(s) | 1 | 2x x- =

(t) |2 3 | 3x x- = +

The Hyperbola


or= =

EXAMPLE

Sketch1

y x=

Solution



x -3 -2 -121-

41- 0

41

21 1 2 3

y 3

1-

2

1- -1 -2 -4 mdash 4 2 1

2

1

3

1

Class Discussion



y is never 0 Why






but never touches)



Investigation




equations




1 y xk

=

21

y x k= +

31

y x k

=+


EXAMPLES


3=

-

] g


Solution



So x 3






be zero

CONTINUED







1

y 0 3

3=

-

= -

-3

4

3

2

1

5

-2

-1

-4

-5

y

-1-2 4 521-3-4

x

x = 3

y = 0

Asymptotes

3

2 Sketch y x2 4

1= -

+

Solution




x

x

2 4

2

-

-






( ) y

2 0 41

41

= -+

= -


a translation of3

y x

=





y

-2

x

-

1

4


a=

+


domain x xb



1 For each graph



exists


(a)2

y x=

(b)1

y x= -

(c) f xx 1

1=

+] g

(d) f xx 2

3=

-

] g

(e)3 6

1 y

x=

+

(f) f xx 3

2= -

-

] g

(g) f xx 1

4=

-

] g

(h)1

2 y

x= -

+

(i) f xx6 3

2=

-

] g

(j)2

6 y

x= -

+

2 Show that f x x2

=] g is an odd

function



(a) f xx2 5

1=

+

] g for x2 2 -

(b)3

1 y

x=

+ for x2 0 -

(c) f xx2 4

5=

-

] g for x3 1 -

57 Exercises




(d) f xx 4

3= -

-

] g for x3 3 -

(e)3 1

2 y

x= -

+ for x0 5



(a)3

y x= for y 1 3

(b)2

y x= - for y 221

- -

(c) f xx 1

1=

-

] g for y 171

- -

(d) f xx2 1

3= -

+

] g for

y 131

- -

(e)3 2

6 y

x=

- for y 1

21

6



Circle gate





Proof

r y

x

( x y)

y

x








c a b

r x y

2 2 2

2 2 2

`

= +

= +

EXAMPLE



Solution


y

x

-2

-2 2

2


one place

y

x

2

2

2

-2

The radius is 4

CONTINUED






Domain 2 2 x x -

Range 2 2 y y -


range y r y r -



Proof


shown

y

x

(a b)

x

y

r

( x y)

a

b x - a

y - b




ndash ndash

c a b

r x a y b

2 2 2

2 2 2

= +

= +] ^g h




EXAMPLES



Solution



y

x 9

9

-9

-9



Domain 9 9 x x -

Range 9 9 y y -



Solution


ndash

ndash ndash

x y

x y

1 2 4

1 2 2

2

2 2

+ + =

+ - =

2

2

] ^

] ]_

g h

g gi

So 1 2a b= = - and 2r =

CONTINUED







y

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

(1 -2)



Domain 1 3 x x -

Range 4 0 y y -


Solution


a b2 1= - = and r 3=

Substituting

ndash ndash

ndash

ndash

x a y b r

x y

x y

2 1 3

2 1 9

2 2 2

2 2 2

2 2

+ =

- - + =

+ + =

] ^]] ^

] ^

g hg g hg h


ndash

ndash

a b a ab b

x x x

x x

a b a ab b

y y y

y y

2

2 2 2 2

4 4

2

1 2 1 1

2 1

So

So

2 2 2

2 2 2

2

2 2 2

2 2 2

2

+ = + +

+ = + +

= + +

= - +

= - +

= - +

]] ] ]]^ ^ ]

gg g ggh h g


ndash

x x y y

x x y y

x x y y

x x y y

4 4 2 1 9

4 2 5 9

4 2 5 9

4 2 4 0

9 9

2

2

2

2

+ + + - + =

+ + - + =

+ + + =

+ + - - =

- -


in Chapter 2




Investigation


two functions





circle






for all values

y

x r

r

-r


Proof

ndash

x y r

y r x y r x

2 2 2

2 2 2

2 2

+ =

=

= -



semi-circles





y 0 for all values

y

x r

-

r

-r


EXAMPLES


1 f x x92

= -] g

Solution



y

x 3

3

-3

Domain 3 3 x x -

Range 0 3 y y




2 y x4 2= - -

Solution



y

x 2

-2

-2

Domain 2 2 x x -

Range 2 0 y y -




range

(a) 9x y 2 2+ =

(b) x y 16 02 2+ =-


(d) 1 9x y 2 2+ + =

] g

(e) ndashx y 2 1 12 2+ + =] ^g h



below the x -axis



range

(a) 25 y x2= - -

(b) 1 y x2= -

(c) 36 y x2= -

(d) 64 y x2= - -

(e) 7 y x2= - -



each circle

(a) 100x y 2 2+ =

(b) 5x y 2 2+ =


(d) ndashx y 5 6 492 2+ + =] ^g h

(e) ndashx y 3 812 2+ =^ h

58 Exercises






grouping symbols)



Centre(c) 1 5-







Centre(j) 4 7- -

^ h and radius 3

Other Graphs




EXAMPLES


Solution



eg When 0x =

y 3

1

c=

=

x

y

1

3

3

1

31

When1

1

= -

=

=

=

-

x 3- 2- 1- 0 1 2 3

y 271

91

31

1 3 9 27



You will meet these

graphs again in the

HSC Course





other point

When

x

y

1

3

3

1

=

=

=

y

x

1

2

1

3







Solution



x minus2 minus1 0 05 1 2 3 4

y minus03 0 03 05 06

y

x

1

2

1 2 3 4

-1




You learned about


stages of maths





range y y 02



range all real y

Cubic function


EXAMPLE



Solution



y minus25 minus6 1 2 3 10 29

y

x

1

1

-2 2 3 4

-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4



If you already

know the shape of


3






Domain and range


EXAMPLE


Solution




function are

f

f

1 1 2

1 21

4 4 2

64 2

66

3

3

- = - +

= - +

=

= +

= +

=

] ]

] ]

g g

g g


these points

y

x

(-1 1)

(4 66)

Range 1 66 y y










positive number

EXAMPLE


Solution


x 4So $

$


(or zero) So y 0$


59 Exercises


(a) 4 3 y x= +

(b) f x 4= -] g

(c) 3x =



(f) f x xx 12 2= - -] g

(g) 64x y 2 2+ =

(h) f t t 4

3=

-] g

(i) ( ) g 2

5zz

= +

(j) | |f x x=] g


(a) y x=

(b) 2 y x= -

(c) | |f x x2 3= -] g

(d) | | 2 y x= -

(e) f x x2 5= - +] g

(f) | | y x5= -

(g) 2 y x=

(h) y 5x= -

(i) f x xx 1

= +] g

(j)2

4 3 y x

x= -


(a) y x x 5 2= -] g


= +-

(d) g x x x164 2= -] g

(e) 49x y 2 2+ =

You may like to


by dividing by x




4 (a) Solve x1 02$-


f x x1 2= -] g


(a) 2 y x x2= - -

(b) g t t t 62= +] g



in each case


x3 3 -


x2 3 -


x2 1 - (d)


x1 5


0 4x


x3 3 -


x1 1 -


x2 3 -


x4 4 -

(j) y x x7 62= - + - in the

domain 0 7x


function y x 1

3=

+





x=] g


function

find its range(b)


plane

(a) f x x4=] g

(b) y x3= -

(c) y x 34= -

(d) 2 p x x3=] g

(e) 1 g x x3= +] g

(f) 100x y 2 2+ =

(g) 2 1 y x= +


y x 1= -









(b) f x x 22= -] g

(c)1

y x=

(d) f x x3=] g

(e) f x 3x=] g

13 (a) Solve x4 02$-


(i) 4 y x2= -

(ii) y x4

2= - -




DID YOU KNOW




a light



Limits






a a1

1

03

Z

=

=

3

3

-


lim a 0x=

x 3-



asymptote

EXAMPLES

1 Find lim x

x x5x 0

2+

Solution


which is undefined


( )

lim lim

lim

xx x

x

x x

x

5 5

5

5

x x

x

0

2

0 1

1

0

+=

+

= +

=

] g




2 Find limx

x

4

22

-

-

x 2

Solution


lim lim

lim

x

x

x x

x

x

4

2

2 2

2

21

41

2 1

1

-

-=

+ -

-

=+

=

x x2 2

x 2

^ _h i

3 Find limh

h x hx h2 72 2+ -

h 0

Solution

lim lim

lim

h

h x hx h

h

h hx x

hx x

x

2 7 2 7

2 7

7

2 2 2

2

2

+ -=

+ -

= + -

= -

h 0

h h0 0

^ h

Continuity





EXAMPLES

1 Find lim

x

x

1

12

-

-

x 1


11

y x

x2

=-

- Sketch the curve

Solution


xx2

-

- gives

00

CONTINUED




( )

lim lim

limx

x

x

x x

x11

1

1 1

1

2

x x

x

1

2

1

1

-

-=

-

+ -

= +

=

-

] ]g g

11

y xx2

=-





y xx

x

x x

x

11

1

1 1

1

2

=-

-

=

+

= +

-

-^ ^h h


2 Find limx

x x2

2x 2

2

+

+ -

-


x x2

22

= + -

+

Solution


x x2

22

+

+ - gives

00

lim lim

lim

xx x

x

x x

x

22

2

1 2

1

3

x x

x

2

2

2

2

+

+ -=

+

- +

=

= -

-

- -

-

^^ ^

^

hh h

h2 y

x

x xx

y x

x

x

x

22

2

2

1

1


=+

+ -= -

=+

= -

+ -^ ^h h


Remember that x 1




1 Find(a) lim x 52

+x 4

(b) lim t 7-t 3 -

(c) lim x x2 43+ -

x 2

(d) lim xx x32

+

x 0

(e) limh

h h

2

22

-

- -

h 2

(f) lim y

y

5

1253

-

-

y 5

(g) limx

x x

12 12

+

+ +

x 1-

(h) limx

x x4

2 82

+

+ -

x 4 -

(i) limc

c

4

22

-

-

c 2

(j) limx x

x 12

-

-

x 1

(k) lim h

h h h2 73 2+ -

h 0

(l) limh

hx hx h32 2- +

h 0

(m) limh

hx h x hx h2 3 53 2 2- + -

h 0

(n) lim x c x c 3 3

-

-

x c



discontinuous

(a) 3 y x2= -

(b)1

1 y

x=

+

(c) f x x 1= -] g

(d)4

1 y

x2=

+

(e)4

1

y x2=

-



(a)3

y xx x2

= +

(b)33

y x

x x2

=+

+

(c)1

5 4 y

xx x2

=+

+ +

510 Exercises




Regions

Class Investigation


them all







EXAMPLE


Solution



y

x =3

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4







region

EXAMPLE


Solution



included


shown

y

x

1

1

-2 2 3 4

-3

-4

-5

2

3

4

5

-1

-2

-3-4

-1 y = -1



y -intercept 1-






EXAMPLES


1 y x 2$ +

Solution






y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0



2 x y 2 3 61-

Solution


region


side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g








3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g






2 y x= - +] g when x 2 01+

ie y x 2= - - when x 21 -

-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2


-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

y = - x - 2

y = x + 2



So 0f x =] g

0 | 2 |

0 2

2

x

x

x

= +

= +

- =


(0) | 0 2 |

2

f = +

=

There is only one

solution for the


Can you see why





-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

If you know how to

sketch the graph of




Investigation




are their equations





1 | |f x k x=] g

2 | |f x x k= +] g

3 | |f x x k= +] g







EXAMPLES

Solve

1 |2 1 | 3x - =

Solution



is x 1 2= -

2 |2 1 | 3 2x x= -+

Solution



3 | 1 | 2x 1+

Solution






and then check them







each function

(a) | | y x=

(b) | |f x x 7= +] g

(c) | |f x x 2= -] g

(d) 5 | | y x=

(e) | |f x x 3= - +] g

(f) | 6 | y x= +

(g) | |f x x3 2= -] g

(h) | 5 4 | y x= +

(i) | 7 1 | y x= -

(j) | |f x x2 9= +] g


plane

(a) | | y x=

(b) | |f x x 1= +] g

(c) | |f x x 3= -] g

(d) 2 | | y x=

(e) | |f x x= -] g

(f) | 1 | y x= +

(g) | |f x x 1= - -] g

(h) | 2 3 | y x= -

(i) | 4 2 | y x= +

(j) | |f x x3 1= +] g


each function

(a) | 1 | y x= -

(b) | |f x x 8= -] g

(c) | |f x x2 5= +] g

(d) 2 | | 3 y x= -

(e) | |f x x 3= - -] g


function is

increasing(i)

decreasing(ii)

(a) | 2 | y x= -

(b) | |f x x 2= +

] g

(c) | |f x x2 3= -] g

(d) 4 | | 1 y x= -

(e) | |f x x= -] g


of each function

(a) | | y x= for x2 2 -

(b) | |f x x 4= - -] g for

x4 3 -

(c) | |f x x 4= +] g for x7 2 -

(d) | 2 5 | y x= - for x3 3 -

(e) | |f x x= -] g for x1 1 -


function increasing

(a) | 3 | y x= +

(b) | |f x x 4= - +] g

(c) | |f x x 9= -] g

(d) | | y x 2 1= - -

(e) | |f x x 2= - +] g

56 Exercises




7 Solve graphically

(a) | | 3x =

(b) | |x 12

(c) | |x 2

(d) | 2 | 1x + =

(e) | 3 | 0x- =

(f) |2 3 | 1x - =

(g) | |x 1 41-

(h) | |x 1 3+

(i) | |x 2 22-

(j) | |x 3 1$-

(k) | |x2 3 5+

(l) | |x2 1 1$-

(m) |3 1 | 3x x- = +

(n) |3 2 | 4x x- = -

(o) | 1 | 1x x- = +

(p) | 3 | 2 2x x+ = + (q) |2 1 | 1x x+ = -

(r) |2 5 | 3x x- = -

(s) | 1 | 2x x- =

(t) |2 3 | 3x x- = +

The Hyperbola


or= =

EXAMPLE

Sketch1

y x=

Solution



x -3 -2 -121-

41- 0

41

21 1 2 3

y 3

1-

2

1- -1 -2 -4 mdash 4 2 1

2

1

3

1

Class Discussion



y is never 0 Why






but never touches)



Investigation




equations




1 y xk

=

21

y x k= +

31

y x k

=+


EXAMPLES


3=

-

] g


Solution



So x 3






be zero

CONTINUED







1

y 0 3

3=

-

= -

-3

4

3

2

1

5

-2

-1

-4

-5

y

-1-2 4 521-3-4

x

x = 3

y = 0

Asymptotes

3

2 Sketch y x2 4

1= -

+

Solution




x

x

2 4

2

-

-






( ) y

2 0 41

41

= -+

= -


a translation of3

y x

=





y

-2

x

-

1

4


a=

+


domain x xb



1 For each graph



exists


(a)2

y x=

(b)1

y x= -

(c) f xx 1

1=

+] g

(d) f xx 2

3=

-

] g

(e)3 6

1 y

x=

+

(f) f xx 3

2= -

-

] g

(g) f xx 1

4=

-

] g

(h)1

2 y

x= -

+

(i) f xx6 3

2=

-

] g

(j)2

6 y

x= -

+

2 Show that f x x2

=] g is an odd

function



(a) f xx2 5

1=

+

] g for x2 2 -

(b)3

1 y

x=

+ for x2 0 -

(c) f xx2 4

5=

-

] g for x3 1 -

57 Exercises




(d) f xx 4

3= -

-

] g for x3 3 -

(e)3 1

2 y

x= -

+ for x0 5



(a)3

y x= for y 1 3

(b)2

y x= - for y 221

- -

(c) f xx 1

1=

-

] g for y 171

- -

(d) f xx2 1

3= -

+

] g for

y 131

- -

(e)3 2

6 y

x=

- for y 1

21

6



Circle gate





Proof

r y

x

( x y)

y

x








c a b

r x y

2 2 2

2 2 2

`

= +

= +

EXAMPLE



Solution


y

x

-2

-2 2

2


one place

y

x

2

2

2

-2

The radius is 4

CONTINUED






Domain 2 2 x x -

Range 2 2 y y -


range y r y r -



Proof


shown

y

x

(a b)

x

y

r

( x y)

a

b x - a

y - b




ndash ndash

c a b

r x a y b

2 2 2

2 2 2

= +

= +] ^g h




EXAMPLES



Solution



y

x 9

9

-9

-9



Domain 9 9 x x -

Range 9 9 y y -



Solution


ndash

ndash ndash

x y

x y

1 2 4

1 2 2

2

2 2

+ + =

+ - =

2

2

] ^

] ]_

g h

g gi

So 1 2a b= = - and 2r =

CONTINUED







y

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

(1 -2)



Domain 1 3 x x -

Range 4 0 y y -


Solution


a b2 1= - = and r 3=

Substituting

ndash ndash

ndash

ndash

x a y b r

x y

x y

2 1 3

2 1 9

2 2 2

2 2 2

2 2

+ =

- - + =

+ + =

] ^]] ^

] ^

g hg g hg h


ndash

ndash

a b a ab b

x x x

x x

a b a ab b

y y y

y y

2

2 2 2 2

4 4

2

1 2 1 1

2 1

So

So

2 2 2

2 2 2

2

2 2 2

2 2 2

2

+ = + +

+ = + +

= + +

= - +

= - +

= - +

]] ] ]]^ ^ ]

gg g ggh h g


ndash

x x y y

x x y y

x x y y

x x y y

4 4 2 1 9

4 2 5 9

4 2 5 9

4 2 4 0

9 9

2

2

2

2

+ + + - + =

+ + - + =

+ + + =

+ + - - =

- -


in Chapter 2




Investigation


two functions





circle






for all values

y

x r

r

-r


Proof

ndash

x y r

y r x y r x

2 2 2

2 2 2

2 2

+ =

=

= -



semi-circles





y 0 for all values

y

x r

-

r

-r


EXAMPLES


1 f x x92

= -] g

Solution



y

x 3

3

-3

Domain 3 3 x x -

Range 0 3 y y




2 y x4 2= - -

Solution



y

x 2

-2

-2

Domain 2 2 x x -

Range 2 0 y y -




range

(a) 9x y 2 2+ =

(b) x y 16 02 2+ =-


(d) 1 9x y 2 2+ + =

] g

(e) ndashx y 2 1 12 2+ + =] ^g h



below the x -axis



range

(a) 25 y x2= - -

(b) 1 y x2= -

(c) 36 y x2= -

(d) 64 y x2= - -

(e) 7 y x2= - -



each circle

(a) 100x y 2 2+ =

(b) 5x y 2 2+ =


(d) ndashx y 5 6 492 2+ + =] ^g h

(e) ndashx y 3 812 2+ =^ h

58 Exercises






grouping symbols)



Centre(c) 1 5-







Centre(j) 4 7- -

^ h and radius 3

Other Graphs




EXAMPLES


Solution



eg When 0x =

y 3

1

c=

=

x

y

1

3

3

1

31

When1

1

= -

=

=

=

-

x 3- 2- 1- 0 1 2 3

y 271

91

31

1 3 9 27



You will meet these

graphs again in the

HSC Course





other point

When

x

y

1

3

3

1

=

=

=

y

x

1

2

1

3







Solution



x minus2 minus1 0 05 1 2 3 4

y minus03 0 03 05 06

y

x

1

2

1 2 3 4

-1




You learned about


stages of maths





range y y 02



range all real y

Cubic function


EXAMPLE



Solution



y minus25 minus6 1 2 3 10 29

y

x

1

1

-2 2 3 4

-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4



If you already

know the shape of


3






Domain and range


EXAMPLE


Solution




function are

f

f

1 1 2

1 21

4 4 2

64 2

66

3

3

- = - +

= - +

=

= +

= +

=

] ]

] ]

g g

g g


these points

y

x

(-1 1)

(4 66)

Range 1 66 y y










positive number

EXAMPLE


Solution


x 4So $

$


(or zero) So y 0$


59 Exercises


(a) 4 3 y x= +

(b) f x 4= -] g

(c) 3x =



(f) f x xx 12 2= - -] g

(g) 64x y 2 2+ =

(h) f t t 4

3=

-] g

(i) ( ) g 2

5zz

= +

(j) | |f x x=] g


(a) y x=

(b) 2 y x= -

(c) | |f x x2 3= -] g

(d) | | 2 y x= -

(e) f x x2 5= - +] g

(f) | | y x5= -

(g) 2 y x=

(h) y 5x= -

(i) f x xx 1

= +] g

(j)2

4 3 y x

x= -


(a) y x x 5 2= -] g


= +-

(d) g x x x164 2= -] g

(e) 49x y 2 2+ =

You may like to


by dividing by x




4 (a) Solve x1 02$-


f x x1 2= -] g


(a) 2 y x x2= - -

(b) g t t t 62= +] g



in each case


x3 3 -


x2 3 -


x2 1 - (d)


x1 5


0 4x


x3 3 -


x1 1 -


x2 3 -


x4 4 -

(j) y x x7 62= - + - in the

domain 0 7x


function y x 1

3=

+





x=] g


function

find its range(b)


plane

(a) f x x4=] g

(b) y x3= -

(c) y x 34= -

(d) 2 p x x3=] g

(e) 1 g x x3= +] g

(f) 100x y 2 2+ =

(g) 2 1 y x= +


y x 1= -









(b) f x x 22= -] g

(c)1

y x=

(d) f x x3=] g

(e) f x 3x=] g

13 (a) Solve x4 02$-


(i) 4 y x2= -

(ii) y x4

2= - -




DID YOU KNOW




a light



Limits






a a1

1

03

Z

=

=

3

3

-


lim a 0x=

x 3-



asymptote

EXAMPLES

1 Find lim x

x x5x 0

2+

Solution


which is undefined


( )

lim lim

lim

xx x

x

x x

x

5 5

5

5

x x

x

0

2

0 1

1

0

+=

+

= +

=

] g




2 Find limx

x

4

22

-

-

x 2

Solution


lim lim

lim

x

x

x x

x

x

4

2

2 2

2

21

41

2 1

1

-

-=

+ -

-

=+

=

x x2 2

x 2

^ _h i

3 Find limh

h x hx h2 72 2+ -

h 0

Solution

lim lim

lim

h

h x hx h

h

h hx x

hx x

x

2 7 2 7

2 7

7

2 2 2

2

2

+ -=

+ -

= + -

= -

h 0

h h0 0

^ h

Continuity





EXAMPLES

1 Find lim

x

x

1

12

-

-

x 1


11

y x

x2

=-

- Sketch the curve

Solution


xx2

-

- gives

00

CONTINUED




( )

lim lim

limx

x

x

x x

x11

1

1 1

1

2

x x

x

1

2

1

1

-

-=

-

+ -

= +

=

-

] ]g g

11

y xx2

=-





y xx

x

x x

x

11

1

1 1

1

2

=-

-

=

+

= +

-

-^ ^h h


2 Find limx

x x2

2x 2

2

+

+ -

-


x x2

22

= + -

+

Solution


x x2

22

+

+ - gives

00

lim lim

lim

xx x

x

x x

x

22

2

1 2

1

3

x x

x

2

2

2

2

+

+ -=

+

- +

=

= -

-

- -

-

^^ ^

^

hh h

h2 y

x

x xx

y x

x

x

x

22

2

2

1

1


=+

+ -= -

=+

= -

+ -^ ^h h


Remember that x 1




1 Find(a) lim x 52

+x 4

(b) lim t 7-t 3 -

(c) lim x x2 43+ -

x 2

(d) lim xx x32

+

x 0

(e) limh

h h

2

22

-

- -

h 2

(f) lim y

y

5

1253

-

-

y 5

(g) limx

x x

12 12

+

+ +

x 1-

(h) limx

x x4

2 82

+

+ -

x 4 -

(i) limc

c

4

22

-

-

c 2

(j) limx x

x 12

-

-

x 1

(k) lim h

h h h2 73 2+ -

h 0

(l) limh

hx hx h32 2- +

h 0

(m) limh

hx h x hx h2 3 53 2 2- + -

h 0

(n) lim x c x c 3 3

-

-

x c



discontinuous

(a) 3 y x2= -

(b)1

1 y

x=

+

(c) f x x 1= -] g

(d)4

1 y

x2=

+

(e)4

1

y x2=

-



(a)3

y xx x2

= +

(b)33

y x

x x2

=+

+

(c)1

5 4 y

xx x2

=+

+ +

510 Exercises




Regions

Class Investigation


them all







EXAMPLE


Solution



y

x =3

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4







region

EXAMPLE


Solution



included


shown

y

x

1

1

-2 2 3 4

-3

-4

-5

2

3

4

5

-1

-2

-3-4

-1 y = -1



y -intercept 1-






EXAMPLES


1 y x 2$ +

Solution






y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0



2 x y 2 3 61-

Solution


region


side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g








3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g







-3

4

3

2

1

5

-2

-1

-4

-5

y

3-1-2 421-3-4

x

If you know how to

sketch the graph of




Investigation




are their equations





1 | |f x k x=] g

2 | |f x x k= +] g

3 | |f x x k= +] g







EXAMPLES

Solve

1 |2 1 | 3x - =

Solution



is x 1 2= -

2 |2 1 | 3 2x x= -+

Solution



3 | 1 | 2x 1+

Solution






and then check them







each function

(a) | | y x=

(b) | |f x x 7= +] g

(c) | |f x x 2= -] g

(d) 5 | | y x=

(e) | |f x x 3= - +] g

(f) | 6 | y x= +

(g) | |f x x3 2= -] g

(h) | 5 4 | y x= +

(i) | 7 1 | y x= -

(j) | |f x x2 9= +] g


plane

(a) | | y x=

(b) | |f x x 1= +] g

(c) | |f x x 3= -] g

(d) 2 | | y x=

(e) | |f x x= -] g

(f) | 1 | y x= +

(g) | |f x x 1= - -] g

(h) | 2 3 | y x= -

(i) | 4 2 | y x= +

(j) | |f x x3 1= +] g


each function

(a) | 1 | y x= -

(b) | |f x x 8= -] g

(c) | |f x x2 5= +] g

(d) 2 | | 3 y x= -

(e) | |f x x 3= - -] g


function is

increasing(i)

decreasing(ii)

(a) | 2 | y x= -

(b) | |f x x 2= +

] g

(c) | |f x x2 3= -] g

(d) 4 | | 1 y x= -

(e) | |f x x= -] g


of each function

(a) | | y x= for x2 2 -

(b) | |f x x 4= - -] g for

x4 3 -

(c) | |f x x 4= +] g for x7 2 -

(d) | 2 5 | y x= - for x3 3 -

(e) | |f x x= -] g for x1 1 -


function increasing

(a) | 3 | y x= +

(b) | |f x x 4= - +] g

(c) | |f x x 9= -] g

(d) | | y x 2 1= - -

(e) | |f x x 2= - +] g

56 Exercises




7 Solve graphically

(a) | | 3x =

(b) | |x 12

(c) | |x 2

(d) | 2 | 1x + =

(e) | 3 | 0x- =

(f) |2 3 | 1x - =

(g) | |x 1 41-

(h) | |x 1 3+

(i) | |x 2 22-

(j) | |x 3 1$-

(k) | |x2 3 5+

(l) | |x2 1 1$-

(m) |3 1 | 3x x- = +

(n) |3 2 | 4x x- = -

(o) | 1 | 1x x- = +

(p) | 3 | 2 2x x+ = + (q) |2 1 | 1x x+ = -

(r) |2 5 | 3x x- = -

(s) | 1 | 2x x- =

(t) |2 3 | 3x x- = +

The Hyperbola


or= =

EXAMPLE

Sketch1

y x=

Solution



x -3 -2 -121-

41- 0

41

21 1 2 3

y 3

1-

2

1- -1 -2 -4 mdash 4 2 1

2

1

3

1

Class Discussion



y is never 0 Why






but never touches)



Investigation




equations




1 y xk

=

21

y x k= +

31

y x k

=+


EXAMPLES


3=

-

] g


Solution



So x 3






be zero

CONTINUED







1

y 0 3

3=

-

= -

-3

4

3

2

1

5

-2

-1

-4

-5

y

-1-2 4 521-3-4

x

x = 3

y = 0

Asymptotes

3

2 Sketch y x2 4

1= -

+

Solution




x

x

2 4

2

-

-






( ) y

2 0 41

41

= -+

= -


a translation of3

y x

=





y

-2

x

-

1

4


a=

+


domain x xb



1 For each graph



exists


(a)2

y x=

(b)1

y x= -

(c) f xx 1

1=

+] g

(d) f xx 2

3=

-

] g

(e)3 6

1 y

x=

+

(f) f xx 3

2= -

-

] g

(g) f xx 1

4=

-

] g

(h)1

2 y

x= -

+

(i) f xx6 3

2=

-

] g

(j)2

6 y

x= -

+

2 Show that f x x2

=] g is an odd

function



(a) f xx2 5

1=

+

] g for x2 2 -

(b)3

1 y

x=

+ for x2 0 -

(c) f xx2 4

5=

-

] g for x3 1 -

57 Exercises




(d) f xx 4

3= -

-

] g for x3 3 -

(e)3 1

2 y

x= -

+ for x0 5



(a)3

y x= for y 1 3

(b)2

y x= - for y 221

- -

(c) f xx 1

1=

-

] g for y 171

- -

(d) f xx2 1

3= -

+

] g for

y 131

- -

(e)3 2

6 y

x=

- for y 1

21

6



Circle gate





Proof

r y

x

( x y)

y

x








c a b

r x y

2 2 2

2 2 2

`

= +

= +

EXAMPLE



Solution


y

x

-2

-2 2

2


one place

y

x

2

2

2

-2

The radius is 4

CONTINUED






Domain 2 2 x x -

Range 2 2 y y -


range y r y r -



Proof


shown

y

x

(a b)

x

y

r

( x y)

a

b x - a

y - b




ndash ndash

c a b

r x a y b

2 2 2

2 2 2

= +

= +] ^g h




EXAMPLES



Solution



y

x 9

9

-9

-9



Domain 9 9 x x -

Range 9 9 y y -



Solution


ndash

ndash ndash

x y

x y

1 2 4

1 2 2

2

2 2

+ + =

+ - =

2

2

] ^

] ]_

g h

g gi

So 1 2a b= = - and 2r =

CONTINUED







y

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

(1 -2)



Domain 1 3 x x -

Range 4 0 y y -


Solution


a b2 1= - = and r 3=

Substituting

ndash ndash

ndash

ndash

x a y b r

x y

x y

2 1 3

2 1 9

2 2 2

2 2 2

2 2

+ =

- - + =

+ + =

] ^]] ^

] ^

g hg g hg h


ndash

ndash

a b a ab b

x x x

x x

a b a ab b

y y y

y y

2

2 2 2 2

4 4

2

1 2 1 1

2 1

So

So

2 2 2

2 2 2

2

2 2 2

2 2 2

2

+ = + +

+ = + +

= + +

= - +

= - +

= - +

]] ] ]]^ ^ ]

gg g ggh h g


ndash

x x y y

x x y y

x x y y

x x y y

4 4 2 1 9

4 2 5 9

4 2 5 9

4 2 4 0

9 9

2

2

2

2

+ + + - + =

+ + - + =

+ + + =

+ + - - =

- -


in Chapter 2




Investigation


two functions





circle






for all values

y

x r

r

-r


Proof

ndash

x y r

y r x y r x

2 2 2

2 2 2

2 2

+ =

=

= -



semi-circles





y 0 for all values

y

x r

-

r

-r


EXAMPLES


1 f x x92

= -] g

Solution



y

x 3

3

-3

Domain 3 3 x x -

Range 0 3 y y




2 y x4 2= - -

Solution



y

x 2

-2

-2

Domain 2 2 x x -

Range 2 0 y y -




range

(a) 9x y 2 2+ =

(b) x y 16 02 2+ =-


(d) 1 9x y 2 2+ + =

] g

(e) ndashx y 2 1 12 2+ + =] ^g h



below the x -axis



range

(a) 25 y x2= - -

(b) 1 y x2= -

(c) 36 y x2= -

(d) 64 y x2= - -

(e) 7 y x2= - -



each circle

(a) 100x y 2 2+ =

(b) 5x y 2 2+ =


(d) ndashx y 5 6 492 2+ + =] ^g h

(e) ndashx y 3 812 2+ =^ h

58 Exercises






grouping symbols)



Centre(c) 1 5-







Centre(j) 4 7- -

^ h and radius 3

Other Graphs




EXAMPLES


Solution



eg When 0x =

y 3

1

c=

=

x

y

1

3

3

1

31

When1

1

= -

=

=

=

-

x 3- 2- 1- 0 1 2 3

y 271

91

31

1 3 9 27



You will meet these

graphs again in the

HSC Course





other point

When

x

y

1

3

3

1

=

=

=

y

x

1

2

1

3







Solution



x minus2 minus1 0 05 1 2 3 4

y minus03 0 03 05 06

y

x

1

2

1 2 3 4

-1




You learned about


stages of maths





range y y 02



range all real y

Cubic function


EXAMPLE



Solution



y minus25 minus6 1 2 3 10 29

y

x

1

1

-2 2 3 4

-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4



If you already

know the shape of


3






Domain and range


EXAMPLE


Solution




function are

f

f

1 1 2

1 21

4 4 2

64 2

66

3

3

- = - +

= - +

=

= +

= +

=

] ]

] ]

g g

g g


these points

y

x

(-1 1)

(4 66)

Range 1 66 y y










positive number

EXAMPLE


Solution


x 4So $

$


(or zero) So y 0$


59 Exercises


(a) 4 3 y x= +

(b) f x 4= -] g

(c) 3x =



(f) f x xx 12 2= - -] g

(g) 64x y 2 2+ =

(h) f t t 4

3=

-] g

(i) ( ) g 2

5zz

= +

(j) | |f x x=] g


(a) y x=

(b) 2 y x= -

(c) | |f x x2 3= -] g

(d) | | 2 y x= -

(e) f x x2 5= - +] g

(f) | | y x5= -

(g) 2 y x=

(h) y 5x= -

(i) f x xx 1

= +] g

(j)2

4 3 y x

x= -


(a) y x x 5 2= -] g


= +-

(d) g x x x164 2= -] g

(e) 49x y 2 2+ =

You may like to


by dividing by x




4 (a) Solve x1 02$-


f x x1 2= -] g


(a) 2 y x x2= - -

(b) g t t t 62= +] g



in each case


x3 3 -


x2 3 -


x2 1 - (d)


x1 5


0 4x


x3 3 -


x1 1 -


x2 3 -


x4 4 -

(j) y x x7 62= - + - in the

domain 0 7x


function y x 1

3=

+





x=] g


function

find its range(b)


plane

(a) f x x4=] g

(b) y x3= -

(c) y x 34= -

(d) 2 p x x3=] g

(e) 1 g x x3= +] g

(f) 100x y 2 2+ =

(g) 2 1 y x= +


y x 1= -









(b) f x x 22= -] g

(c)1

y x=

(d) f x x3=] g

(e) f x 3x=] g

13 (a) Solve x4 02$-


(i) 4 y x2= -

(ii) y x4

2= - -




DID YOU KNOW




a light



Limits






a a1

1

03

Z

=

=

3

3

-


lim a 0x=

x 3-



asymptote

EXAMPLES

1 Find lim x

x x5x 0

2+

Solution


which is undefined


( )

lim lim

lim

xx x

x

x x

x

5 5

5

5

x x

x

0

2

0 1

1

0

+=

+

= +

=

] g




2 Find limx

x

4

22

-

-

x 2

Solution


lim lim

lim

x

x

x x

x

x

4

2

2 2

2

21

41

2 1

1

-

-=

+ -

-

=+

=

x x2 2

x 2

^ _h i

3 Find limh

h x hx h2 72 2+ -

h 0

Solution

lim lim

lim

h

h x hx h

h

h hx x

hx x

x

2 7 2 7

2 7

7

2 2 2

2

2

+ -=

+ -

= + -

= -

h 0

h h0 0

^ h

Continuity





EXAMPLES

1 Find lim

x

x

1

12

-

-

x 1


11

y x

x2

=-

- Sketch the curve

Solution


xx2

-

- gives

00

CONTINUED




( )

lim lim

limx

x

x

x x

x11

1

1 1

1

2

x x

x

1

2

1

1

-

-=

-

+ -

= +

=

-

] ]g g

11

y xx2

=-





y xx

x

x x

x

11

1

1 1

1

2

=-

-

=

+

= +

-

-^ ^h h


2 Find limx

x x2

2x 2

2

+

+ -

-


x x2

22

= + -

+

Solution


x x2

22

+

+ - gives

00

lim lim

lim

xx x

x

x x

x

22

2

1 2

1

3

x x

x

2

2

2

2

+

+ -=

+

- +

=

= -

-

- -

-

^^ ^

^

hh h

h2 y

x

x xx

y x

x

x

x

22

2

2

1

1


=+

+ -= -

=+

= -

+ -^ ^h h


Remember that x 1




1 Find(a) lim x 52

+x 4

(b) lim t 7-t 3 -

(c) lim x x2 43+ -

x 2

(d) lim xx x32

+

x 0

(e) limh

h h

2

22

-

- -

h 2

(f) lim y

y

5

1253

-

-

y 5

(g) limx

x x

12 12

+

+ +

x 1-

(h) limx

x x4

2 82

+

+ -

x 4 -

(i) limc

c

4

22

-

-

c 2

(j) limx x

x 12

-

-

x 1

(k) lim h

h h h2 73 2+ -

h 0

(l) limh

hx hx h32 2- +

h 0

(m) limh

hx h x hx h2 3 53 2 2- + -

h 0

(n) lim x c x c 3 3

-

-

x c



discontinuous

(a) 3 y x2= -

(b)1

1 y

x=

+

(c) f x x 1= -] g

(d)4

1 y

x2=

+

(e)4

1

y x2=

-



(a)3

y xx x2

= +

(b)33

y x

x x2

=+

+

(c)1

5 4 y

xx x2

=+

+ +

510 Exercises




Regions

Class Investigation


them all







EXAMPLE


Solution



y

x =3

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4







region

EXAMPLE


Solution



included


shown

y

x

1

1

-2 2 3 4

-3

-4

-5

2

3

4

5

-1

-2

-3-4

-1 y = -1



y -intercept 1-






EXAMPLES


1 y x 2$ +

Solution






y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0



2 x y 2 3 61-

Solution


region


side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g








3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g






EXAMPLES

Solve

1 |2 1 | 3x - =

Solution



is x 1 2= -

2 |2 1 | 3 2x x= -+

Solution



3 | 1 | 2x 1+

Solution






and then check them







each function

(a) | | y x=

(b) | |f x x 7= +] g

(c) | |f x x 2= -] g

(d) 5 | | y x=

(e) | |f x x 3= - +] g

(f) | 6 | y x= +

(g) | |f x x3 2= -] g

(h) | 5 4 | y x= +

(i) | 7 1 | y x= -

(j) | |f x x2 9= +] g


plane

(a) | | y x=

(b) | |f x x 1= +] g

(c) | |f x x 3= -] g

(d) 2 | | y x=

(e) | |f x x= -] g

(f) | 1 | y x= +

(g) | |f x x 1= - -] g

(h) | 2 3 | y x= -

(i) | 4 2 | y x= +

(j) | |f x x3 1= +] g


each function

(a) | 1 | y x= -

(b) | |f x x 8= -] g

(c) | |f x x2 5= +] g

(d) 2 | | 3 y x= -

(e) | |f x x 3= - -] g


function is

increasing(i)

decreasing(ii)

(a) | 2 | y x= -

(b) | |f x x 2= +

] g

(c) | |f x x2 3= -] g

(d) 4 | | 1 y x= -

(e) | |f x x= -] g


of each function

(a) | | y x= for x2 2 -

(b) | |f x x 4= - -] g for

x4 3 -

(c) | |f x x 4= +] g for x7 2 -

(d) | 2 5 | y x= - for x3 3 -

(e) | |f x x= -] g for x1 1 -


function increasing

(a) | 3 | y x= +

(b) | |f x x 4= - +] g

(c) | |f x x 9= -] g

(d) | | y x 2 1= - -

(e) | |f x x 2= - +] g

56 Exercises




7 Solve graphically

(a) | | 3x =

(b) | |x 12

(c) | |x 2

(d) | 2 | 1x + =

(e) | 3 | 0x- =

(f) |2 3 | 1x - =

(g) | |x 1 41-

(h) | |x 1 3+

(i) | |x 2 22-

(j) | |x 3 1$-

(k) | |x2 3 5+

(l) | |x2 1 1$-

(m) |3 1 | 3x x- = +

(n) |3 2 | 4x x- = -

(o) | 1 | 1x x- = +

(p) | 3 | 2 2x x+ = + (q) |2 1 | 1x x+ = -

(r) |2 5 | 3x x- = -

(s) | 1 | 2x x- =

(t) |2 3 | 3x x- = +

The Hyperbola


or= =

EXAMPLE

Sketch1

y x=

Solution



x -3 -2 -121-

41- 0

41

21 1 2 3

y 3

1-

2

1- -1 -2 -4 mdash 4 2 1

2

1

3

1

Class Discussion



y is never 0 Why






but never touches)



Investigation




equations




1 y xk

=

21

y x k= +

31

y x k

=+


EXAMPLES


3=

-

] g


Solution



So x 3






be zero

CONTINUED







1

y 0 3

3=

-

= -

-3

4

3

2

1

5

-2

-1

-4

-5

y

-1-2 4 521-3-4

x

x = 3

y = 0

Asymptotes

3

2 Sketch y x2 4

1= -

+

Solution




x

x

2 4

2

-

-






( ) y

2 0 41

41

= -+

= -


a translation of3

y x

=





y

-2

x

-

1

4


a=

+


domain x xb



1 For each graph



exists


(a)2

y x=

(b)1

y x= -

(c) f xx 1

1=

+] g

(d) f xx 2

3=

-

] g

(e)3 6

1 y

x=

+

(f) f xx 3

2= -

-

] g

(g) f xx 1

4=

-

] g

(h)1

2 y

x= -

+

(i) f xx6 3

2=

-

] g

(j)2

6 y

x= -

+

2 Show that f x x2

=] g is an odd

function



(a) f xx2 5

1=

+

] g for x2 2 -

(b)3

1 y

x=

+ for x2 0 -

(c) f xx2 4

5=

-

] g for x3 1 -

57 Exercises




(d) f xx 4

3= -

-

] g for x3 3 -

(e)3 1

2 y

x= -

+ for x0 5



(a)3

y x= for y 1 3

(b)2

y x= - for y 221

- -

(c) f xx 1

1=

-

] g for y 171

- -

(d) f xx2 1

3= -

+

] g for

y 131

- -

(e)3 2

6 y

x=

- for y 1

21

6



Circle gate





Proof

r y

x

( x y)

y

x








c a b

r x y

2 2 2

2 2 2

`

= +

= +

EXAMPLE



Solution


y

x

-2

-2 2

2


one place

y

x

2

2

2

-2

The radius is 4

CONTINUED






Domain 2 2 x x -

Range 2 2 y y -


range y r y r -



Proof


shown

y

x

(a b)

x

y

r

( x y)

a

b x - a

y - b




ndash ndash

c a b

r x a y b

2 2 2

2 2 2

= +

= +] ^g h




EXAMPLES



Solution



y

x 9

9

-9

-9



Domain 9 9 x x -

Range 9 9 y y -



Solution


ndash

ndash ndash

x y

x y

1 2 4

1 2 2

2

2 2

+ + =

+ - =

2

2

] ^

] ]_

g h

g gi

So 1 2a b= = - and 2r =

CONTINUED







y

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

(1 -2)



Domain 1 3 x x -

Range 4 0 y y -


Solution


a b2 1= - = and r 3=

Substituting

ndash ndash

ndash

ndash

x a y b r

x y

x y

2 1 3

2 1 9

2 2 2

2 2 2

2 2

+ =

- - + =

+ + =

] ^]] ^

] ^

g hg g hg h


ndash

ndash

a b a ab b

x x x

x x

a b a ab b

y y y

y y

2

2 2 2 2

4 4

2

1 2 1 1

2 1

So

So

2 2 2

2 2 2

2

2 2 2

2 2 2

2

+ = + +

+ = + +

= + +

= - +

= - +

= - +

]] ] ]]^ ^ ]

gg g ggh h g


ndash

x x y y

x x y y

x x y y

x x y y

4 4 2 1 9

4 2 5 9

4 2 5 9

4 2 4 0

9 9

2

2

2

2

+ + + - + =

+ + - + =

+ + + =

+ + - - =

- -


in Chapter 2




Investigation


two functions





circle






for all values

y

x r

r

-r


Proof

ndash

x y r

y r x y r x

2 2 2

2 2 2

2 2

+ =

=

= -



semi-circles





y 0 for all values

y

x r

-

r

-r


EXAMPLES


1 f x x92

= -] g

Solution



y

x 3

3

-3

Domain 3 3 x x -

Range 0 3 y y




2 y x4 2= - -

Solution



y

x 2

-2

-2

Domain 2 2 x x -

Range 2 0 y y -




range

(a) 9x y 2 2+ =

(b) x y 16 02 2+ =-


(d) 1 9x y 2 2+ + =

] g

(e) ndashx y 2 1 12 2+ + =] ^g h



below the x -axis



range

(a) 25 y x2= - -

(b) 1 y x2= -

(c) 36 y x2= -

(d) 64 y x2= - -

(e) 7 y x2= - -



each circle

(a) 100x y 2 2+ =

(b) 5x y 2 2+ =


(d) ndashx y 5 6 492 2+ + =] ^g h

(e) ndashx y 3 812 2+ =^ h

58 Exercises






grouping symbols)



Centre(c) 1 5-







Centre(j) 4 7- -

^ h and radius 3

Other Graphs




EXAMPLES


Solution



eg When 0x =

y 3

1

c=

=

x

y

1

3

3

1

31

When1

1

= -

=

=

=

-

x 3- 2- 1- 0 1 2 3

y 271

91

31

1 3 9 27



You will meet these

graphs again in the

HSC Course





other point

When

x

y

1

3

3

1

=

=

=

y

x

1

2

1

3







Solution



x minus2 minus1 0 05 1 2 3 4

y minus03 0 03 05 06

y

x

1

2

1 2 3 4

-1




You learned about


stages of maths





range y y 02



range all real y

Cubic function


EXAMPLE



Solution



y minus25 minus6 1 2 3 10 29

y

x

1

1

-2 2 3 4

-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4



If you already

know the shape of


3






Domain and range


EXAMPLE


Solution




function are

f

f

1 1 2

1 21

4 4 2

64 2

66

3

3

- = - +

= - +

=

= +

= +

=

] ]

] ]

g g

g g


these points

y

x

(-1 1)

(4 66)

Range 1 66 y y










positive number

EXAMPLE


Solution


x 4So $

$


(or zero) So y 0$


59 Exercises


(a) 4 3 y x= +

(b) f x 4= -] g

(c) 3x =



(f) f x xx 12 2= - -] g

(g) 64x y 2 2+ =

(h) f t t 4

3=

-] g

(i) ( ) g 2

5zz

= +

(j) | |f x x=] g


(a) y x=

(b) 2 y x= -

(c) | |f x x2 3= -] g

(d) | | 2 y x= -

(e) f x x2 5= - +] g

(f) | | y x5= -

(g) 2 y x=

(h) y 5x= -

(i) f x xx 1

= +] g

(j)2

4 3 y x

x= -


(a) y x x 5 2= -] g


= +-

(d) g x x x164 2= -] g

(e) 49x y 2 2+ =

You may like to


by dividing by x




4 (a) Solve x1 02$-


f x x1 2= -] g


(a) 2 y x x2= - -

(b) g t t t 62= +] g



in each case


x3 3 -


x2 3 -


x2 1 - (d)


x1 5


0 4x


x3 3 -


x1 1 -


x2 3 -


x4 4 -

(j) y x x7 62= - + - in the

domain 0 7x


function y x 1

3=

+





x=] g


function

find its range(b)


plane

(a) f x x4=] g

(b) y x3= -

(c) y x 34= -

(d) 2 p x x3=] g

(e) 1 g x x3= +] g

(f) 100x y 2 2+ =

(g) 2 1 y x= +


y x 1= -









(b) f x x 22= -] g

(c)1

y x=

(d) f x x3=] g

(e) f x 3x=] g

13 (a) Solve x4 02$-


(i) 4 y x2= -

(ii) y x4

2= - -




DID YOU KNOW




a light



Limits






a a1

1

03

Z

=

=

3

3

-


lim a 0x=

x 3-



asymptote

EXAMPLES

1 Find lim x

x x5x 0

2+

Solution


which is undefined


( )

lim lim

lim

xx x

x

x x

x

5 5

5

5

x x

x

0

2

0 1

1

0

+=

+

= +

=

] g




2 Find limx

x

4

22

-

-

x 2

Solution


lim lim

lim

x

x

x x

x

x

4

2

2 2

2

21

41

2 1

1

-

-=

+ -

-

=+

=

x x2 2

x 2

^ _h i

3 Find limh

h x hx h2 72 2+ -

h 0

Solution

lim lim

lim

h

h x hx h

h

h hx x

hx x

x

2 7 2 7

2 7

7

2 2 2

2

2

+ -=

+ -

= + -

= -

h 0

h h0 0

^ h

Continuity





EXAMPLES

1 Find lim

x

x

1

12

-

-

x 1


11

y x

x2

=-

- Sketch the curve

Solution


xx2

-

- gives

00

CONTINUED




( )

lim lim

limx

x

x

x x

x11

1

1 1

1

2

x x

x

1

2

1

1

-

-=

-

+ -

= +

=

-

] ]g g

11

y xx2

=-





y xx

x

x x

x

11

1

1 1

1

2

=-

-

=

+

= +

-

-^ ^h h


2 Find limx

x x2

2x 2

2

+

+ -

-


x x2

22

= + -

+

Solution


x x2

22

+

+ - gives

00

lim lim

lim

xx x

x

x x

x

22

2

1 2

1

3

x x

x

2

2

2

2

+

+ -=

+

- +

=

= -

-

- -

-

^^ ^

^

hh h

h2 y

x

x xx

y x

x

x

x

22

2

2

1

1


=+

+ -= -

=+

= -

+ -^ ^h h


Remember that x 1




1 Find(a) lim x 52

+x 4

(b) lim t 7-t 3 -

(c) lim x x2 43+ -

x 2

(d) lim xx x32

+

x 0

(e) limh

h h

2

22

-

- -

h 2

(f) lim y

y

5

1253

-

-

y 5

(g) limx

x x

12 12

+

+ +

x 1-

(h) limx

x x4

2 82

+

+ -

x 4 -

(i) limc

c

4

22

-

-

c 2

(j) limx x

x 12

-

-

x 1

(k) lim h

h h h2 73 2+ -

h 0

(l) limh

hx hx h32 2- +

h 0

(m) limh

hx h x hx h2 3 53 2 2- + -

h 0

(n) lim x c x c 3 3

-

-

x c



discontinuous

(a) 3 y x2= -

(b)1

1 y

x=

+

(c) f x x 1= -] g

(d)4

1 y

x2=

+

(e)4

1

y x2=

-



(a)3

y xx x2

= +

(b)33

y x

x x2

=+

+

(c)1

5 4 y

xx x2

=+

+ +

510 Exercises




Regions

Class Investigation


them all







EXAMPLE


Solution



y

x =3

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4







region

EXAMPLE


Solution



included


shown

y

x

1

1

-2 2 3 4

-3

-4

-5

2

3

4

5

-1

-2

-3-4

-1 y = -1



y -intercept 1-






EXAMPLES


1 y x 2$ +

Solution






y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0



2 x y 2 3 61-

Solution


region


side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g








3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g









each function

(a) | | y x=

(b) | |f x x 7= +] g

(c) | |f x x 2= -] g

(d) 5 | | y x=

(e) | |f x x 3= - +] g

(f) | 6 | y x= +

(g) | |f x x3 2= -] g

(h) | 5 4 | y x= +

(i) | 7 1 | y x= -

(j) | |f x x2 9= +] g


plane

(a) | | y x=

(b) | |f x x 1= +] g

(c) | |f x x 3= -] g

(d) 2 | | y x=

(e) | |f x x= -] g

(f) | 1 | y x= +

(g) | |f x x 1= - -] g

(h) | 2 3 | y x= -

(i) | 4 2 | y x= +

(j) | |f x x3 1= +] g


each function

(a) | 1 | y x= -

(b) | |f x x 8= -] g

(c) | |f x x2 5= +] g

(d) 2 | | 3 y x= -

(e) | |f x x 3= - -] g


function is

increasing(i)

decreasing(ii)

(a) | 2 | y x= -

(b) | |f x x 2= +

] g

(c) | |f x x2 3= -] g

(d) 4 | | 1 y x= -

(e) | |f x x= -] g


of each function

(a) | | y x= for x2 2 -

(b) | |f x x 4= - -] g for

x4 3 -

(c) | |f x x 4= +] g for x7 2 -

(d) | 2 5 | y x= - for x3 3 -

(e) | |f x x= -] g for x1 1 -


function increasing

(a) | 3 | y x= +

(b) | |f x x 4= - +] g

(c) | |f x x 9= -] g

(d) | | y x 2 1= - -

(e) | |f x x 2= - +] g

56 Exercises




7 Solve graphically

(a) | | 3x =

(b) | |x 12

(c) | |x 2

(d) | 2 | 1x + =

(e) | 3 | 0x- =

(f) |2 3 | 1x - =

(g) | |x 1 41-

(h) | |x 1 3+

(i) | |x 2 22-

(j) | |x 3 1$-

(k) | |x2 3 5+

(l) | |x2 1 1$-

(m) |3 1 | 3x x- = +

(n) |3 2 | 4x x- = -

(o) | 1 | 1x x- = +

(p) | 3 | 2 2x x+ = + (q) |2 1 | 1x x+ = -

(r) |2 5 | 3x x- = -

(s) | 1 | 2x x- =

(t) |2 3 | 3x x- = +

The Hyperbola


or= =

EXAMPLE

Sketch1

y x=

Solution



x -3 -2 -121-

41- 0

41

21 1 2 3

y 3

1-

2

1- -1 -2 -4 mdash 4 2 1

2

1

3

1

Class Discussion



y is never 0 Why






but never touches)



Investigation




equations




1 y xk

=

21

y x k= +

31

y x k

=+


EXAMPLES


3=

-

] g


Solution



So x 3






be zero

CONTINUED







1

y 0 3

3=

-

= -

-3

4

3

2

1

5

-2

-1

-4

-5

y

-1-2 4 521-3-4

x

x = 3

y = 0

Asymptotes

3

2 Sketch y x2 4

1= -

+

Solution




x

x

2 4

2

-

-






( ) y

2 0 41

41

= -+

= -


a translation of3

y x

=





y

-2

x

-

1

4


a=

+


domain x xb



1 For each graph



exists


(a)2

y x=

(b)1

y x= -

(c) f xx 1

1=

+] g

(d) f xx 2

3=

-

] g

(e)3 6

1 y

x=

+

(f) f xx 3

2= -

-

] g

(g) f xx 1

4=

-

] g

(h)1

2 y

x= -

+

(i) f xx6 3

2=

-

] g

(j)2

6 y

x= -

+

2 Show that f x x2

=] g is an odd

function



(a) f xx2 5

1=

+

] g for x2 2 -

(b)3

1 y

x=

+ for x2 0 -

(c) f xx2 4

5=

-

] g for x3 1 -

57 Exercises




(d) f xx 4

3= -

-

] g for x3 3 -

(e)3 1

2 y

x= -

+ for x0 5



(a)3

y x= for y 1 3

(b)2

y x= - for y 221

- -

(c) f xx 1

1=

-

] g for y 171

- -

(d) f xx2 1

3= -

+

] g for

y 131

- -

(e)3 2

6 y

x=

- for y 1

21

6



Circle gate





Proof

r y

x

( x y)

y

x








c a b

r x y

2 2 2

2 2 2

`

= +

= +

EXAMPLE



Solution


y

x

-2

-2 2

2


one place

y

x

2

2

2

-2

The radius is 4

CONTINUED






Domain 2 2 x x -

Range 2 2 y y -


range y r y r -



Proof


shown

y

x

(a b)

x

y

r

( x y)

a

b x - a

y - b




ndash ndash

c a b

r x a y b

2 2 2

2 2 2

= +

= +] ^g h




EXAMPLES



Solution



y

x 9

9

-9

-9



Domain 9 9 x x -

Range 9 9 y y -



Solution


ndash

ndash ndash

x y

x y

1 2 4

1 2 2

2

2 2

+ + =

+ - =

2

2

] ^

] ]_

g h

g gi

So 1 2a b= = - and 2r =

CONTINUED







y

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

(1 -2)



Domain 1 3 x x -

Range 4 0 y y -


Solution


a b2 1= - = and r 3=

Substituting

ndash ndash

ndash

ndash

x a y b r

x y

x y

2 1 3

2 1 9

2 2 2

2 2 2

2 2

+ =

- - + =

+ + =

] ^]] ^

] ^

g hg g hg h


ndash

ndash

a b a ab b

x x x

x x

a b a ab b

y y y

y y

2

2 2 2 2

4 4

2

1 2 1 1

2 1

So

So

2 2 2

2 2 2

2

2 2 2

2 2 2

2

+ = + +

+ = + +

= + +

= - +

= - +

= - +

]] ] ]]^ ^ ]

gg g ggh h g


ndash

x x y y

x x y y

x x y y

x x y y

4 4 2 1 9

4 2 5 9

4 2 5 9

4 2 4 0

9 9

2

2

2

2

+ + + - + =

+ + - + =

+ + + =

+ + - - =

- -


in Chapter 2




Investigation


two functions





circle






for all values

y

x r

r

-r


Proof

ndash

x y r

y r x y r x

2 2 2

2 2 2

2 2

+ =

=

= -



semi-circles





y 0 for all values

y

x r

-

r

-r


EXAMPLES


1 f x x92

= -] g

Solution



y

x 3

3

-3

Domain 3 3 x x -

Range 0 3 y y




2 y x4 2= - -

Solution



y

x 2

-2

-2

Domain 2 2 x x -

Range 2 0 y y -




range

(a) 9x y 2 2+ =

(b) x y 16 02 2+ =-


(d) 1 9x y 2 2+ + =

] g

(e) ndashx y 2 1 12 2+ + =] ^g h



below the x -axis



range

(a) 25 y x2= - -

(b) 1 y x2= -

(c) 36 y x2= -

(d) 64 y x2= - -

(e) 7 y x2= - -



each circle

(a) 100x y 2 2+ =

(b) 5x y 2 2+ =


(d) ndashx y 5 6 492 2+ + =] ^g h

(e) ndashx y 3 812 2+ =^ h

58 Exercises






grouping symbols)



Centre(c) 1 5-







Centre(j) 4 7- -

^ h and radius 3

Other Graphs




EXAMPLES


Solution



eg When 0x =

y 3

1

c=

=

x

y

1

3

3

1

31

When1

1

= -

=

=

=

-

x 3- 2- 1- 0 1 2 3

y 271

91

31

1 3 9 27



You will meet these

graphs again in the

HSC Course





other point

When

x

y

1

3

3

1

=

=

=

y

x

1

2

1

3







Solution



x minus2 minus1 0 05 1 2 3 4

y minus03 0 03 05 06

y

x

1

2

1 2 3 4

-1




You learned about


stages of maths





range y y 02



range all real y

Cubic function


EXAMPLE



Solution



y minus25 minus6 1 2 3 10 29

y

x

1

1

-2 2 3 4

-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4



If you already

know the shape of


3






Domain and range


EXAMPLE


Solution




function are

f

f

1 1 2

1 21

4 4 2

64 2

66

3

3

- = - +

= - +

=

= +

= +

=

] ]

] ]

g g

g g


these points

y

x

(-1 1)

(4 66)

Range 1 66 y y










positive number

EXAMPLE


Solution


x 4So $

$


(or zero) So y 0$


59 Exercises


(a) 4 3 y x= +

(b) f x 4= -] g

(c) 3x =



(f) f x xx 12 2= - -] g

(g) 64x y 2 2+ =

(h) f t t 4

3=

-] g

(i) ( ) g 2

5zz

= +

(j) | |f x x=] g


(a) y x=

(b) 2 y x= -

(c) | |f x x2 3= -] g

(d) | | 2 y x= -

(e) f x x2 5= - +] g

(f) | | y x5= -

(g) 2 y x=

(h) y 5x= -

(i) f x xx 1

= +] g

(j)2

4 3 y x

x= -


(a) y x x 5 2= -] g


= +-

(d) g x x x164 2= -] g

(e) 49x y 2 2+ =

You may like to


by dividing by x




4 (a) Solve x1 02$-


f x x1 2= -] g


(a) 2 y x x2= - -

(b) g t t t 62= +] g



in each case


x3 3 -


x2 3 -


x2 1 - (d)


x1 5


0 4x


x3 3 -


x1 1 -


x2 3 -


x4 4 -

(j) y x x7 62= - + - in the

domain 0 7x


function y x 1

3=

+





x=] g


function

find its range(b)


plane

(a) f x x4=] g

(b) y x3= -

(c) y x 34= -

(d) 2 p x x3=] g

(e) 1 g x x3= +] g

(f) 100x y 2 2+ =

(g) 2 1 y x= +


y x 1= -









(b) f x x 22= -] g

(c)1

y x=

(d) f x x3=] g

(e) f x 3x=] g

13 (a) Solve x4 02$-


(i) 4 y x2= -

(ii) y x4

2= - -




DID YOU KNOW




a light



Limits






a a1

1

03

Z

=

=

3

3

-


lim a 0x=

x 3-



asymptote

EXAMPLES

1 Find lim x

x x5x 0

2+

Solution


which is undefined


( )

lim lim

lim

xx x

x

x x

x

5 5

5

5

x x

x

0

2

0 1

1

0

+=

+

= +

=

] g




2 Find limx

x

4

22

-

-

x 2

Solution


lim lim

lim

x

x

x x

x

x

4

2

2 2

2

21

41

2 1

1

-

-=

+ -

-

=+

=

x x2 2

x 2

^ _h i

3 Find limh

h x hx h2 72 2+ -

h 0

Solution

lim lim

lim

h

h x hx h

h

h hx x

hx x

x

2 7 2 7

2 7

7

2 2 2

2

2

+ -=

+ -

= + -

= -

h 0

h h0 0

^ h

Continuity





EXAMPLES

1 Find lim

x

x

1

12

-

-

x 1


11

y x

x2

=-

- Sketch the curve

Solution


xx2

-

- gives

00

CONTINUED




( )

lim lim

limx

x

x

x x

x11

1

1 1

1

2

x x

x

1

2

1

1

-

-=

-

+ -

= +

=

-

] ]g g

11

y xx2

=-





y xx

x

x x

x

11

1

1 1

1

2

=-

-

=

+

= +

-

-^ ^h h


2 Find limx

x x2

2x 2

2

+

+ -

-


x x2

22

= + -

+

Solution


x x2

22

+

+ - gives

00

lim lim

lim

xx x

x

x x

x

22

2

1 2

1

3

x x

x

2

2

2

2

+

+ -=

+

- +

=

= -

-

- -

-

^^ ^

^

hh h

h2 y

x

x xx

y x

x

x

x

22

2

2

1

1


=+

+ -= -

=+

= -

+ -^ ^h h


Remember that x 1




1 Find(a) lim x 52

+x 4

(b) lim t 7-t 3 -

(c) lim x x2 43+ -

x 2

(d) lim xx x32

+

x 0

(e) limh

h h

2

22

-

- -

h 2

(f) lim y

y

5

1253

-

-

y 5

(g) limx

x x

12 12

+

+ +

x 1-

(h) limx

x x4

2 82

+

+ -

x 4 -

(i) limc

c

4

22

-

-

c 2

(j) limx x

x 12

-

-

x 1

(k) lim h

h h h2 73 2+ -

h 0

(l) limh

hx hx h32 2- +

h 0

(m) limh

hx h x hx h2 3 53 2 2- + -

h 0

(n) lim x c x c 3 3

-

-

x c



discontinuous

(a) 3 y x2= -

(b)1

1 y

x=

+

(c) f x x 1= -] g

(d)4

1 y

x2=

+

(e)4

1

y x2=

-



(a)3

y xx x2

= +

(b)33

y x

x x2

=+

+

(c)1

5 4 y

xx x2

=+

+ +

510 Exercises




Regions

Class Investigation


them all







EXAMPLE


Solution



y

x =3

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4







region

EXAMPLE


Solution



included


shown

y

x

1

1

-2 2 3 4

-3

-4

-5

2

3

4

5

-1

-2

-3-4

-1 y = -1



y -intercept 1-






EXAMPLES


1 y x 2$ +

Solution






y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0



2 x y 2 3 61-

Solution


region


side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g








3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g






7 Solve graphically

(a) | | 3x =

(b) | |x 12

(c) | |x 2

(d) | 2 | 1x + =

(e) | 3 | 0x- =

(f) |2 3 | 1x - =

(g) | |x 1 41-

(h) | |x 1 3+

(i) | |x 2 22-

(j) | |x 3 1$-

(k) | |x2 3 5+

(l) | |x2 1 1$-

(m) |3 1 | 3x x- = +

(n) |3 2 | 4x x- = -

(o) | 1 | 1x x- = +

(p) | 3 | 2 2x x+ = + (q) |2 1 | 1x x+ = -

(r) |2 5 | 3x x- = -

(s) | 1 | 2x x- =

(t) |2 3 | 3x x- = +

The Hyperbola


or= =

EXAMPLE

Sketch1

y x=

Solution



x -3 -2 -121-

41- 0

41

21 1 2 3

y 3

1-

2

1- -1 -2 -4 mdash 4 2 1

2

1

3

1

Class Discussion



y is never 0 Why






but never touches)



Investigation




equations




1 y xk

=

21

y x k= +

31

y x k

=+


EXAMPLES


3=

-

] g


Solution



So x 3






be zero

CONTINUED







1

y 0 3

3=

-

= -

-3

4

3

2

1

5

-2

-1

-4

-5

y

-1-2 4 521-3-4

x

x = 3

y = 0

Asymptotes

3

2 Sketch y x2 4

1= -

+

Solution




x

x

2 4

2

-

-






( ) y

2 0 41

41

= -+

= -


a translation of3

y x

=





y

-2

x

-

1

4


a=

+


domain x xb



1 For each graph



exists


(a)2

y x=

(b)1

y x= -

(c) f xx 1

1=

+] g

(d) f xx 2

3=

-

] g

(e)3 6

1 y

x=

+

(f) f xx 3

2= -

-

] g

(g) f xx 1

4=

-

] g

(h)1

2 y

x= -

+

(i) f xx6 3

2=

-

] g

(j)2

6 y

x= -

+

2 Show that f x x2

=] g is an odd

function



(a) f xx2 5

1=

+

] g for x2 2 -

(b)3

1 y

x=

+ for x2 0 -

(c) f xx2 4

5=

-

] g for x3 1 -

57 Exercises




(d) f xx 4

3= -

-

] g for x3 3 -

(e)3 1

2 y

x= -

+ for x0 5



(a)3

y x= for y 1 3

(b)2

y x= - for y 221

- -

(c) f xx 1

1=

-

] g for y 171

- -

(d) f xx2 1

3= -

+

] g for

y 131

- -

(e)3 2

6 y

x=

- for y 1

21

6



Circle gate





Proof

r y

x

( x y)

y

x








c a b

r x y

2 2 2

2 2 2

`

= +

= +

EXAMPLE



Solution


y

x

-2

-2 2

2


one place

y

x

2

2

2

-2

The radius is 4

CONTINUED






Domain 2 2 x x -

Range 2 2 y y -


range y r y r -



Proof


shown

y

x

(a b)

x

y

r

( x y)

a

b x - a

y - b




ndash ndash

c a b

r x a y b

2 2 2

2 2 2

= +

= +] ^g h




EXAMPLES



Solution



y

x 9

9

-9

-9



Domain 9 9 x x -

Range 9 9 y y -



Solution


ndash

ndash ndash

x y

x y

1 2 4

1 2 2

2

2 2

+ + =

+ - =

2

2

] ^

] ]_

g h

g gi

So 1 2a b= = - and 2r =

CONTINUED







y

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

(1 -2)



Domain 1 3 x x -

Range 4 0 y y -


Solution


a b2 1= - = and r 3=

Substituting

ndash ndash

ndash

ndash

x a y b r

x y

x y

2 1 3

2 1 9

2 2 2

2 2 2

2 2

+ =

- - + =

+ + =

] ^]] ^

] ^

g hg g hg h


ndash

ndash

a b a ab b

x x x

x x

a b a ab b

y y y

y y

2

2 2 2 2

4 4

2

1 2 1 1

2 1

So

So

2 2 2

2 2 2

2

2 2 2

2 2 2

2

+ = + +

+ = + +

= + +

= - +

= - +

= - +

]] ] ]]^ ^ ]

gg g ggh h g


ndash

x x y y

x x y y

x x y y

x x y y

4 4 2 1 9

4 2 5 9

4 2 5 9

4 2 4 0

9 9

2

2

2

2

+ + + - + =

+ + - + =

+ + + =

+ + - - =

- -


in Chapter 2




Investigation


two functions





circle






for all values

y

x r

r

-r


Proof

ndash

x y r

y r x y r x

2 2 2

2 2 2

2 2

+ =

=

= -



semi-circles





y 0 for all values

y

x r

-

r

-r


EXAMPLES


1 f x x92

= -] g

Solution



y

x 3

3

-3

Domain 3 3 x x -

Range 0 3 y y




2 y x4 2= - -

Solution



y

x 2

-2

-2

Domain 2 2 x x -

Range 2 0 y y -




range

(a) 9x y 2 2+ =

(b) x y 16 02 2+ =-


(d) 1 9x y 2 2+ + =

] g

(e) ndashx y 2 1 12 2+ + =] ^g h



below the x -axis



range

(a) 25 y x2= - -

(b) 1 y x2= -

(c) 36 y x2= -

(d) 64 y x2= - -

(e) 7 y x2= - -



each circle

(a) 100x y 2 2+ =

(b) 5x y 2 2+ =


(d) ndashx y 5 6 492 2+ + =] ^g h

(e) ndashx y 3 812 2+ =^ h

58 Exercises






grouping symbols)



Centre(c) 1 5-







Centre(j) 4 7- -

^ h and radius 3

Other Graphs




EXAMPLES


Solution



eg When 0x =

y 3

1

c=

=

x

y

1

3

3

1

31

When1

1

= -

=

=

=

-

x 3- 2- 1- 0 1 2 3

y 271

91

31

1 3 9 27



You will meet these

graphs again in the

HSC Course





other point

When

x

y

1

3

3

1

=

=

=

y

x

1

2

1

3







Solution



x minus2 minus1 0 05 1 2 3 4

y minus03 0 03 05 06

y

x

1

2

1 2 3 4

-1




You learned about


stages of maths





range y y 02



range all real y

Cubic function


EXAMPLE



Solution



y minus25 minus6 1 2 3 10 29

y

x

1

1

-2 2 3 4

-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4



If you already

know the shape of


3






Domain and range


EXAMPLE


Solution




function are

f

f

1 1 2

1 21

4 4 2

64 2

66

3

3

- = - +

= - +

=

= +

= +

=

] ]

] ]

g g

g g


these points

y

x

(-1 1)

(4 66)

Range 1 66 y y










positive number

EXAMPLE


Solution


x 4So $

$


(or zero) So y 0$


59 Exercises


(a) 4 3 y x= +

(b) f x 4= -] g

(c) 3x =



(f) f x xx 12 2= - -] g

(g) 64x y 2 2+ =

(h) f t t 4

3=

-] g

(i) ( ) g 2

5zz

= +

(j) | |f x x=] g


(a) y x=

(b) 2 y x= -

(c) | |f x x2 3= -] g

(d) | | 2 y x= -

(e) f x x2 5= - +] g

(f) | | y x5= -

(g) 2 y x=

(h) y 5x= -

(i) f x xx 1

= +] g

(j)2

4 3 y x

x= -


(a) y x x 5 2= -] g


= +-

(d) g x x x164 2= -] g

(e) 49x y 2 2+ =

You may like to


by dividing by x




4 (a) Solve x1 02$-


f x x1 2= -] g


(a) 2 y x x2= - -

(b) g t t t 62= +] g



in each case


x3 3 -


x2 3 -


x2 1 - (d)


x1 5


0 4x


x3 3 -


x1 1 -


x2 3 -


x4 4 -

(j) y x x7 62= - + - in the

domain 0 7x


function y x 1

3=

+





x=] g


function

find its range(b)


plane

(a) f x x4=] g

(b) y x3= -

(c) y x 34= -

(d) 2 p x x3=] g

(e) 1 g x x3= +] g

(f) 100x y 2 2+ =

(g) 2 1 y x= +


y x 1= -









(b) f x x 22= -] g

(c)1

y x=

(d) f x x3=] g

(e) f x 3x=] g

13 (a) Solve x4 02$-


(i) 4 y x2= -

(ii) y x4

2= - -




DID YOU KNOW




a light



Limits






a a1

1

03

Z

=

=

3

3

-


lim a 0x=

x 3-



asymptote

EXAMPLES

1 Find lim x

x x5x 0

2+

Solution


which is undefined


( )

lim lim

lim

xx x

x

x x

x

5 5

5

5

x x

x

0

2

0 1

1

0

+=

+

= +

=

] g




2 Find limx

x

4

22

-

-

x 2

Solution


lim lim

lim

x

x

x x

x

x

4

2

2 2

2

21

41

2 1

1

-

-=

+ -

-

=+

=

x x2 2

x 2

^ _h i

3 Find limh

h x hx h2 72 2+ -

h 0

Solution

lim lim

lim

h

h x hx h

h

h hx x

hx x

x

2 7 2 7

2 7

7

2 2 2

2

2

+ -=

+ -

= + -

= -

h 0

h h0 0

^ h

Continuity





EXAMPLES

1 Find lim

x

x

1

12

-

-

x 1


11

y x

x2

=-

- Sketch the curve

Solution


xx2

-

- gives

00

CONTINUED




( )

lim lim

limx

x

x

x x

x11

1

1 1

1

2

x x

x

1

2

1

1

-

-=

-

+ -

= +

=

-

] ]g g

11

y xx2

=-





y xx

x

x x

x

11

1

1 1

1

2

=-

-

=

+

= +

-

-^ ^h h


2 Find limx

x x2

2x 2

2

+

+ -

-


x x2

22

= + -

+

Solution


x x2

22

+

+ - gives

00

lim lim

lim

xx x

x

x x

x

22

2

1 2

1

3

x x

x

2

2

2

2

+

+ -=

+

- +

=

= -

-

- -

-

^^ ^

^

hh h

h2 y

x

x xx

y x

x

x

x

22

2

2

1

1


=+

+ -= -

=+

= -

+ -^ ^h h


Remember that x 1




1 Find(a) lim x 52

+x 4

(b) lim t 7-t 3 -

(c) lim x x2 43+ -

x 2

(d) lim xx x32

+

x 0

(e) limh

h h

2

22

-

- -

h 2

(f) lim y

y

5

1253

-

-

y 5

(g) limx

x x

12 12

+

+ +

x 1-

(h) limx

x x4

2 82

+

+ -

x 4 -

(i) limc

c

4

22

-

-

c 2

(j) limx x

x 12

-

-

x 1

(k) lim h

h h h2 73 2+ -

h 0

(l) limh

hx hx h32 2- +

h 0

(m) limh

hx h x hx h2 3 53 2 2- + -

h 0

(n) lim x c x c 3 3

-

-

x c



discontinuous

(a) 3 y x2= -

(b)1

1 y

x=

+

(c) f x x 1= -] g

(d)4

1 y

x2=

+

(e)4

1

y x2=

-



(a)3

y xx x2

= +

(b)33

y x

x x2

=+

+

(c)1

5 4 y

xx x2

=+

+ +

510 Exercises




Regions

Class Investigation


them all







EXAMPLE


Solution



y

x =3

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4







region

EXAMPLE


Solution



included


shown

y

x

1

1

-2 2 3 4

-3

-4

-5

2

3

4

5

-1

-2

-3-4

-1 y = -1



y -intercept 1-






EXAMPLES


1 y x 2$ +

Solution






y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0



2 x y 2 3 61-

Solution


region


side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g








3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g








but never touches)



Investigation




equations




1 y xk

=

21

y x k= +

31

y x k

=+


EXAMPLES


3=

-

] g


Solution



So x 3






be zero

CONTINUED







1

y 0 3

3=

-

= -

-3

4

3

2

1

5

-2

-1

-4

-5

y

-1-2 4 521-3-4

x

x = 3

y = 0

Asymptotes

3

2 Sketch y x2 4

1= -

+

Solution




x

x

2 4

2

-

-






( ) y

2 0 41

41

= -+

= -


a translation of3

y x

=





y

-2

x

-

1

4


a=

+


domain x xb



1 For each graph



exists


(a)2

y x=

(b)1

y x= -

(c) f xx 1

1=

+] g

(d) f xx 2

3=

-

] g

(e)3 6

1 y

x=

+

(f) f xx 3

2= -

-

] g

(g) f xx 1

4=

-

] g

(h)1

2 y

x= -

+

(i) f xx6 3

2=

-

] g

(j)2

6 y

x= -

+

2 Show that f x x2

=] g is an odd

function



(a) f xx2 5

1=

+

] g for x2 2 -

(b)3

1 y

x=

+ for x2 0 -

(c) f xx2 4

5=

-

] g for x3 1 -

57 Exercises




(d) f xx 4

3= -

-

] g for x3 3 -

(e)3 1

2 y

x= -

+ for x0 5



(a)3

y x= for y 1 3

(b)2

y x= - for y 221

- -

(c) f xx 1

1=

-

] g for y 171

- -

(d) f xx2 1

3= -

+

] g for

y 131

- -

(e)3 2

6 y

x=

- for y 1

21

6



Circle gate





Proof

r y

x

( x y)

y

x








c a b

r x y

2 2 2

2 2 2

`

= +

= +

EXAMPLE



Solution


y

x

-2

-2 2

2


one place

y

x

2

2

2

-2

The radius is 4

CONTINUED






Domain 2 2 x x -

Range 2 2 y y -


range y r y r -



Proof


shown

y

x

(a b)

x

y

r

( x y)

a

b x - a

y - b




ndash ndash

c a b

r x a y b

2 2 2

2 2 2

= +

= +] ^g h




EXAMPLES



Solution



y

x 9

9

-9

-9



Domain 9 9 x x -

Range 9 9 y y -



Solution


ndash

ndash ndash

x y

x y

1 2 4

1 2 2

2

2 2

+ + =

+ - =

2

2

] ^

] ]_

g h

g gi

So 1 2a b= = - and 2r =

CONTINUED







y

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

(1 -2)



Domain 1 3 x x -

Range 4 0 y y -


Solution


a b2 1= - = and r 3=

Substituting

ndash ndash

ndash

ndash

x a y b r

x y

x y

2 1 3

2 1 9

2 2 2

2 2 2

2 2

+ =

- - + =

+ + =

] ^]] ^

] ^

g hg g hg h


ndash

ndash

a b a ab b

x x x

x x

a b a ab b

y y y

y y

2

2 2 2 2

4 4

2

1 2 1 1

2 1

So

So

2 2 2

2 2 2

2

2 2 2

2 2 2

2

+ = + +

+ = + +

= + +

= - +

= - +

= - +

]] ] ]]^ ^ ]

gg g ggh h g


ndash

x x y y

x x y y

x x y y

x x y y

4 4 2 1 9

4 2 5 9

4 2 5 9

4 2 4 0

9 9

2

2

2

2

+ + + - + =

+ + - + =

+ + + =

+ + - - =

- -


in Chapter 2




Investigation


two functions





circle






for all values

y

x r

r

-r


Proof

ndash

x y r

y r x y r x

2 2 2

2 2 2

2 2

+ =

=

= -



semi-circles





y 0 for all values

y

x r

-

r

-r


EXAMPLES


1 f x x92

= -] g

Solution



y

x 3

3

-3

Domain 3 3 x x -

Range 0 3 y y




2 y x4 2= - -

Solution



y

x 2

-2

-2

Domain 2 2 x x -

Range 2 0 y y -




range

(a) 9x y 2 2+ =

(b) x y 16 02 2+ =-


(d) 1 9x y 2 2+ + =

] g

(e) ndashx y 2 1 12 2+ + =] ^g h



below the x -axis



range

(a) 25 y x2= - -

(b) 1 y x2= -

(c) 36 y x2= -

(d) 64 y x2= - -

(e) 7 y x2= - -



each circle

(a) 100x y 2 2+ =

(b) 5x y 2 2+ =


(d) ndashx y 5 6 492 2+ + =] ^g h

(e) ndashx y 3 812 2+ =^ h

58 Exercises






grouping symbols)



Centre(c) 1 5-







Centre(j) 4 7- -

^ h and radius 3

Other Graphs




EXAMPLES


Solution



eg When 0x =

y 3

1

c=

=

x

y

1

3

3

1

31

When1

1

= -

=

=

=

-

x 3- 2- 1- 0 1 2 3

y 271

91

31

1 3 9 27



You will meet these

graphs again in the

HSC Course





other point

When

x

y

1

3

3

1

=

=

=

y

x

1

2

1

3







Solution



x minus2 minus1 0 05 1 2 3 4

y minus03 0 03 05 06

y

x

1

2

1 2 3 4

-1




You learned about


stages of maths





range y y 02



range all real y

Cubic function


EXAMPLE



Solution



y minus25 minus6 1 2 3 10 29

y

x

1

1

-2 2 3 4

-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4



If you already

know the shape of


3






Domain and range


EXAMPLE


Solution




function are

f

f

1 1 2

1 21

4 4 2

64 2

66

3

3

- = - +

= - +

=

= +

= +

=

] ]

] ]

g g

g g


these points

y

x

(-1 1)

(4 66)

Range 1 66 y y










positive number

EXAMPLE


Solution


x 4So $

$


(or zero) So y 0$


59 Exercises


(a) 4 3 y x= +

(b) f x 4= -] g

(c) 3x =



(f) f x xx 12 2= - -] g

(g) 64x y 2 2+ =

(h) f t t 4

3=

-] g

(i) ( ) g 2

5zz

= +

(j) | |f x x=] g


(a) y x=

(b) 2 y x= -

(c) | |f x x2 3= -] g

(d) | | 2 y x= -

(e) f x x2 5= - +] g

(f) | | y x5= -

(g) 2 y x=

(h) y 5x= -

(i) f x xx 1

= +] g

(j)2

4 3 y x

x= -


(a) y x x 5 2= -] g


= +-

(d) g x x x164 2= -] g

(e) 49x y 2 2+ =

You may like to


by dividing by x




4 (a) Solve x1 02$-


f x x1 2= -] g


(a) 2 y x x2= - -

(b) g t t t 62= +] g



in each case


x3 3 -


x2 3 -


x2 1 - (d)


x1 5


0 4x


x3 3 -


x1 1 -


x2 3 -


x4 4 -

(j) y x x7 62= - + - in the

domain 0 7x


function y x 1

3=

+





x=] g


function

find its range(b)


plane

(a) f x x4=] g

(b) y x3= -

(c) y x 34= -

(d) 2 p x x3=] g

(e) 1 g x x3= +] g

(f) 100x y 2 2+ =

(g) 2 1 y x= +


y x 1= -









(b) f x x 22= -] g

(c)1

y x=

(d) f x x3=] g

(e) f x 3x=] g

13 (a) Solve x4 02$-


(i) 4 y x2= -

(ii) y x4

2= - -




DID YOU KNOW




a light



Limits






a a1

1

03

Z

=

=

3

3

-


lim a 0x=

x 3-



asymptote

EXAMPLES

1 Find lim x

x x5x 0

2+

Solution


which is undefined


( )

lim lim

lim

xx x

x

x x

x

5 5

5

5

x x

x

0

2

0 1

1

0

+=

+

= +

=

] g




2 Find limx

x

4

22

-

-

x 2

Solution


lim lim

lim

x

x

x x

x

x

4

2

2 2

2

21

41

2 1

1

-

-=

+ -

-

=+

=

x x2 2

x 2

^ _h i

3 Find limh

h x hx h2 72 2+ -

h 0

Solution

lim lim

lim

h

h x hx h

h

h hx x

hx x

x

2 7 2 7

2 7

7

2 2 2

2

2

+ -=

+ -

= + -

= -

h 0

h h0 0

^ h

Continuity





EXAMPLES

1 Find lim

x

x

1

12

-

-

x 1


11

y x

x2

=-

- Sketch the curve

Solution


xx2

-

- gives

00

CONTINUED




( )

lim lim

limx

x

x

x x

x11

1

1 1

1

2

x x

x

1

2

1

1

-

-=

-

+ -

= +

=

-

] ]g g

11

y xx2

=-





y xx

x

x x

x

11

1

1 1

1

2

=-

-

=

+

= +

-

-^ ^h h


2 Find limx

x x2

2x 2

2

+

+ -

-


x x2

22

= + -

+

Solution


x x2

22

+

+ - gives

00

lim lim

lim

xx x

x

x x

x

22

2

1 2

1

3

x x

x

2

2

2

2

+

+ -=

+

- +

=

= -

-

- -

-

^^ ^

^

hh h

h2 y

x

x xx

y x

x

x

x

22

2

2

1

1


=+

+ -= -

=+

= -

+ -^ ^h h


Remember that x 1




1 Find(a) lim x 52

+x 4

(b) lim t 7-t 3 -

(c) lim x x2 43+ -

x 2

(d) lim xx x32

+

x 0

(e) limh

h h

2

22

-

- -

h 2

(f) lim y

y

5

1253

-

-

y 5

(g) limx

x x

12 12

+

+ +

x 1-

(h) limx

x x4

2 82

+

+ -

x 4 -

(i) limc

c

4

22

-

-

c 2

(j) limx x

x 12

-

-

x 1

(k) lim h

h h h2 73 2+ -

h 0

(l) limh

hx hx h32 2- +

h 0

(m) limh

hx h x hx h2 3 53 2 2- + -

h 0

(n) lim x c x c 3 3

-

-

x c



discontinuous

(a) 3 y x2= -

(b)1

1 y

x=

+

(c) f x x 1= -] g

(d)4

1 y

x2=

+

(e)4

1

y x2=

-



(a)3

y xx x2

= +

(b)33

y x

x x2

=+

+

(c)1

5 4 y

xx x2

=+

+ +

510 Exercises




Regions

Class Investigation


them all







EXAMPLE


Solution



y

x =3

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4







region

EXAMPLE


Solution



included


shown

y

x

1

1

-2 2 3 4

-3

-4

-5

2

3

4

5

-1

-2

-3-4

-1 y = -1



y -intercept 1-






EXAMPLES


1 y x 2$ +

Solution






y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0



2 x y 2 3 61-

Solution


region


side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g








3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g









1

y 0 3

3=

-

= -

-3

4

3

2

1

5

-2

-1

-4

-5

y

-1-2 4 521-3-4

x

x = 3

y = 0

Asymptotes

3

2 Sketch y x2 4

1= -

+

Solution




x

x

2 4

2

-

-






( ) y

2 0 41

41

= -+

= -


a translation of3

y x

=





y

-2

x

-

1

4


a=

+


domain x xb



1 For each graph



exists


(a)2

y x=

(b)1

y x= -

(c) f xx 1

1=

+] g

(d) f xx 2

3=

-

] g

(e)3 6

1 y

x=

+

(f) f xx 3

2= -

-

] g

(g) f xx 1

4=

-

] g

(h)1

2 y

x= -

+

(i) f xx6 3

2=

-

] g

(j)2

6 y

x= -

+

2 Show that f x x2

=] g is an odd

function



(a) f xx2 5

1=

+

] g for x2 2 -

(b)3

1 y

x=

+ for x2 0 -

(c) f xx2 4

5=

-

] g for x3 1 -

57 Exercises




(d) f xx 4

3= -

-

] g for x3 3 -

(e)3 1

2 y

x= -

+ for x0 5



(a)3

y x= for y 1 3

(b)2

y x= - for y 221

- -

(c) f xx 1

1=

-

] g for y 171

- -

(d) f xx2 1

3= -

+

] g for

y 131

- -

(e)3 2

6 y

x=

- for y 1

21

6



Circle gate





Proof

r y

x

( x y)

y

x








c a b

r x y

2 2 2

2 2 2

`

= +

= +

EXAMPLE



Solution


y

x

-2

-2 2

2


one place

y

x

2

2

2

-2

The radius is 4

CONTINUED






Domain 2 2 x x -

Range 2 2 y y -


range y r y r -



Proof


shown

y

x

(a b)

x

y

r

( x y)

a

b x - a

y - b




ndash ndash

c a b

r x a y b

2 2 2

2 2 2

= +

= +] ^g h




EXAMPLES



Solution



y

x 9

9

-9

-9



Domain 9 9 x x -

Range 9 9 y y -



Solution


ndash

ndash ndash

x y

x y

1 2 4

1 2 2

2

2 2

+ + =

+ - =

2

2

] ^

] ]_

g h

g gi

So 1 2a b= = - and 2r =

CONTINUED







y

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

(1 -2)



Domain 1 3 x x -

Range 4 0 y y -


Solution


a b2 1= - = and r 3=

Substituting

ndash ndash

ndash

ndash

x a y b r

x y

x y

2 1 3

2 1 9

2 2 2

2 2 2

2 2

+ =

- - + =

+ + =

] ^]] ^

] ^

g hg g hg h


ndash

ndash

a b a ab b

x x x

x x

a b a ab b

y y y

y y

2

2 2 2 2

4 4

2

1 2 1 1

2 1

So

So

2 2 2

2 2 2

2

2 2 2

2 2 2

2

+ = + +

+ = + +

= + +

= - +

= - +

= - +

]] ] ]]^ ^ ]

gg g ggh h g


ndash

x x y y

x x y y

x x y y

x x y y

4 4 2 1 9

4 2 5 9

4 2 5 9

4 2 4 0

9 9

2

2

2

2

+ + + - + =

+ + - + =

+ + + =

+ + - - =

- -


in Chapter 2




Investigation


two functions





circle






for all values

y

x r

r

-r


Proof

ndash

x y r

y r x y r x

2 2 2

2 2 2

2 2

+ =

=

= -



semi-circles





y 0 for all values

y

x r

-

r

-r


EXAMPLES


1 f x x92

= -] g

Solution



y

x 3

3

-3

Domain 3 3 x x -

Range 0 3 y y




2 y x4 2= - -

Solution



y

x 2

-2

-2

Domain 2 2 x x -

Range 2 0 y y -




range

(a) 9x y 2 2+ =

(b) x y 16 02 2+ =-


(d) 1 9x y 2 2+ + =

] g

(e) ndashx y 2 1 12 2+ + =] ^g h



below the x -axis



range

(a) 25 y x2= - -

(b) 1 y x2= -

(c) 36 y x2= -

(d) 64 y x2= - -

(e) 7 y x2= - -



each circle

(a) 100x y 2 2+ =

(b) 5x y 2 2+ =


(d) ndashx y 5 6 492 2+ + =] ^g h

(e) ndashx y 3 812 2+ =^ h

58 Exercises






grouping symbols)



Centre(c) 1 5-







Centre(j) 4 7- -

^ h and radius 3

Other Graphs




EXAMPLES


Solution



eg When 0x =

y 3

1

c=

=

x

y

1

3

3

1

31

When1

1

= -

=

=

=

-

x 3- 2- 1- 0 1 2 3

y 271

91

31

1 3 9 27



You will meet these

graphs again in the

HSC Course





other point

When

x

y

1

3

3

1

=

=

=

y

x

1

2

1

3







Solution



x minus2 minus1 0 05 1 2 3 4

y minus03 0 03 05 06

y

x

1

2

1 2 3 4

-1




You learned about


stages of maths





range y y 02



range all real y

Cubic function


EXAMPLE



Solution



y minus25 minus6 1 2 3 10 29

y

x

1

1

-2 2 3 4

-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4



If you already

know the shape of


3






Domain and range


EXAMPLE


Solution




function are

f

f

1 1 2

1 21

4 4 2

64 2

66

3

3

- = - +

= - +

=

= +

= +

=

] ]

] ]

g g

g g


these points

y

x

(-1 1)

(4 66)

Range 1 66 y y










positive number

EXAMPLE


Solution


x 4So $

$


(or zero) So y 0$


59 Exercises


(a) 4 3 y x= +

(b) f x 4= -] g

(c) 3x =



(f) f x xx 12 2= - -] g

(g) 64x y 2 2+ =

(h) f t t 4

3=

-] g

(i) ( ) g 2

5zz

= +

(j) | |f x x=] g


(a) y x=

(b) 2 y x= -

(c) | |f x x2 3= -] g

(d) | | 2 y x= -

(e) f x x2 5= - +] g

(f) | | y x5= -

(g) 2 y x=

(h) y 5x= -

(i) f x xx 1

= +] g

(j)2

4 3 y x

x= -


(a) y x x 5 2= -] g


= +-

(d) g x x x164 2= -] g

(e) 49x y 2 2+ =

You may like to


by dividing by x




4 (a) Solve x1 02$-


f x x1 2= -] g


(a) 2 y x x2= - -

(b) g t t t 62= +] g



in each case


x3 3 -


x2 3 -


x2 1 - (d)


x1 5


0 4x


x3 3 -


x1 1 -


x2 3 -


x4 4 -

(j) y x x7 62= - + - in the

domain 0 7x


function y x 1

3=

+





x=] g


function

find its range(b)


plane

(a) f x x4=] g

(b) y x3= -

(c) y x 34= -

(d) 2 p x x3=] g

(e) 1 g x x3= +] g

(f) 100x y 2 2+ =

(g) 2 1 y x= +


y x 1= -









(b) f x x 22= -] g

(c)1

y x=

(d) f x x3=] g

(e) f x 3x=] g

13 (a) Solve x4 02$-


(i) 4 y x2= -

(ii) y x4

2= - -




DID YOU KNOW




a light



Limits






a a1

1

03

Z

=

=

3

3

-


lim a 0x=

x 3-



asymptote

EXAMPLES

1 Find lim x

x x5x 0

2+

Solution


which is undefined


( )

lim lim

lim

xx x

x

x x

x

5 5

5

5

x x

x

0

2

0 1

1

0

+=

+

= +

=

] g




2 Find limx

x

4

22

-

-

x 2

Solution


lim lim

lim

x

x

x x

x

x

4

2

2 2

2

21

41

2 1

1

-

-=

+ -

-

=+

=

x x2 2

x 2

^ _h i

3 Find limh

h x hx h2 72 2+ -

h 0

Solution

lim lim

lim

h

h x hx h

h

h hx x

hx x

x

2 7 2 7

2 7

7

2 2 2

2

2

+ -=

+ -

= + -

= -

h 0

h h0 0

^ h

Continuity





EXAMPLES

1 Find lim

x

x

1

12

-

-

x 1


11

y x

x2

=-

- Sketch the curve

Solution


xx2

-

- gives

00

CONTINUED




( )

lim lim

limx

x

x

x x

x11

1

1 1

1

2

x x

x

1

2

1

1

-

-=

-

+ -

= +

=

-

] ]g g

11

y xx2

=-





y xx

x

x x

x

11

1

1 1

1

2

=-

-

=

+

= +

-

-^ ^h h


2 Find limx

x x2

2x 2

2

+

+ -

-


x x2

22

= + -

+

Solution


x x2

22

+

+ - gives

00

lim lim

lim

xx x

x

x x

x

22

2

1 2

1

3

x x

x

2

2

2

2

+

+ -=

+

- +

=

= -

-

- -

-

^^ ^

^

hh h

h2 y

x

x xx

y x

x

x

x

22

2

2

1

1


=+

+ -= -

=+

= -

+ -^ ^h h


Remember that x 1




1 Find(a) lim x 52

+x 4

(b) lim t 7-t 3 -

(c) lim x x2 43+ -

x 2

(d) lim xx x32

+

x 0

(e) limh

h h

2

22

-

- -

h 2

(f) lim y

y

5

1253

-

-

y 5

(g) limx

x x

12 12

+

+ +

x 1-

(h) limx

x x4

2 82

+

+ -

x 4 -

(i) limc

c

4

22

-

-

c 2

(j) limx x

x 12

-

-

x 1

(k) lim h

h h h2 73 2+ -

h 0

(l) limh

hx hx h32 2- +

h 0

(m) limh

hx h x hx h2 3 53 2 2- + -

h 0

(n) lim x c x c 3 3

-

-

x c



discontinuous

(a) 3 y x2= -

(b)1

1 y

x=

+

(c) f x x 1= -] g

(d)4

1 y

x2=

+

(e)4

1

y x2=

-



(a)3

y xx x2

= +

(b)33

y x

x x2

=+

+

(c)1

5 4 y

xx x2

=+

+ +

510 Exercises




Regions

Class Investigation


them all







EXAMPLE


Solution



y

x =3

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4







region

EXAMPLE


Solution



included


shown

y

x

1

1

-2 2 3 4

-3

-4

-5

2

3

4

5

-1

-2

-3-4

-1 y = -1



y -intercept 1-






EXAMPLES


1 y x 2$ +

Solution






y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0



2 x y 2 3 61-

Solution


region


side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g








3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g






y

-2

x

-

1

4


a=

+


domain x xb



1 For each graph



exists


(a)2

y x=

(b)1

y x= -

(c) f xx 1

1=

+] g

(d) f xx 2

3=

-

] g

(e)3 6

1 y

x=

+

(f) f xx 3

2= -

-

] g

(g) f xx 1

4=

-

] g

(h)1

2 y

x= -

+

(i) f xx6 3

2=

-

] g

(j)2

6 y

x= -

+

2 Show that f x x2

=] g is an odd

function



(a) f xx2 5

1=

+

] g for x2 2 -

(b)3

1 y

x=

+ for x2 0 -

(c) f xx2 4

5=

-

] g for x3 1 -

57 Exercises




(d) f xx 4

3= -

-

] g for x3 3 -

(e)3 1

2 y

x= -

+ for x0 5



(a)3

y x= for y 1 3

(b)2

y x= - for y 221

- -

(c) f xx 1

1=

-

] g for y 171

- -

(d) f xx2 1

3= -

+

] g for

y 131

- -

(e)3 2

6 y

x=

- for y 1

21

6



Circle gate





Proof

r y

x

( x y)

y

x








c a b

r x y

2 2 2

2 2 2

`

= +

= +

EXAMPLE



Solution


y

x

-2

-2 2

2


one place

y

x

2

2

2

-2

The radius is 4

CONTINUED






Domain 2 2 x x -

Range 2 2 y y -


range y r y r -



Proof


shown

y

x

(a b)

x

y

r

( x y)

a

b x - a

y - b




ndash ndash

c a b

r x a y b

2 2 2

2 2 2

= +

= +] ^g h




EXAMPLES



Solution



y

x 9

9

-9

-9



Domain 9 9 x x -

Range 9 9 y y -



Solution


ndash

ndash ndash

x y

x y

1 2 4

1 2 2

2

2 2

+ + =

+ - =

2

2

] ^

] ]_

g h

g gi

So 1 2a b= = - and 2r =

CONTINUED







y

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

(1 -2)



Domain 1 3 x x -

Range 4 0 y y -


Solution


a b2 1= - = and r 3=

Substituting

ndash ndash

ndash

ndash

x a y b r

x y

x y

2 1 3

2 1 9

2 2 2

2 2 2

2 2

+ =

- - + =

+ + =

] ^]] ^

] ^

g hg g hg h


ndash

ndash

a b a ab b

x x x

x x

a b a ab b

y y y

y y

2

2 2 2 2

4 4

2

1 2 1 1

2 1

So

So

2 2 2

2 2 2

2

2 2 2

2 2 2

2

+ = + +

+ = + +

= + +

= - +

= - +

= - +

]] ] ]]^ ^ ]

gg g ggh h g


ndash

x x y y

x x y y

x x y y

x x y y

4 4 2 1 9

4 2 5 9

4 2 5 9

4 2 4 0

9 9

2

2

2

2

+ + + - + =

+ + - + =

+ + + =

+ + - - =

- -


in Chapter 2




Investigation


two functions





circle






for all values

y

x r

r

-r


Proof

ndash

x y r

y r x y r x

2 2 2

2 2 2

2 2

+ =

=

= -



semi-circles





y 0 for all values

y

x r

-

r

-r


EXAMPLES


1 f x x92

= -] g

Solution



y

x 3

3

-3

Domain 3 3 x x -

Range 0 3 y y




2 y x4 2= - -

Solution



y

x 2

-2

-2

Domain 2 2 x x -

Range 2 0 y y -




range

(a) 9x y 2 2+ =

(b) x y 16 02 2+ =-


(d) 1 9x y 2 2+ + =

] g

(e) ndashx y 2 1 12 2+ + =] ^g h



below the x -axis



range

(a) 25 y x2= - -

(b) 1 y x2= -

(c) 36 y x2= -

(d) 64 y x2= - -

(e) 7 y x2= - -



each circle

(a) 100x y 2 2+ =

(b) 5x y 2 2+ =


(d) ndashx y 5 6 492 2+ + =] ^g h

(e) ndashx y 3 812 2+ =^ h

58 Exercises






grouping symbols)



Centre(c) 1 5-







Centre(j) 4 7- -

^ h and radius 3

Other Graphs




EXAMPLES


Solution



eg When 0x =

y 3

1

c=

=

x

y

1

3

3

1

31

When1

1

= -

=

=

=

-

x 3- 2- 1- 0 1 2 3

y 271

91

31

1 3 9 27



You will meet these

graphs again in the

HSC Course





other point

When

x

y

1

3

3

1

=

=

=

y

x

1

2

1

3







Solution



x minus2 minus1 0 05 1 2 3 4

y minus03 0 03 05 06

y

x

1

2

1 2 3 4

-1




You learned about


stages of maths





range y y 02



range all real y

Cubic function


EXAMPLE



Solution



y minus25 minus6 1 2 3 10 29

y

x

1

1

-2 2 3 4

-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4



If you already

know the shape of


3






Domain and range


EXAMPLE


Solution




function are

f

f

1 1 2

1 21

4 4 2

64 2

66

3

3

- = - +

= - +

=

= +

= +

=

] ]

] ]

g g

g g


these points

y

x

(-1 1)

(4 66)

Range 1 66 y y










positive number

EXAMPLE


Solution


x 4So $

$


(or zero) So y 0$


59 Exercises


(a) 4 3 y x= +

(b) f x 4= -] g

(c) 3x =



(f) f x xx 12 2= - -] g

(g) 64x y 2 2+ =

(h) f t t 4

3=

-] g

(i) ( ) g 2

5zz

= +

(j) | |f x x=] g


(a) y x=

(b) 2 y x= -

(c) | |f x x2 3= -] g

(d) | | 2 y x= -

(e) f x x2 5= - +] g

(f) | | y x5= -

(g) 2 y x=

(h) y 5x= -

(i) f x xx 1

= +] g

(j)2

4 3 y x

x= -


(a) y x x 5 2= -] g


= +-

(d) g x x x164 2= -] g

(e) 49x y 2 2+ =

You may like to


by dividing by x




4 (a) Solve x1 02$-


f x x1 2= -] g


(a) 2 y x x2= - -

(b) g t t t 62= +] g



in each case


x3 3 -


x2 3 -


x2 1 - (d)


x1 5


0 4x


x3 3 -


x1 1 -


x2 3 -


x4 4 -

(j) y x x7 62= - + - in the

domain 0 7x


function y x 1

3=

+





x=] g


function

find its range(b)


plane

(a) f x x4=] g

(b) y x3= -

(c) y x 34= -

(d) 2 p x x3=] g

(e) 1 g x x3= +] g

(f) 100x y 2 2+ =

(g) 2 1 y x= +


y x 1= -









(b) f x x 22= -] g

(c)1

y x=

(d) f x x3=] g

(e) f x 3x=] g

13 (a) Solve x4 02$-


(i) 4 y x2= -

(ii) y x4

2= - -




DID YOU KNOW




a light



Limits






a a1

1

03

Z

=

=

3

3

-


lim a 0x=

x 3-



asymptote

EXAMPLES

1 Find lim x

x x5x 0

2+

Solution


which is undefined


( )

lim lim

lim

xx x

x

x x

x

5 5

5

5

x x

x

0

2

0 1

1

0

+=

+

= +

=

] g




2 Find limx

x

4

22

-

-

x 2

Solution


lim lim

lim

x

x

x x

x

x

4

2

2 2

2

21

41

2 1

1

-

-=

+ -

-

=+

=

x x2 2

x 2

^ _h i

3 Find limh

h x hx h2 72 2+ -

h 0

Solution

lim lim

lim

h

h x hx h

h

h hx x

hx x

x

2 7 2 7

2 7

7

2 2 2

2

2

+ -=

+ -

= + -

= -

h 0

h h0 0

^ h

Continuity





EXAMPLES

1 Find lim

x

x

1

12

-

-

x 1


11

y x

x2

=-

- Sketch the curve

Solution


xx2

-

- gives

00

CONTINUED




( )

lim lim

limx

x

x

x x

x11

1

1 1

1

2

x x

x

1

2

1

1

-

-=

-

+ -

= +

=

-

] ]g g

11

y xx2

=-





y xx

x

x x

x

11

1

1 1

1

2

=-

-

=

+

= +

-

-^ ^h h


2 Find limx

x x2

2x 2

2

+

+ -

-


x x2

22

= + -

+

Solution


x x2

22

+

+ - gives

00

lim lim

lim

xx x

x

x x

x

22

2

1 2

1

3

x x

x

2

2

2

2

+

+ -=

+

- +

=

= -

-

- -

-

^^ ^

^

hh h

h2 y

x

x xx

y x

x

x

x

22

2

2

1

1


=+

+ -= -

=+

= -

+ -^ ^h h


Remember that x 1




1 Find(a) lim x 52

+x 4

(b) lim t 7-t 3 -

(c) lim x x2 43+ -

x 2

(d) lim xx x32

+

x 0

(e) limh

h h

2

22

-

- -

h 2

(f) lim y

y

5

1253

-

-

y 5

(g) limx

x x

12 12

+

+ +

x 1-

(h) limx

x x4

2 82

+

+ -

x 4 -

(i) limc

c

4

22

-

-

c 2

(j) limx x

x 12

-

-

x 1

(k) lim h

h h h2 73 2+ -

h 0

(l) limh

hx hx h32 2- +

h 0

(m) limh

hx h x hx h2 3 53 2 2- + -

h 0

(n) lim x c x c 3 3

-

-

x c



discontinuous

(a) 3 y x2= -

(b)1

1 y

x=

+

(c) f x x 1= -] g

(d)4

1 y

x2=

+

(e)4

1

y x2=

-



(a)3

y xx x2

= +

(b)33

y x

x x2

=+

+

(c)1

5 4 y

xx x2

=+

+ +

510 Exercises




Regions

Class Investigation


them all







EXAMPLE


Solution



y

x =3

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4







region

EXAMPLE


Solution



included


shown

y

x

1

1

-2 2 3 4

-3

-4

-5

2

3

4

5

-1

-2

-3-4

-1 y = -1



y -intercept 1-






EXAMPLES


1 y x 2$ +

Solution






y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0



2 x y 2 3 61-

Solution


region


side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g








3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g






(d) f xx 4

3= -

-

] g for x3 3 -

(e)3 1

2 y

x= -

+ for x0 5



(a)3

y x= for y 1 3

(b)2

y x= - for y 221

- -

(c) f xx 1

1=

-

] g for y 171

- -

(d) f xx2 1

3= -

+

] g for

y 131

- -

(e)3 2

6 y

x=

- for y 1

21

6



Circle gate





Proof

r y

x

( x y)

y

x








c a b

r x y

2 2 2

2 2 2

`

= +

= +

EXAMPLE



Solution


y

x

-2

-2 2

2


one place

y

x

2

2

2

-2

The radius is 4

CONTINUED






Domain 2 2 x x -

Range 2 2 y y -


range y r y r -



Proof


shown

y

x

(a b)

x

y

r

( x y)

a

b x - a

y - b




ndash ndash

c a b

r x a y b

2 2 2

2 2 2

= +

= +] ^g h




EXAMPLES



Solution



y

x 9

9

-9

-9



Domain 9 9 x x -

Range 9 9 y y -



Solution


ndash

ndash ndash

x y

x y

1 2 4

1 2 2

2

2 2

+ + =

+ - =

2

2

] ^

] ]_

g h

g gi

So 1 2a b= = - and 2r =

CONTINUED







y

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

(1 -2)



Domain 1 3 x x -

Range 4 0 y y -


Solution


a b2 1= - = and r 3=

Substituting

ndash ndash

ndash

ndash

x a y b r

x y

x y

2 1 3

2 1 9

2 2 2

2 2 2

2 2

+ =

- - + =

+ + =

] ^]] ^

] ^

g hg g hg h


ndash

ndash

a b a ab b

x x x

x x

a b a ab b

y y y

y y

2

2 2 2 2

4 4

2

1 2 1 1

2 1

So

So

2 2 2

2 2 2

2

2 2 2

2 2 2

2

+ = + +

+ = + +

= + +

= - +

= - +

= - +

]] ] ]]^ ^ ]

gg g ggh h g


ndash

x x y y

x x y y

x x y y

x x y y

4 4 2 1 9

4 2 5 9

4 2 5 9

4 2 4 0

9 9

2

2

2

2

+ + + - + =

+ + - + =

+ + + =

+ + - - =

- -


in Chapter 2




Investigation


two functions





circle






for all values

y

x r

r

-r


Proof

ndash

x y r

y r x y r x

2 2 2

2 2 2

2 2

+ =

=

= -



semi-circles





y 0 for all values

y

x r

-

r

-r


EXAMPLES


1 f x x92

= -] g

Solution



y

x 3

3

-3

Domain 3 3 x x -

Range 0 3 y y




2 y x4 2= - -

Solution



y

x 2

-2

-2

Domain 2 2 x x -

Range 2 0 y y -




range

(a) 9x y 2 2+ =

(b) x y 16 02 2+ =-


(d) 1 9x y 2 2+ + =

] g

(e) ndashx y 2 1 12 2+ + =] ^g h



below the x -axis



range

(a) 25 y x2= - -

(b) 1 y x2= -

(c) 36 y x2= -

(d) 64 y x2= - -

(e) 7 y x2= - -



each circle

(a) 100x y 2 2+ =

(b) 5x y 2 2+ =


(d) ndashx y 5 6 492 2+ + =] ^g h

(e) ndashx y 3 812 2+ =^ h

58 Exercises






grouping symbols)



Centre(c) 1 5-







Centre(j) 4 7- -

^ h and radius 3

Other Graphs




EXAMPLES


Solution



eg When 0x =

y 3

1

c=

=

x

y

1

3

3

1

31

When1

1

= -

=

=

=

-

x 3- 2- 1- 0 1 2 3

y 271

91

31

1 3 9 27



You will meet these

graphs again in the

HSC Course





other point

When

x

y

1

3

3

1

=

=

=

y

x

1

2

1

3







Solution



x minus2 minus1 0 05 1 2 3 4

y minus03 0 03 05 06

y

x

1

2

1 2 3 4

-1




You learned about


stages of maths





range y y 02



range all real y

Cubic function


EXAMPLE



Solution



y minus25 minus6 1 2 3 10 29

y

x

1

1

-2 2 3 4

-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4



If you already

know the shape of


3






Domain and range


EXAMPLE


Solution




function are

f

f

1 1 2

1 21

4 4 2

64 2

66

3

3

- = - +

= - +

=

= +

= +

=

] ]

] ]

g g

g g


these points

y

x

(-1 1)

(4 66)

Range 1 66 y y










positive number

EXAMPLE


Solution


x 4So $

$


(or zero) So y 0$


59 Exercises


(a) 4 3 y x= +

(b) f x 4= -] g

(c) 3x =



(f) f x xx 12 2= - -] g

(g) 64x y 2 2+ =

(h) f t t 4

3=

-] g

(i) ( ) g 2

5zz

= +

(j) | |f x x=] g


(a) y x=

(b) 2 y x= -

(c) | |f x x2 3= -] g

(d) | | 2 y x= -

(e) f x x2 5= - +] g

(f) | | y x5= -

(g) 2 y x=

(h) y 5x= -

(i) f x xx 1

= +] g

(j)2

4 3 y x

x= -


(a) y x x 5 2= -] g


= +-

(d) g x x x164 2= -] g

(e) 49x y 2 2+ =

You may like to


by dividing by x




4 (a) Solve x1 02$-


f x x1 2= -] g


(a) 2 y x x2= - -

(b) g t t t 62= +] g



in each case


x3 3 -


x2 3 -


x2 1 - (d)


x1 5


0 4x


x3 3 -


x1 1 -


x2 3 -


x4 4 -

(j) y x x7 62= - + - in the

domain 0 7x


function y x 1

3=

+





x=] g


function

find its range(b)


plane

(a) f x x4=] g

(b) y x3= -

(c) y x 34= -

(d) 2 p x x3=] g

(e) 1 g x x3= +] g

(f) 100x y 2 2+ =

(g) 2 1 y x= +


y x 1= -









(b) f x x 22= -] g

(c)1

y x=

(d) f x x3=] g

(e) f x 3x=] g

13 (a) Solve x4 02$-


(i) 4 y x2= -

(ii) y x4

2= - -




DID YOU KNOW




a light



Limits






a a1

1

03

Z

=

=

3

3

-


lim a 0x=

x 3-



asymptote

EXAMPLES

1 Find lim x

x x5x 0

2+

Solution


which is undefined


( )

lim lim

lim

xx x

x

x x

x

5 5

5

5

x x

x

0

2

0 1

1

0

+=

+

= +

=

] g




2 Find limx

x

4

22

-

-

x 2

Solution


lim lim

lim

x

x

x x

x

x

4

2

2 2

2

21

41

2 1

1

-

-=

+ -

-

=+

=

x x2 2

x 2

^ _h i

3 Find limh

h x hx h2 72 2+ -

h 0

Solution

lim lim

lim

h

h x hx h

h

h hx x

hx x

x

2 7 2 7

2 7

7

2 2 2

2

2

+ -=

+ -

= + -

= -

h 0

h h0 0

^ h

Continuity





EXAMPLES

1 Find lim

x

x

1

12

-

-

x 1


11

y x

x2

=-

- Sketch the curve

Solution


xx2

-

- gives

00

CONTINUED




( )

lim lim

limx

x

x

x x

x11

1

1 1

1

2

x x

x

1

2

1

1

-

-=

-

+ -

= +

=

-

] ]g g

11

y xx2

=-





y xx

x

x x

x

11

1

1 1

1

2

=-

-

=

+

= +

-

-^ ^h h


2 Find limx

x x2

2x 2

2

+

+ -

-


x x2

22

= + -

+

Solution


x x2

22

+

+ - gives

00

lim lim

lim

xx x

x

x x

x

22

2

1 2

1

3

x x

x

2

2

2

2

+

+ -=

+

- +

=

= -

-

- -

-

^^ ^

^

hh h

h2 y

x

x xx

y x

x

x

x

22

2

2

1

1


=+

+ -= -

=+

= -

+ -^ ^h h


Remember that x 1




1 Find(a) lim x 52

+x 4

(b) lim t 7-t 3 -

(c) lim x x2 43+ -

x 2

(d) lim xx x32

+

x 0

(e) limh

h h

2

22

-

- -

h 2

(f) lim y

y

5

1253

-

-

y 5

(g) limx

x x

12 12

+

+ +

x 1-

(h) limx

x x4

2 82

+

+ -

x 4 -

(i) limc

c

4

22

-

-

c 2

(j) limx x

x 12

-

-

x 1

(k) lim h

h h h2 73 2+ -

h 0

(l) limh

hx hx h32 2- +

h 0

(m) limh

hx h x hx h2 3 53 2 2- + -

h 0

(n) lim x c x c 3 3

-

-

x c



discontinuous

(a) 3 y x2= -

(b)1

1 y

x=

+

(c) f x x 1= -] g

(d)4

1 y

x2=

+

(e)4

1

y x2=

-



(a)3

y xx x2

= +

(b)33

y x

x x2

=+

+

(c)1

5 4 y

xx x2

=+

+ +

510 Exercises




Regions

Class Investigation


them all







EXAMPLE


Solution



y

x =3

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4







region

EXAMPLE


Solution



included


shown

y

x

1

1

-2 2 3 4

-3

-4

-5

2

3

4

5

-1

-2

-3-4

-1 y = -1



y -intercept 1-






EXAMPLES


1 y x 2$ +

Solution






y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0



2 x y 2 3 61-

Solution


region


side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g








3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g










c a b

r x y

2 2 2

2 2 2

`

= +

= +

EXAMPLE



Solution


y

x

-2

-2 2

2


one place

y

x

2

2

2

-2

The radius is 4

CONTINUED






Domain 2 2 x x -

Range 2 2 y y -


range y r y r -



Proof


shown

y

x

(a b)

x

y

r

( x y)

a

b x - a

y - b




ndash ndash

c a b

r x a y b

2 2 2

2 2 2

= +

= +] ^g h




EXAMPLES



Solution



y

x 9

9

-9

-9



Domain 9 9 x x -

Range 9 9 y y -



Solution


ndash

ndash ndash

x y

x y

1 2 4

1 2 2

2

2 2

+ + =

+ - =

2

2

] ^

] ]_

g h

g gi

So 1 2a b= = - and 2r =

CONTINUED







y

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

(1 -2)



Domain 1 3 x x -

Range 4 0 y y -


Solution


a b2 1= - = and r 3=

Substituting

ndash ndash

ndash

ndash

x a y b r

x y

x y

2 1 3

2 1 9

2 2 2

2 2 2

2 2

+ =

- - + =

+ + =

] ^]] ^

] ^

g hg g hg h


ndash

ndash

a b a ab b

x x x

x x

a b a ab b

y y y

y y

2

2 2 2 2

4 4

2

1 2 1 1

2 1

So

So

2 2 2

2 2 2

2

2 2 2

2 2 2

2

+ = + +

+ = + +

= + +

= - +

= - +

= - +

]] ] ]]^ ^ ]

gg g ggh h g


ndash

x x y y

x x y y

x x y y

x x y y

4 4 2 1 9

4 2 5 9

4 2 5 9

4 2 4 0

9 9

2

2

2

2

+ + + - + =

+ + - + =

+ + + =

+ + - - =

- -


in Chapter 2




Investigation


two functions





circle






for all values

y

x r

r

-r


Proof

ndash

x y r

y r x y r x

2 2 2

2 2 2

2 2

+ =

=

= -



semi-circles





y 0 for all values

y

x r

-

r

-r


EXAMPLES


1 f x x92

= -] g

Solution



y

x 3

3

-3

Domain 3 3 x x -

Range 0 3 y y




2 y x4 2= - -

Solution



y

x 2

-2

-2

Domain 2 2 x x -

Range 2 0 y y -




range

(a) 9x y 2 2+ =

(b) x y 16 02 2+ =-


(d) 1 9x y 2 2+ + =

] g

(e) ndashx y 2 1 12 2+ + =] ^g h



below the x -axis



range

(a) 25 y x2= - -

(b) 1 y x2= -

(c) 36 y x2= -

(d) 64 y x2= - -

(e) 7 y x2= - -



each circle

(a) 100x y 2 2+ =

(b) 5x y 2 2+ =


(d) ndashx y 5 6 492 2+ + =] ^g h

(e) ndashx y 3 812 2+ =^ h

58 Exercises






grouping symbols)



Centre(c) 1 5-







Centre(j) 4 7- -

^ h and radius 3

Other Graphs




EXAMPLES


Solution



eg When 0x =

y 3

1

c=

=

x

y

1

3

3

1

31

When1

1

= -

=

=

=

-

x 3- 2- 1- 0 1 2 3

y 271

91

31

1 3 9 27



You will meet these

graphs again in the

HSC Course





other point

When

x

y

1

3

3

1

=

=

=

y

x

1

2

1

3







Solution



x minus2 minus1 0 05 1 2 3 4

y minus03 0 03 05 06

y

x

1

2

1 2 3 4

-1




You learned about


stages of maths





range y y 02



range all real y

Cubic function


EXAMPLE



Solution



y minus25 minus6 1 2 3 10 29

y

x

1

1

-2 2 3 4

-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4



If you already

know the shape of


3






Domain and range


EXAMPLE


Solution




function are

f

f

1 1 2

1 21

4 4 2

64 2

66

3

3

- = - +

= - +

=

= +

= +

=

] ]

] ]

g g

g g


these points

y

x

(-1 1)

(4 66)

Range 1 66 y y










positive number

EXAMPLE


Solution


x 4So $

$


(or zero) So y 0$


59 Exercises


(a) 4 3 y x= +

(b) f x 4= -] g

(c) 3x =



(f) f x xx 12 2= - -] g

(g) 64x y 2 2+ =

(h) f t t 4

3=

-] g

(i) ( ) g 2

5zz

= +

(j) | |f x x=] g


(a) y x=

(b) 2 y x= -

(c) | |f x x2 3= -] g

(d) | | 2 y x= -

(e) f x x2 5= - +] g

(f) | | y x5= -

(g) 2 y x=

(h) y 5x= -

(i) f x xx 1

= +] g

(j)2

4 3 y x

x= -


(a) y x x 5 2= -] g


= +-

(d) g x x x164 2= -] g

(e) 49x y 2 2+ =

You may like to


by dividing by x




4 (a) Solve x1 02$-


f x x1 2= -] g


(a) 2 y x x2= - -

(b) g t t t 62= +] g



in each case


x3 3 -


x2 3 -


x2 1 - (d)


x1 5


0 4x


x3 3 -


x1 1 -


x2 3 -


x4 4 -

(j) y x x7 62= - + - in the

domain 0 7x


function y x 1

3=

+





x=] g


function

find its range(b)


plane

(a) f x x4=] g

(b) y x3= -

(c) y x 34= -

(d) 2 p x x3=] g

(e) 1 g x x3= +] g

(f) 100x y 2 2+ =

(g) 2 1 y x= +


y x 1= -









(b) f x x 22= -] g

(c)1

y x=

(d) f x x3=] g

(e) f x 3x=] g

13 (a) Solve x4 02$-


(i) 4 y x2= -

(ii) y x4

2= - -




DID YOU KNOW




a light



Limits






a a1

1

03

Z

=

=

3

3

-


lim a 0x=

x 3-



asymptote

EXAMPLES

1 Find lim x

x x5x 0

2+

Solution


which is undefined


( )

lim lim

lim

xx x

x

x x

x

5 5

5

5

x x

x

0

2

0 1

1

0

+=

+

= +

=

] g




2 Find limx

x

4

22

-

-

x 2

Solution


lim lim

lim

x

x

x x

x

x

4

2

2 2

2

21

41

2 1

1

-

-=

+ -

-

=+

=

x x2 2

x 2

^ _h i

3 Find limh

h x hx h2 72 2+ -

h 0

Solution

lim lim

lim

h

h x hx h

h

h hx x

hx x

x

2 7 2 7

2 7

7

2 2 2

2

2

+ -=

+ -

= + -

= -

h 0

h h0 0

^ h

Continuity





EXAMPLES

1 Find lim

x

x

1

12

-

-

x 1


11

y x

x2

=-

- Sketch the curve

Solution


xx2

-

- gives

00

CONTINUED




( )

lim lim

limx

x

x

x x

x11

1

1 1

1

2

x x

x

1

2

1

1

-

-=

-

+ -

= +

=

-

] ]g g

11

y xx2

=-





y xx

x

x x

x

11

1

1 1

1

2

=-

-

=

+

= +

-

-^ ^h h


2 Find limx

x x2

2x 2

2

+

+ -

-


x x2

22

= + -

+

Solution


x x2

22

+

+ - gives

00

lim lim

lim

xx x

x

x x

x

22

2

1 2

1

3

x x

x

2

2

2

2

+

+ -=

+

- +

=

= -

-

- -

-

^^ ^

^

hh h

h2 y

x

x xx

y x

x

x

x

22

2

2

1

1


=+

+ -= -

=+

= -

+ -^ ^h h


Remember that x 1




1 Find(a) lim x 52

+x 4

(b) lim t 7-t 3 -

(c) lim x x2 43+ -

x 2

(d) lim xx x32

+

x 0

(e) limh

h h

2

22

-

- -

h 2

(f) lim y

y

5

1253

-

-

y 5

(g) limx

x x

12 12

+

+ +

x 1-

(h) limx

x x4

2 82

+

+ -

x 4 -

(i) limc

c

4

22

-

-

c 2

(j) limx x

x 12

-

-

x 1

(k) lim h

h h h2 73 2+ -

h 0

(l) limh

hx hx h32 2- +

h 0

(m) limh

hx h x hx h2 3 53 2 2- + -

h 0

(n) lim x c x c 3 3

-

-

x c



discontinuous

(a) 3 y x2= -

(b)1

1 y

x=

+

(c) f x x 1= -] g

(d)4

1 y

x2=

+

(e)4

1

y x2=

-



(a)3

y xx x2

= +

(b)33

y x

x x2

=+

+

(c)1

5 4 y

xx x2

=+

+ +

510 Exercises




Regions

Class Investigation


them all







EXAMPLE


Solution



y

x =3

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4







region

EXAMPLE


Solution



included


shown

y

x

1

1

-2 2 3 4

-3

-4

-5

2

3

4

5

-1

-2

-3-4

-1 y = -1



y -intercept 1-






EXAMPLES


1 y x 2$ +

Solution






y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0



2 x y 2 3 61-

Solution


region


side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g








3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g








Domain 2 2 x x -

Range 2 2 y y -


range y r y r -



Proof


shown

y

x

(a b)

x

y

r

( x y)

a

b x - a

y - b




ndash ndash

c a b

r x a y b

2 2 2

2 2 2

= +

= +] ^g h




EXAMPLES



Solution



y

x 9

9

-9

-9



Domain 9 9 x x -

Range 9 9 y y -



Solution


ndash

ndash ndash

x y

x y

1 2 4

1 2 2

2

2 2

+ + =

+ - =

2

2

] ^

] ]_

g h

g gi

So 1 2a b= = - and 2r =

CONTINUED







y

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

(1 -2)



Domain 1 3 x x -

Range 4 0 y y -


Solution


a b2 1= - = and r 3=

Substituting

ndash ndash

ndash

ndash

x a y b r

x y

x y

2 1 3

2 1 9

2 2 2

2 2 2

2 2

+ =

- - + =

+ + =

] ^]] ^

] ^

g hg g hg h


ndash

ndash

a b a ab b

x x x

x x

a b a ab b

y y y

y y

2

2 2 2 2

4 4

2

1 2 1 1

2 1

So

So

2 2 2

2 2 2

2

2 2 2

2 2 2

2

+ = + +

+ = + +

= + +

= - +

= - +

= - +

]] ] ]]^ ^ ]

gg g ggh h g


ndash

x x y y

x x y y

x x y y

x x y y

4 4 2 1 9

4 2 5 9

4 2 5 9

4 2 4 0

9 9

2

2

2

2

+ + + - + =

+ + - + =

+ + + =

+ + - - =

- -


in Chapter 2




Investigation


two functions





circle






for all values

y

x r

r

-r


Proof

ndash

x y r

y r x y r x

2 2 2

2 2 2

2 2

+ =

=

= -



semi-circles





y 0 for all values

y

x r

-

r

-r


EXAMPLES


1 f x x92

= -] g

Solution



y

x 3

3

-3

Domain 3 3 x x -

Range 0 3 y y




2 y x4 2= - -

Solution



y

x 2

-2

-2

Domain 2 2 x x -

Range 2 0 y y -




range

(a) 9x y 2 2+ =

(b) x y 16 02 2+ =-


(d) 1 9x y 2 2+ + =

] g

(e) ndashx y 2 1 12 2+ + =] ^g h



below the x -axis



range

(a) 25 y x2= - -

(b) 1 y x2= -

(c) 36 y x2= -

(d) 64 y x2= - -

(e) 7 y x2= - -



each circle

(a) 100x y 2 2+ =

(b) 5x y 2 2+ =


(d) ndashx y 5 6 492 2+ + =] ^g h

(e) ndashx y 3 812 2+ =^ h

58 Exercises






grouping symbols)



Centre(c) 1 5-







Centre(j) 4 7- -

^ h and radius 3

Other Graphs




EXAMPLES


Solution



eg When 0x =

y 3

1

c=

=

x

y

1

3

3

1

31

When1

1

= -

=

=

=

-

x 3- 2- 1- 0 1 2 3

y 271

91

31

1 3 9 27



You will meet these

graphs again in the

HSC Course





other point

When

x

y

1

3

3

1

=

=

=

y

x

1

2

1

3







Solution



x minus2 minus1 0 05 1 2 3 4

y minus03 0 03 05 06

y

x

1

2

1 2 3 4

-1




You learned about


stages of maths





range y y 02



range all real y

Cubic function


EXAMPLE



Solution



y minus25 minus6 1 2 3 10 29

y

x

1

1

-2 2 3 4

-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4



If you already

know the shape of


3






Domain and range


EXAMPLE


Solution




function are

f

f

1 1 2

1 21

4 4 2

64 2

66

3

3

- = - +

= - +

=

= +

= +

=

] ]

] ]

g g

g g


these points

y

x

(-1 1)

(4 66)

Range 1 66 y y










positive number

EXAMPLE


Solution


x 4So $

$


(or zero) So y 0$


59 Exercises


(a) 4 3 y x= +

(b) f x 4= -] g

(c) 3x =



(f) f x xx 12 2= - -] g

(g) 64x y 2 2+ =

(h) f t t 4

3=

-] g

(i) ( ) g 2

5zz

= +

(j) | |f x x=] g


(a) y x=

(b) 2 y x= -

(c) | |f x x2 3= -] g

(d) | | 2 y x= -

(e) f x x2 5= - +] g

(f) | | y x5= -

(g) 2 y x=

(h) y 5x= -

(i) f x xx 1

= +] g

(j)2

4 3 y x

x= -


(a) y x x 5 2= -] g


= +-

(d) g x x x164 2= -] g

(e) 49x y 2 2+ =

You may like to


by dividing by x




4 (a) Solve x1 02$-


f x x1 2= -] g


(a) 2 y x x2= - -

(b) g t t t 62= +] g



in each case


x3 3 -


x2 3 -


x2 1 - (d)


x1 5


0 4x


x3 3 -


x1 1 -


x2 3 -


x4 4 -

(j) y x x7 62= - + - in the

domain 0 7x


function y x 1

3=

+





x=] g


function

find its range(b)


plane

(a) f x x4=] g

(b) y x3= -

(c) y x 34= -

(d) 2 p x x3=] g

(e) 1 g x x3= +] g

(f) 100x y 2 2+ =

(g) 2 1 y x= +


y x 1= -









(b) f x x 22= -] g

(c)1

y x=

(d) f x x3=] g

(e) f x 3x=] g

13 (a) Solve x4 02$-


(i) 4 y x2= -

(ii) y x4

2= - -




DID YOU KNOW




a light



Limits






a a1

1

03

Z

=

=

3

3

-


lim a 0x=

x 3-



asymptote

EXAMPLES

1 Find lim x

x x5x 0

2+

Solution


which is undefined


( )

lim lim

lim

xx x

x

x x

x

5 5

5

5

x x

x

0

2

0 1

1

0

+=

+

= +

=

] g




2 Find limx

x

4

22

-

-

x 2

Solution


lim lim

lim

x

x

x x

x

x

4

2

2 2

2

21

41

2 1

1

-

-=

+ -

-

=+

=

x x2 2

x 2

^ _h i

3 Find limh

h x hx h2 72 2+ -

h 0

Solution

lim lim

lim

h

h x hx h

h

h hx x

hx x

x

2 7 2 7

2 7

7

2 2 2

2

2

+ -=

+ -

= + -

= -

h 0

h h0 0

^ h

Continuity





EXAMPLES

1 Find lim

x

x

1

12

-

-

x 1


11

y x

x2

=-

- Sketch the curve

Solution


xx2

-

- gives

00

CONTINUED




( )

lim lim

limx

x

x

x x

x11

1

1 1

1

2

x x

x

1

2

1

1

-

-=

-

+ -

= +

=

-

] ]g g

11

y xx2

=-





y xx

x

x x

x

11

1

1 1

1

2

=-

-

=

+

= +

-

-^ ^h h


2 Find limx

x x2

2x 2

2

+

+ -

-


x x2

22

= + -

+

Solution


x x2

22

+

+ - gives

00

lim lim

lim

xx x

x

x x

x

22

2

1 2

1

3

x x

x

2

2

2

2

+

+ -=

+

- +

=

= -

-

- -

-

^^ ^

^

hh h

h2 y

x

x xx

y x

x

x

x

22

2

2

1

1


=+

+ -= -

=+

= -

+ -^ ^h h


Remember that x 1




1 Find(a) lim x 52

+x 4

(b) lim t 7-t 3 -

(c) lim x x2 43+ -

x 2

(d) lim xx x32

+

x 0

(e) limh

h h

2

22

-

- -

h 2

(f) lim y

y

5

1253

-

-

y 5

(g) limx

x x

12 12

+

+ +

x 1-

(h) limx

x x4

2 82

+

+ -

x 4 -

(i) limc

c

4

22

-

-

c 2

(j) limx x

x 12

-

-

x 1

(k) lim h

h h h2 73 2+ -

h 0

(l) limh

hx hx h32 2- +

h 0

(m) limh

hx h x hx h2 3 53 2 2- + -

h 0

(n) lim x c x c 3 3

-

-

x c



discontinuous

(a) 3 y x2= -

(b)1

1 y

x=

+

(c) f x x 1= -] g

(d)4

1 y

x2=

+

(e)4

1

y x2=

-



(a)3

y xx x2

= +

(b)33

y x

x x2

=+

+

(c)1

5 4 y

xx x2

=+

+ +

510 Exercises




Regions

Class Investigation


them all







EXAMPLE


Solution



y

x =3

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4







region

EXAMPLE


Solution



included


shown

y

x

1

1

-2 2 3 4

-3

-4

-5

2

3

4

5

-1

-2

-3-4

-1 y = -1



y -intercept 1-






EXAMPLES


1 y x 2$ +

Solution






y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0



2 x y 2 3 61-

Solution


region


side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g








3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g






EXAMPLES



Solution



y

x 9

9

-9

-9



Domain 9 9 x x -

Range 9 9 y y -



Solution


ndash

ndash ndash

x y

x y

1 2 4

1 2 2

2

2 2

+ + =

+ - =

2

2

] ^

] ]_

g h

g gi

So 1 2a b= = - and 2r =

CONTINUED







y

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

(1 -2)



Domain 1 3 x x -

Range 4 0 y y -


Solution


a b2 1= - = and r 3=

Substituting

ndash ndash

ndash

ndash

x a y b r

x y

x y

2 1 3

2 1 9

2 2 2

2 2 2

2 2

+ =

- - + =

+ + =

] ^]] ^

] ^

g hg g hg h


ndash

ndash

a b a ab b

x x x

x x

a b a ab b

y y y

y y

2

2 2 2 2

4 4

2

1 2 1 1

2 1

So

So

2 2 2

2 2 2

2

2 2 2

2 2 2

2

+ = + +

+ = + +

= + +

= - +

= - +

= - +

]] ] ]]^ ^ ]

gg g ggh h g


ndash

x x y y

x x y y

x x y y

x x y y

4 4 2 1 9

4 2 5 9

4 2 5 9

4 2 4 0

9 9

2

2

2

2

+ + + - + =

+ + - + =

+ + + =

+ + - - =

- -


in Chapter 2




Investigation


two functions





circle






for all values

y

x r

r

-r


Proof

ndash

x y r

y r x y r x

2 2 2

2 2 2

2 2

+ =

=

= -



semi-circles





y 0 for all values

y

x r

-

r

-r


EXAMPLES


1 f x x92

= -] g

Solution



y

x 3

3

-3

Domain 3 3 x x -

Range 0 3 y y




2 y x4 2= - -

Solution



y

x 2

-2

-2

Domain 2 2 x x -

Range 2 0 y y -




range

(a) 9x y 2 2+ =

(b) x y 16 02 2+ =-


(d) 1 9x y 2 2+ + =

] g

(e) ndashx y 2 1 12 2+ + =] ^g h



below the x -axis



range

(a) 25 y x2= - -

(b) 1 y x2= -

(c) 36 y x2= -

(d) 64 y x2= - -

(e) 7 y x2= - -



each circle

(a) 100x y 2 2+ =

(b) 5x y 2 2+ =


(d) ndashx y 5 6 492 2+ + =] ^g h

(e) ndashx y 3 812 2+ =^ h

58 Exercises






grouping symbols)



Centre(c) 1 5-







Centre(j) 4 7- -

^ h and radius 3

Other Graphs




EXAMPLES


Solution



eg When 0x =

y 3

1

c=

=

x

y

1

3

3

1

31

When1

1

= -

=

=

=

-

x 3- 2- 1- 0 1 2 3

y 271

91

31

1 3 9 27



You will meet these

graphs again in the

HSC Course





other point

When

x

y

1

3

3

1

=

=

=

y

x

1

2

1

3







Solution



x minus2 minus1 0 05 1 2 3 4

y minus03 0 03 05 06

y

x

1

2

1 2 3 4

-1




You learned about


stages of maths





range y y 02



range all real y

Cubic function


EXAMPLE



Solution



y minus25 minus6 1 2 3 10 29

y

x

1

1

-2 2 3 4

-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4



If you already

know the shape of


3






Domain and range


EXAMPLE


Solution




function are

f

f

1 1 2

1 21

4 4 2

64 2

66

3

3

- = - +

= - +

=

= +

= +

=

] ]

] ]

g g

g g


these points

y

x

(-1 1)

(4 66)

Range 1 66 y y










positive number

EXAMPLE


Solution


x 4So $

$


(or zero) So y 0$


59 Exercises


(a) 4 3 y x= +

(b) f x 4= -] g

(c) 3x =



(f) f x xx 12 2= - -] g

(g) 64x y 2 2+ =

(h) f t t 4

3=

-] g

(i) ( ) g 2

5zz

= +

(j) | |f x x=] g


(a) y x=

(b) 2 y x= -

(c) | |f x x2 3= -] g

(d) | | 2 y x= -

(e) f x x2 5= - +] g

(f) | | y x5= -

(g) 2 y x=

(h) y 5x= -

(i) f x xx 1

= +] g

(j)2

4 3 y x

x= -


(a) y x x 5 2= -] g


= +-

(d) g x x x164 2= -] g

(e) 49x y 2 2+ =

You may like to


by dividing by x




4 (a) Solve x1 02$-


f x x1 2= -] g


(a) 2 y x x2= - -

(b) g t t t 62= +] g



in each case


x3 3 -


x2 3 -


x2 1 - (d)


x1 5


0 4x


x3 3 -


x1 1 -


x2 3 -


x4 4 -

(j) y x x7 62= - + - in the

domain 0 7x


function y x 1

3=

+





x=] g


function

find its range(b)


plane

(a) f x x4=] g

(b) y x3= -

(c) y x 34= -

(d) 2 p x x3=] g

(e) 1 g x x3= +] g

(f) 100x y 2 2+ =

(g) 2 1 y x= +


y x 1= -









(b) f x x 22= -] g

(c)1

y x=

(d) f x x3=] g

(e) f x 3x=] g

13 (a) Solve x4 02$-


(i) 4 y x2= -

(ii) y x4

2= - -




DID YOU KNOW




a light



Limits






a a1

1

03

Z

=

=

3

3

-


lim a 0x=

x 3-



asymptote

EXAMPLES

1 Find lim x

x x5x 0

2+

Solution


which is undefined


( )

lim lim

lim

xx x

x

x x

x

5 5

5

5

x x

x

0

2

0 1

1

0

+=

+

= +

=

] g




2 Find limx

x

4

22

-

-

x 2

Solution


lim lim

lim

x

x

x x

x

x

4

2

2 2

2

21

41

2 1

1

-

-=

+ -

-

=+

=

x x2 2

x 2

^ _h i

3 Find limh

h x hx h2 72 2+ -

h 0

Solution

lim lim

lim

h

h x hx h

h

h hx x

hx x

x

2 7 2 7

2 7

7

2 2 2

2

2

+ -=

+ -

= + -

= -

h 0

h h0 0

^ h

Continuity





EXAMPLES

1 Find lim

x

x

1

12

-

-

x 1


11

y x

x2

=-

- Sketch the curve

Solution


xx2

-

- gives

00

CONTINUED




( )

lim lim

limx

x

x

x x

x11

1

1 1

1

2

x x

x

1

2

1

1

-

-=

-

+ -

= +

=

-

] ]g g

11

y xx2

=-





y xx

x

x x

x

11

1

1 1

1

2

=-

-

=

+

= +

-

-^ ^h h


2 Find limx

x x2

2x 2

2

+

+ -

-


x x2

22

= + -

+

Solution


x x2

22

+

+ - gives

00

lim lim

lim

xx x

x

x x

x

22

2

1 2

1

3

x x

x

2

2

2

2

+

+ -=

+

- +

=

= -

-

- -

-

^^ ^

^

hh h

h2 y

x

x xx

y x

x

x

x

22

2

2

1

1


=+

+ -= -

=+

= -

+ -^ ^h h


Remember that x 1




1 Find(a) lim x 52

+x 4

(b) lim t 7-t 3 -

(c) lim x x2 43+ -

x 2

(d) lim xx x32

+

x 0

(e) limh

h h

2

22

-

- -

h 2

(f) lim y

y

5

1253

-

-

y 5

(g) limx

x x

12 12

+

+ +

x 1-

(h) limx

x x4

2 82

+

+ -

x 4 -

(i) limc

c

4

22

-

-

c 2

(j) limx x

x 12

-

-

x 1

(k) lim h

h h h2 73 2+ -

h 0

(l) limh

hx hx h32 2- +

h 0

(m) limh

hx h x hx h2 3 53 2 2- + -

h 0

(n) lim x c x c 3 3

-

-

x c



discontinuous

(a) 3 y x2= -

(b)1

1 y

x=

+

(c) f x x 1= -] g

(d)4

1 y

x2=

+

(e)4

1

y x2=

-



(a)3

y xx x2

= +

(b)33

y x

x x2

=+

+

(c)1

5 4 y

xx x2

=+

+ +

510 Exercises




Regions

Class Investigation


them all







EXAMPLE


Solution



y

x =3

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4







region

EXAMPLE


Solution



included


shown

y

x

1

1

-2 2 3 4

-3

-4

-5

2

3

4

5

-1

-2

-3-4

-1 y = -1



y -intercept 1-






EXAMPLES


1 y x 2$ +

Solution






y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0



2 x y 2 3 61-

Solution


region


side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g








3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g









y

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

(1 -2)



Domain 1 3 x x -

Range 4 0 y y -


Solution


a b2 1= - = and r 3=

Substituting

ndash ndash

ndash

ndash

x a y b r

x y

x y

2 1 3

2 1 9

2 2 2

2 2 2

2 2

+ =

- - + =

+ + =

] ^]] ^

] ^

g hg g hg h


ndash

ndash

a b a ab b

x x x

x x

a b a ab b

y y y

y y

2

2 2 2 2

4 4

2

1 2 1 1

2 1

So

So

2 2 2

2 2 2

2

2 2 2

2 2 2

2

+ = + +

+ = + +

= + +

= - +

= - +

= - +

]] ] ]]^ ^ ]

gg g ggh h g


ndash

x x y y

x x y y

x x y y

x x y y

4 4 2 1 9

4 2 5 9

4 2 5 9

4 2 4 0

9 9

2

2

2

2

+ + + - + =

+ + - + =

+ + + =

+ + - - =

- -


in Chapter 2




Investigation


two functions





circle






for all values

y

x r

r

-r


Proof

ndash

x y r

y r x y r x

2 2 2

2 2 2

2 2

+ =

=

= -



semi-circles





y 0 for all values

y

x r

-

r

-r


EXAMPLES


1 f x x92

= -] g

Solution



y

x 3

3

-3

Domain 3 3 x x -

Range 0 3 y y




2 y x4 2= - -

Solution



y

x 2

-2

-2

Domain 2 2 x x -

Range 2 0 y y -




range

(a) 9x y 2 2+ =

(b) x y 16 02 2+ =-


(d) 1 9x y 2 2+ + =

] g

(e) ndashx y 2 1 12 2+ + =] ^g h



below the x -axis



range

(a) 25 y x2= - -

(b) 1 y x2= -

(c) 36 y x2= -

(d) 64 y x2= - -

(e) 7 y x2= - -



each circle

(a) 100x y 2 2+ =

(b) 5x y 2 2+ =


(d) ndashx y 5 6 492 2+ + =] ^g h

(e) ndashx y 3 812 2+ =^ h

58 Exercises






grouping symbols)



Centre(c) 1 5-







Centre(j) 4 7- -

^ h and radius 3

Other Graphs




EXAMPLES


Solution



eg When 0x =

y 3

1

c=

=

x

y

1

3

3

1

31

When1

1

= -

=

=

=

-

x 3- 2- 1- 0 1 2 3

y 271

91

31

1 3 9 27



You will meet these

graphs again in the

HSC Course





other point

When

x

y

1

3

3

1

=

=

=

y

x

1

2

1

3







Solution



x minus2 minus1 0 05 1 2 3 4

y minus03 0 03 05 06

y

x

1

2

1 2 3 4

-1




You learned about


stages of maths





range y y 02



range all real y

Cubic function


EXAMPLE



Solution



y minus25 minus6 1 2 3 10 29

y

x

1

1

-2 2 3 4

-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4



If you already

know the shape of


3






Domain and range


EXAMPLE


Solution




function are

f

f

1 1 2

1 21

4 4 2

64 2

66

3

3

- = - +

= - +

=

= +

= +

=

] ]

] ]

g g

g g


these points

y

x

(-1 1)

(4 66)

Range 1 66 y y










positive number

EXAMPLE


Solution


x 4So $

$


(or zero) So y 0$


59 Exercises


(a) 4 3 y x= +

(b) f x 4= -] g

(c) 3x =



(f) f x xx 12 2= - -] g

(g) 64x y 2 2+ =

(h) f t t 4

3=

-] g

(i) ( ) g 2

5zz

= +

(j) | |f x x=] g


(a) y x=

(b) 2 y x= -

(c) | |f x x2 3= -] g

(d) | | 2 y x= -

(e) f x x2 5= - +] g

(f) | | y x5= -

(g) 2 y x=

(h) y 5x= -

(i) f x xx 1

= +] g

(j)2

4 3 y x

x= -


(a) y x x 5 2= -] g


= +-

(d) g x x x164 2= -] g

(e) 49x y 2 2+ =

You may like to


by dividing by x




4 (a) Solve x1 02$-


f x x1 2= -] g


(a) 2 y x x2= - -

(b) g t t t 62= +] g



in each case


x3 3 -


x2 3 -


x2 1 - (d)


x1 5


0 4x


x3 3 -


x1 1 -


x2 3 -


x4 4 -

(j) y x x7 62= - + - in the

domain 0 7x


function y x 1

3=

+





x=] g


function

find its range(b)


plane

(a) f x x4=] g

(b) y x3= -

(c) y x 34= -

(d) 2 p x x3=] g

(e) 1 g x x3= +] g

(f) 100x y 2 2+ =

(g) 2 1 y x= +


y x 1= -









(b) f x x 22= -] g

(c)1

y x=

(d) f x x3=] g

(e) f x 3x=] g

13 (a) Solve x4 02$-


(i) 4 y x2= -

(ii) y x4

2= - -




DID YOU KNOW




a light



Limits






a a1

1

03

Z

=

=

3

3

-


lim a 0x=

x 3-



asymptote

EXAMPLES

1 Find lim x

x x5x 0

2+

Solution


which is undefined


( )

lim lim

lim

xx x

x

x x

x

5 5

5

5

x x

x

0

2

0 1

1

0

+=

+

= +

=

] g




2 Find limx

x

4

22

-

-

x 2

Solution


lim lim

lim

x

x

x x

x

x

4

2

2 2

2

21

41

2 1

1

-

-=

+ -

-

=+

=

x x2 2

x 2

^ _h i

3 Find limh

h x hx h2 72 2+ -

h 0

Solution

lim lim

lim

h

h x hx h

h

h hx x

hx x

x

2 7 2 7

2 7

7

2 2 2

2

2

+ -=

+ -

= + -

= -

h 0

h h0 0

^ h

Continuity





EXAMPLES

1 Find lim

x

x

1

12

-

-

x 1


11

y x

x2

=-

- Sketch the curve

Solution


xx2

-

- gives

00

CONTINUED




( )

lim lim

limx

x

x

x x

x11

1

1 1

1

2

x x

x

1

2

1

1

-

-=

-

+ -

= +

=

-

] ]g g

11

y xx2

=-





y xx

x

x x

x

11

1

1 1

1

2

=-

-

=

+

= +

-

-^ ^h h


2 Find limx

x x2

2x 2

2

+

+ -

-


x x2

22

= + -

+

Solution


x x2

22

+

+ - gives

00

lim lim

lim

xx x

x

x x

x

22

2

1 2

1

3

x x

x

2

2

2

2

+

+ -=

+

- +

=

= -

-

- -

-

^^ ^

^

hh h

h2 y

x

x xx

y x

x

x

x

22

2

2

1

1


=+

+ -= -

=+

= -

+ -^ ^h h


Remember that x 1




1 Find(a) lim x 52

+x 4

(b) lim t 7-t 3 -

(c) lim x x2 43+ -

x 2

(d) lim xx x32

+

x 0

(e) limh

h h

2

22

-

- -

h 2

(f) lim y

y

5

1253

-

-

y 5

(g) limx

x x

12 12

+

+ +

x 1-

(h) limx

x x4

2 82

+

+ -

x 4 -

(i) limc

c

4

22

-

-

c 2

(j) limx x

x 12

-

-

x 1

(k) lim h

h h h2 73 2+ -

h 0

(l) limh

hx hx h32 2- +

h 0

(m) limh

hx h x hx h2 3 53 2 2- + -

h 0

(n) lim x c x c 3 3

-

-

x c



discontinuous

(a) 3 y x2= -

(b)1

1 y

x=

+

(c) f x x 1= -] g

(d)4

1 y

x2=

+

(e)4

1

y x2=

-



(a)3

y xx x2

= +

(b)33

y x

x x2

=+

+

(c)1

5 4 y

xx x2

=+

+ +

510 Exercises




Regions

Class Investigation


them all







EXAMPLE


Solution



y

x =3

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4







region

EXAMPLE


Solution



included


shown

y

x

1

1

-2 2 3 4

-3

-4

-5

2

3

4

5

-1

-2

-3-4

-1 y = -1



y -intercept 1-






EXAMPLES


1 y x 2$ +

Solution






y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0



2 x y 2 3 61-

Solution


region


side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g








3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g






Investigation


two functions





circle






for all values

y

x r

r

-r


Proof

ndash

x y r

y r x y r x

2 2 2

2 2 2

2 2

+ =

=

= -



semi-circles





y 0 for all values

y

x r

-

r

-r


EXAMPLES


1 f x x92

= -] g

Solution



y

x 3

3

-3

Domain 3 3 x x -

Range 0 3 y y




2 y x4 2= - -

Solution



y

x 2

-2

-2

Domain 2 2 x x -

Range 2 0 y y -




range

(a) 9x y 2 2+ =

(b) x y 16 02 2+ =-


(d) 1 9x y 2 2+ + =

] g

(e) ndashx y 2 1 12 2+ + =] ^g h



below the x -axis



range

(a) 25 y x2= - -

(b) 1 y x2= -

(c) 36 y x2= -

(d) 64 y x2= - -

(e) 7 y x2= - -



each circle

(a) 100x y 2 2+ =

(b) 5x y 2 2+ =


(d) ndashx y 5 6 492 2+ + =] ^g h

(e) ndashx y 3 812 2+ =^ h

58 Exercises






grouping symbols)



Centre(c) 1 5-







Centre(j) 4 7- -

^ h and radius 3

Other Graphs




EXAMPLES


Solution



eg When 0x =

y 3

1

c=

=

x

y

1

3

3

1

31

When1

1

= -

=

=

=

-

x 3- 2- 1- 0 1 2 3

y 271

91

31

1 3 9 27



You will meet these

graphs again in the

HSC Course





other point

When

x

y

1

3

3

1

=

=

=

y

x

1

2

1

3







Solution



x minus2 minus1 0 05 1 2 3 4

y minus03 0 03 05 06

y

x

1

2

1 2 3 4

-1




You learned about


stages of maths





range y y 02



range all real y

Cubic function


EXAMPLE



Solution



y minus25 minus6 1 2 3 10 29

y

x

1

1

-2 2 3 4

-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4



If you already

know the shape of


3






Domain and range


EXAMPLE


Solution




function are

f

f

1 1 2

1 21

4 4 2

64 2

66

3

3

- = - +

= - +

=

= +

= +

=

] ]

] ]

g g

g g


these points

y

x

(-1 1)

(4 66)

Range 1 66 y y










positive number

EXAMPLE


Solution


x 4So $

$


(or zero) So y 0$


59 Exercises


(a) 4 3 y x= +

(b) f x 4= -] g

(c) 3x =



(f) f x xx 12 2= - -] g

(g) 64x y 2 2+ =

(h) f t t 4

3=

-] g

(i) ( ) g 2

5zz

= +

(j) | |f x x=] g


(a) y x=

(b) 2 y x= -

(c) | |f x x2 3= -] g

(d) | | 2 y x= -

(e) f x x2 5= - +] g

(f) | | y x5= -

(g) 2 y x=

(h) y 5x= -

(i) f x xx 1

= +] g

(j)2

4 3 y x

x= -


(a) y x x 5 2= -] g


= +-

(d) g x x x164 2= -] g

(e) 49x y 2 2+ =

You may like to


by dividing by x




4 (a) Solve x1 02$-


f x x1 2= -] g


(a) 2 y x x2= - -

(b) g t t t 62= +] g



in each case


x3 3 -


x2 3 -


x2 1 - (d)


x1 5


0 4x


x3 3 -


x1 1 -


x2 3 -


x4 4 -

(j) y x x7 62= - + - in the

domain 0 7x


function y x 1

3=

+





x=] g


function

find its range(b)


plane

(a) f x x4=] g

(b) y x3= -

(c) y x 34= -

(d) 2 p x x3=] g

(e) 1 g x x3= +] g

(f) 100x y 2 2+ =

(g) 2 1 y x= +


y x 1= -









(b) f x x 22= -] g

(c)1

y x=

(d) f x x3=] g

(e) f x 3x=] g

13 (a) Solve x4 02$-


(i) 4 y x2= -

(ii) y x4

2= - -




DID YOU KNOW




a light



Limits






a a1

1

03

Z

=

=

3

3

-


lim a 0x=

x 3-



asymptote

EXAMPLES

1 Find lim x

x x5x 0

2+

Solution


which is undefined


( )

lim lim

lim

xx x

x

x x

x

5 5

5

5

x x

x

0

2

0 1

1

0

+=

+

= +

=

] g




2 Find limx

x

4

22

-

-

x 2

Solution


lim lim

lim

x

x

x x

x

x

4

2

2 2

2

21

41

2 1

1

-

-=

+ -

-

=+

=

x x2 2

x 2

^ _h i

3 Find limh

h x hx h2 72 2+ -

h 0

Solution

lim lim

lim

h

h x hx h

h

h hx x

hx x

x

2 7 2 7

2 7

7

2 2 2

2

2

+ -=

+ -

= + -

= -

h 0

h h0 0

^ h

Continuity





EXAMPLES

1 Find lim

x

x

1

12

-

-

x 1


11

y x

x2

=-

- Sketch the curve

Solution


xx2

-

- gives

00

CONTINUED




( )

lim lim

limx

x

x

x x

x11

1

1 1

1

2

x x

x

1

2

1

1

-

-=

-

+ -

= +

=

-

] ]g g

11

y xx2

=-





y xx

x

x x

x

11

1

1 1

1

2

=-

-

=

+

= +

-

-^ ^h h


2 Find limx

x x2

2x 2

2

+

+ -

-


x x2

22

= + -

+

Solution


x x2

22

+

+ - gives

00

lim lim

lim

xx x

x

x x

x

22

2

1 2

1

3

x x

x

2

2

2

2

+

+ -=

+

- +

=

= -

-

- -

-

^^ ^

^

hh h

h2 y

x

x xx

y x

x

x

x

22

2

2

1

1


=+

+ -= -

=+

= -

+ -^ ^h h


Remember that x 1




1 Find(a) lim x 52

+x 4

(b) lim t 7-t 3 -

(c) lim x x2 43+ -

x 2

(d) lim xx x32

+

x 0

(e) limh

h h

2

22

-

- -

h 2

(f) lim y

y

5

1253

-

-

y 5

(g) limx

x x

12 12

+

+ +

x 1-

(h) limx

x x4

2 82

+

+ -

x 4 -

(i) limc

c

4

22

-

-

c 2

(j) limx x

x 12

-

-

x 1

(k) lim h

h h h2 73 2+ -

h 0

(l) limh

hx hx h32 2- +

h 0

(m) limh

hx h x hx h2 3 53 2 2- + -

h 0

(n) lim x c x c 3 3

-

-

x c



discontinuous

(a) 3 y x2= -

(b)1

1 y

x=

+

(c) f x x 1= -] g

(d)4

1 y

x2=

+

(e)4

1

y x2=

-



(a)3

y xx x2

= +

(b)33

y x

x x2

=+

+

(c)1

5 4 y

xx x2

=+

+ +

510 Exercises




Regions

Class Investigation


them all







EXAMPLE


Solution



y

x =3

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4







region

EXAMPLE


Solution



included


shown

y

x

1

1

-2 2 3 4

-3

-4

-5

2

3

4

5

-1

-2

-3-4

-1 y = -1



y -intercept 1-






EXAMPLES


1 y x 2$ +

Solution






y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0



2 x y 2 3 61-

Solution


region


side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g








3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g







y 0 for all values

y

x r

-

r

-r


EXAMPLES


1 f x x92

= -] g

Solution



y

x 3

3

-3

Domain 3 3 x x -

Range 0 3 y y




2 y x4 2= - -

Solution



y

x 2

-2

-2

Domain 2 2 x x -

Range 2 0 y y -




range

(a) 9x y 2 2+ =

(b) x y 16 02 2+ =-


(d) 1 9x y 2 2+ + =

] g

(e) ndashx y 2 1 12 2+ + =] ^g h



below the x -axis



range

(a) 25 y x2= - -

(b) 1 y x2= -

(c) 36 y x2= -

(d) 64 y x2= - -

(e) 7 y x2= - -



each circle

(a) 100x y 2 2+ =

(b) 5x y 2 2+ =


(d) ndashx y 5 6 492 2+ + =] ^g h

(e) ndashx y 3 812 2+ =^ h

58 Exercises






grouping symbols)



Centre(c) 1 5-







Centre(j) 4 7- -

^ h and radius 3

Other Graphs




EXAMPLES


Solution



eg When 0x =

y 3

1

c=

=

x

y

1

3

3

1

31

When1

1

= -

=

=

=

-

x 3- 2- 1- 0 1 2 3

y 271

91

31

1 3 9 27



You will meet these

graphs again in the

HSC Course





other point

When

x

y

1

3

3

1

=

=

=

y

x

1

2

1

3







Solution



x minus2 minus1 0 05 1 2 3 4

y minus03 0 03 05 06

y

x

1

2

1 2 3 4

-1




You learned about


stages of maths





range y y 02



range all real y

Cubic function


EXAMPLE



Solution



y minus25 minus6 1 2 3 10 29

y

x

1

1

-2 2 3 4

-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4



If you already

know the shape of


3






Domain and range


EXAMPLE


Solution




function are

f

f

1 1 2

1 21

4 4 2

64 2

66

3

3

- = - +

= - +

=

= +

= +

=

] ]

] ]

g g

g g


these points

y

x

(-1 1)

(4 66)

Range 1 66 y y










positive number

EXAMPLE


Solution


x 4So $

$


(or zero) So y 0$


59 Exercises


(a) 4 3 y x= +

(b) f x 4= -] g

(c) 3x =



(f) f x xx 12 2= - -] g

(g) 64x y 2 2+ =

(h) f t t 4

3=

-] g

(i) ( ) g 2

5zz

= +

(j) | |f x x=] g


(a) y x=

(b) 2 y x= -

(c) | |f x x2 3= -] g

(d) | | 2 y x= -

(e) f x x2 5= - +] g

(f) | | y x5= -

(g) 2 y x=

(h) y 5x= -

(i) f x xx 1

= +] g

(j)2

4 3 y x

x= -


(a) y x x 5 2= -] g


= +-

(d) g x x x164 2= -] g

(e) 49x y 2 2+ =

You may like to


by dividing by x




4 (a) Solve x1 02$-


f x x1 2= -] g


(a) 2 y x x2= - -

(b) g t t t 62= +] g



in each case


x3 3 -


x2 3 -


x2 1 - (d)


x1 5


0 4x


x3 3 -


x1 1 -


x2 3 -


x4 4 -

(j) y x x7 62= - + - in the

domain 0 7x


function y x 1

3=

+





x=] g


function

find its range(b)


plane

(a) f x x4=] g

(b) y x3= -

(c) y x 34= -

(d) 2 p x x3=] g

(e) 1 g x x3= +] g

(f) 100x y 2 2+ =

(g) 2 1 y x= +


y x 1= -









(b) f x x 22= -] g

(c)1

y x=

(d) f x x3=] g

(e) f x 3x=] g

13 (a) Solve x4 02$-


(i) 4 y x2= -

(ii) y x4

2= - -




DID YOU KNOW




a light



Limits






a a1

1

03

Z

=

=

3

3

-


lim a 0x=

x 3-



asymptote

EXAMPLES

1 Find lim x

x x5x 0

2+

Solution


which is undefined


( )

lim lim

lim

xx x

x

x x

x

5 5

5

5

x x

x

0

2

0 1

1

0

+=

+

= +

=

] g




2 Find limx

x

4

22

-

-

x 2

Solution


lim lim

lim

x

x

x x

x

x

4

2

2 2

2

21

41

2 1

1

-

-=

+ -

-

=+

=

x x2 2

x 2

^ _h i

3 Find limh

h x hx h2 72 2+ -

h 0

Solution

lim lim

lim

h

h x hx h

h

h hx x

hx x

x

2 7 2 7

2 7

7

2 2 2

2

2

+ -=

+ -

= + -

= -

h 0

h h0 0

^ h

Continuity





EXAMPLES

1 Find lim

x

x

1

12

-

-

x 1


11

y x

x2

=-

- Sketch the curve

Solution


xx2

-

- gives

00

CONTINUED




( )

lim lim

limx

x

x

x x

x11

1

1 1

1

2

x x

x

1

2

1

1

-

-=

-

+ -

= +

=

-

] ]g g

11

y xx2

=-





y xx

x

x x

x

11

1

1 1

1

2

=-

-

=

+

= +

-

-^ ^h h


2 Find limx

x x2

2x 2

2

+

+ -

-


x x2

22

= + -

+

Solution


x x2

22

+

+ - gives

00

lim lim

lim

xx x

x

x x

x

22

2

1 2

1

3

x x

x

2

2

2

2

+

+ -=

+

- +

=

= -

-

- -

-

^^ ^

^

hh h

h2 y

x

x xx

y x

x

x

x

22

2

2

1

1


=+

+ -= -

=+

= -

+ -^ ^h h


Remember that x 1




1 Find(a) lim x 52

+x 4

(b) lim t 7-t 3 -

(c) lim x x2 43+ -

x 2

(d) lim xx x32

+

x 0

(e) limh

h h

2

22

-

- -

h 2

(f) lim y

y

5

1253

-

-

y 5

(g) limx

x x

12 12

+

+ +

x 1-

(h) limx

x x4

2 82

+

+ -

x 4 -

(i) limc

c

4

22

-

-

c 2

(j) limx x

x 12

-

-

x 1

(k) lim h

h h h2 73 2+ -

h 0

(l) limh

hx hx h32 2- +

h 0

(m) limh

hx h x hx h2 3 53 2 2- + -

h 0

(n) lim x c x c 3 3

-

-

x c



discontinuous

(a) 3 y x2= -

(b)1

1 y

x=

+

(c) f x x 1= -] g

(d)4

1 y

x2=

+

(e)4

1

y x2=

-



(a)3

y xx x2

= +

(b)33

y x

x x2

=+

+

(c)1

5 4 y

xx x2

=+

+ +

510 Exercises




Regions

Class Investigation


them all







EXAMPLE


Solution



y

x =3

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4







region

EXAMPLE


Solution



included


shown

y

x

1

1

-2 2 3 4

-3

-4

-5

2

3

4

5

-1

-2

-3-4

-1 y = -1



y -intercept 1-






EXAMPLES


1 y x 2$ +

Solution






y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0



2 x y 2 3 61-

Solution


region


side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g








3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g






2 y x4 2= - -

Solution



y

x 2

-2

-2

Domain 2 2 x x -

Range 2 0 y y -




range

(a) 9x y 2 2+ =

(b) x y 16 02 2+ =-


(d) 1 9x y 2 2+ + =

] g

(e) ndashx y 2 1 12 2+ + =] ^g h



below the x -axis



range

(a) 25 y x2= - -

(b) 1 y x2= -

(c) 36 y x2= -

(d) 64 y x2= - -

(e) 7 y x2= - -



each circle

(a) 100x y 2 2+ =

(b) 5x y 2 2+ =


(d) ndashx y 5 6 492 2+ + =] ^g h

(e) ndashx y 3 812 2+ =^ h

58 Exercises






grouping symbols)



Centre(c) 1 5-







Centre(j) 4 7- -

^ h and radius 3

Other Graphs




EXAMPLES


Solution



eg When 0x =

y 3

1

c=

=

x

y

1

3

3

1

31

When1

1

= -

=

=

=

-

x 3- 2- 1- 0 1 2 3

y 271

91

31

1 3 9 27



You will meet these

graphs again in the

HSC Course





other point

When

x

y

1

3

3

1

=

=

=

y

x

1

2

1

3







Solution



x minus2 minus1 0 05 1 2 3 4

y minus03 0 03 05 06

y

x

1

2

1 2 3 4

-1




You learned about


stages of maths





range y y 02



range all real y

Cubic function


EXAMPLE



Solution



y minus25 minus6 1 2 3 10 29

y

x

1

1

-2 2 3 4

-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4



If you already

know the shape of


3






Domain and range


EXAMPLE


Solution




function are

f

f

1 1 2

1 21

4 4 2

64 2

66

3

3

- = - +

= - +

=

= +

= +

=

] ]

] ]

g g

g g


these points

y

x

(-1 1)

(4 66)

Range 1 66 y y










positive number

EXAMPLE


Solution


x 4So $

$


(or zero) So y 0$


59 Exercises


(a) 4 3 y x= +

(b) f x 4= -] g

(c) 3x =



(f) f x xx 12 2= - -] g

(g) 64x y 2 2+ =

(h) f t t 4

3=

-] g

(i) ( ) g 2

5zz

= +

(j) | |f x x=] g


(a) y x=

(b) 2 y x= -

(c) | |f x x2 3= -] g

(d) | | 2 y x= -

(e) f x x2 5= - +] g

(f) | | y x5= -

(g) 2 y x=

(h) y 5x= -

(i) f x xx 1

= +] g

(j)2

4 3 y x

x= -


(a) y x x 5 2= -] g


= +-

(d) g x x x164 2= -] g

(e) 49x y 2 2+ =

You may like to


by dividing by x




4 (a) Solve x1 02$-


f x x1 2= -] g


(a) 2 y x x2= - -

(b) g t t t 62= +] g



in each case


x3 3 -


x2 3 -


x2 1 - (d)


x1 5


0 4x


x3 3 -


x1 1 -


x2 3 -


x4 4 -

(j) y x x7 62= - + - in the

domain 0 7x


function y x 1

3=

+





x=] g


function

find its range(b)


plane

(a) f x x4=] g

(b) y x3= -

(c) y x 34= -

(d) 2 p x x3=] g

(e) 1 g x x3= +] g

(f) 100x y 2 2+ =

(g) 2 1 y x= +


y x 1= -









(b) f x x 22= -] g

(c)1

y x=

(d) f x x3=] g

(e) f x 3x=] g

13 (a) Solve x4 02$-


(i) 4 y x2= -

(ii) y x4

2= - -




DID YOU KNOW




a light



Limits






a a1

1

03

Z

=

=

3

3

-


lim a 0x=

x 3-



asymptote

EXAMPLES

1 Find lim x

x x5x 0

2+

Solution


which is undefined


( )

lim lim

lim

xx x

x

x x

x

5 5

5

5

x x

x

0

2

0 1

1

0

+=

+

= +

=

] g




2 Find limx

x

4

22

-

-

x 2

Solution


lim lim

lim

x

x

x x

x

x

4

2

2 2

2

21

41

2 1

1

-

-=

+ -

-

=+

=

x x2 2

x 2

^ _h i

3 Find limh

h x hx h2 72 2+ -

h 0

Solution

lim lim

lim

h

h x hx h

h

h hx x

hx x

x

2 7 2 7

2 7

7

2 2 2

2

2

+ -=

+ -

= + -

= -

h 0

h h0 0

^ h

Continuity





EXAMPLES

1 Find lim

x

x

1

12

-

-

x 1


11

y x

x2

=-

- Sketch the curve

Solution


xx2

-

- gives

00

CONTINUED




( )

lim lim

limx

x

x

x x

x11

1

1 1

1

2

x x

x

1

2

1

1

-

-=

-

+ -

= +

=

-

] ]g g

11

y xx2

=-





y xx

x

x x

x

11

1

1 1

1

2

=-

-

=

+

= +

-

-^ ^h h


2 Find limx

x x2

2x 2

2

+

+ -

-


x x2

22

= + -

+

Solution


x x2

22

+

+ - gives

00

lim lim

lim

xx x

x

x x

x

22

2

1 2

1

3

x x

x

2

2

2

2

+

+ -=

+

- +

=

= -

-

- -

-

^^ ^

^

hh h

h2 y

x

x xx

y x

x

x

x

22

2

2

1

1


=+

+ -= -

=+

= -

+ -^ ^h h


Remember that x 1




1 Find(a) lim x 52

+x 4

(b) lim t 7-t 3 -

(c) lim x x2 43+ -

x 2

(d) lim xx x32

+

x 0

(e) limh

h h

2

22

-

- -

h 2

(f) lim y

y

5

1253

-

-

y 5

(g) limx

x x

12 12

+

+ +

x 1-

(h) limx

x x4

2 82

+

+ -

x 4 -

(i) limc

c

4

22

-

-

c 2

(j) limx x

x 12

-

-

x 1

(k) lim h

h h h2 73 2+ -

h 0

(l) limh

hx hx h32 2- +

h 0

(m) limh

hx h x hx h2 3 53 2 2- + -

h 0

(n) lim x c x c 3 3

-

-

x c



discontinuous

(a) 3 y x2= -

(b)1

1 y

x=

+

(c) f x x 1= -] g

(d)4

1 y

x2=

+

(e)4

1

y x2=

-



(a)3

y xx x2

= +

(b)33

y x

x x2

=+

+

(c)1

5 4 y

xx x2

=+

+ +

510 Exercises




Regions

Class Investigation


them all







EXAMPLE


Solution



y

x =3

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4







region

EXAMPLE


Solution



included


shown

y

x

1

1

-2 2 3 4

-3

-4

-5

2

3

4

5

-1

-2

-3-4

-1 y = -1



y -intercept 1-






EXAMPLES


1 y x 2$ +

Solution






y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0



2 x y 2 3 61-

Solution


region


side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g








3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g








grouping symbols)



Centre(c) 1 5-







Centre(j) 4 7- -

^ h and radius 3

Other Graphs




EXAMPLES


Solution



eg When 0x =

y 3

1

c=

=

x

y

1

3

3

1

31

When1

1

= -

=

=

=

-

x 3- 2- 1- 0 1 2 3

y 271

91

31

1 3 9 27



You will meet these

graphs again in the

HSC Course





other point

When

x

y

1

3

3

1

=

=

=

y

x

1

2

1

3







Solution



x minus2 minus1 0 05 1 2 3 4

y minus03 0 03 05 06

y

x

1

2

1 2 3 4

-1




You learned about


stages of maths





range y y 02



range all real y

Cubic function


EXAMPLE



Solution



y minus25 minus6 1 2 3 10 29

y

x

1

1

-2 2 3 4

-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4



If you already

know the shape of


3






Domain and range


EXAMPLE


Solution




function are

f

f

1 1 2

1 21

4 4 2

64 2

66

3

3

- = - +

= - +

=

= +

= +

=

] ]

] ]

g g

g g


these points

y

x

(-1 1)

(4 66)

Range 1 66 y y










positive number

EXAMPLE


Solution


x 4So $

$


(or zero) So y 0$


59 Exercises


(a) 4 3 y x= +

(b) f x 4= -] g

(c) 3x =



(f) f x xx 12 2= - -] g

(g) 64x y 2 2+ =

(h) f t t 4

3=

-] g

(i) ( ) g 2

5zz

= +

(j) | |f x x=] g


(a) y x=

(b) 2 y x= -

(c) | |f x x2 3= -] g

(d) | | 2 y x= -

(e) f x x2 5= - +] g

(f) | | y x5= -

(g) 2 y x=

(h) y 5x= -

(i) f x xx 1

= +] g

(j)2

4 3 y x

x= -


(a) y x x 5 2= -] g


= +-

(d) g x x x164 2= -] g

(e) 49x y 2 2+ =

You may like to


by dividing by x




4 (a) Solve x1 02$-


f x x1 2= -] g


(a) 2 y x x2= - -

(b) g t t t 62= +] g



in each case


x3 3 -


x2 3 -


x2 1 - (d)


x1 5


0 4x


x3 3 -


x1 1 -


x2 3 -


x4 4 -

(j) y x x7 62= - + - in the

domain 0 7x


function y x 1

3=

+





x=] g


function

find its range(b)


plane

(a) f x x4=] g

(b) y x3= -

(c) y x 34= -

(d) 2 p x x3=] g

(e) 1 g x x3= +] g

(f) 100x y 2 2+ =

(g) 2 1 y x= +


y x 1= -









(b) f x x 22= -] g

(c)1

y x=

(d) f x x3=] g

(e) f x 3x=] g

13 (a) Solve x4 02$-


(i) 4 y x2= -

(ii) y x4

2= - -




DID YOU KNOW




a light



Limits






a a1

1

03

Z

=

=

3

3

-


lim a 0x=

x 3-



asymptote

EXAMPLES

1 Find lim x

x x5x 0

2+

Solution


which is undefined


( )

lim lim

lim

xx x

x

x x

x

5 5

5

5

x x

x

0

2

0 1

1

0

+=

+

= +

=

] g




2 Find limx

x

4

22

-

-

x 2

Solution


lim lim

lim

x

x

x x

x

x

4

2

2 2

2

21

41

2 1

1

-

-=

+ -

-

=+

=

x x2 2

x 2

^ _h i

3 Find limh

h x hx h2 72 2+ -

h 0

Solution

lim lim

lim

h

h x hx h

h

h hx x

hx x

x

2 7 2 7

2 7

7

2 2 2

2

2

+ -=

+ -

= + -

= -

h 0

h h0 0

^ h

Continuity





EXAMPLES

1 Find lim

x

x

1

12

-

-

x 1


11

y x

x2

=-

- Sketch the curve

Solution


xx2

-

- gives

00

CONTINUED




( )

lim lim

limx

x

x

x x

x11

1

1 1

1

2

x x

x

1

2

1

1

-

-=

-

+ -

= +

=

-

] ]g g

11

y xx2

=-





y xx

x

x x

x

11

1

1 1

1

2

=-

-

=

+

= +

-

-^ ^h h


2 Find limx

x x2

2x 2

2

+

+ -

-


x x2

22

= + -

+

Solution


x x2

22

+

+ - gives

00

lim lim

lim

xx x

x

x x

x

22

2

1 2

1

3

x x

x

2

2

2

2

+

+ -=

+

- +

=

= -

-

- -

-

^^ ^

^

hh h

h2 y

x

x xx

y x

x

x

x

22

2

2

1

1


=+

+ -= -

=+

= -

+ -^ ^h h


Remember that x 1




1 Find(a) lim x 52

+x 4

(b) lim t 7-t 3 -

(c) lim x x2 43+ -

x 2

(d) lim xx x32

+

x 0

(e) limh

h h

2

22

-

- -

h 2

(f) lim y

y

5

1253

-

-

y 5

(g) limx

x x

12 12

+

+ +

x 1-

(h) limx

x x4

2 82

+

+ -

x 4 -

(i) limc

c

4

22

-

-

c 2

(j) limx x

x 12

-

-

x 1

(k) lim h

h h h2 73 2+ -

h 0

(l) limh

hx hx h32 2- +

h 0

(m) limh

hx h x hx h2 3 53 2 2- + -

h 0

(n) lim x c x c 3 3

-

-

x c



discontinuous

(a) 3 y x2= -

(b)1

1 y

x=

+

(c) f x x 1= -] g

(d)4

1 y

x2=

+

(e)4

1

y x2=

-



(a)3

y xx x2

= +

(b)33

y x

x x2

=+

+

(c)1

5 4 y

xx x2

=+

+ +

510 Exercises




Regions

Class Investigation


them all







EXAMPLE


Solution



y

x =3

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4







region

EXAMPLE


Solution



included


shown

y

x

1

1

-2 2 3 4

-3

-4

-5

2

3

4

5

-1

-2

-3-4

-1 y = -1



y -intercept 1-






EXAMPLES


1 y x 2$ +

Solution






y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0



2 x y 2 3 61-

Solution


region


side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g








3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g







other point

When

x

y

1

3

3

1

=

=

=

y

x

1

2

1

3







Solution



x minus2 minus1 0 05 1 2 3 4

y minus03 0 03 05 06

y

x

1

2

1 2 3 4

-1




You learned about


stages of maths





range y y 02



range all real y

Cubic function


EXAMPLE



Solution



y minus25 minus6 1 2 3 10 29

y

x

1

1

-2 2 3 4

-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4



If you already

know the shape of


3






Domain and range


EXAMPLE


Solution




function are

f

f

1 1 2

1 21

4 4 2

64 2

66

3

3

- = - +

= - +

=

= +

= +

=

] ]

] ]

g g

g g


these points

y

x

(-1 1)

(4 66)

Range 1 66 y y










positive number

EXAMPLE


Solution


x 4So $

$


(or zero) So y 0$


59 Exercises


(a) 4 3 y x= +

(b) f x 4= -] g

(c) 3x =



(f) f x xx 12 2= - -] g

(g) 64x y 2 2+ =

(h) f t t 4

3=

-] g

(i) ( ) g 2

5zz

= +

(j) | |f x x=] g


(a) y x=

(b) 2 y x= -

(c) | |f x x2 3= -] g

(d) | | 2 y x= -

(e) f x x2 5= - +] g

(f) | | y x5= -

(g) 2 y x=

(h) y 5x= -

(i) f x xx 1

= +] g

(j)2

4 3 y x

x= -


(a) y x x 5 2= -] g


= +-

(d) g x x x164 2= -] g

(e) 49x y 2 2+ =

You may like to


by dividing by x




4 (a) Solve x1 02$-


f x x1 2= -] g


(a) 2 y x x2= - -

(b) g t t t 62= +] g



in each case


x3 3 -


x2 3 -


x2 1 - (d)


x1 5


0 4x


x3 3 -


x1 1 -


x2 3 -


x4 4 -

(j) y x x7 62= - + - in the

domain 0 7x


function y x 1

3=

+





x=] g


function

find its range(b)


plane

(a) f x x4=] g

(b) y x3= -

(c) y x 34= -

(d) 2 p x x3=] g

(e) 1 g x x3= +] g

(f) 100x y 2 2+ =

(g) 2 1 y x= +


y x 1= -









(b) f x x 22= -] g

(c)1

y x=

(d) f x x3=] g

(e) f x 3x=] g

13 (a) Solve x4 02$-


(i) 4 y x2= -

(ii) y x4

2= - -




DID YOU KNOW




a light



Limits






a a1

1

03

Z

=

=

3

3

-


lim a 0x=

x 3-



asymptote

EXAMPLES

1 Find lim x

x x5x 0

2+

Solution


which is undefined


( )

lim lim

lim

xx x

x

x x

x

5 5

5

5

x x

x

0

2

0 1

1

0

+=

+

= +

=

] g




2 Find limx

x

4

22

-

-

x 2

Solution


lim lim

lim

x

x

x x

x

x

4

2

2 2

2

21

41

2 1

1

-

-=

+ -

-

=+

=

x x2 2

x 2

^ _h i

3 Find limh

h x hx h2 72 2+ -

h 0

Solution

lim lim

lim

h

h x hx h

h

h hx x

hx x

x

2 7 2 7

2 7

7

2 2 2

2

2

+ -=

+ -

= + -

= -

h 0

h h0 0

^ h

Continuity





EXAMPLES

1 Find lim

x

x

1

12

-

-

x 1


11

y x

x2

=-

- Sketch the curve

Solution


xx2

-

- gives

00

CONTINUED




( )

lim lim

limx

x

x

x x

x11

1

1 1

1

2

x x

x

1

2

1

1

-

-=

-

+ -

= +

=

-

] ]g g

11

y xx2

=-





y xx

x

x x

x

11

1

1 1

1

2

=-

-

=

+

= +

-

-^ ^h h


2 Find limx

x x2

2x 2

2

+

+ -

-


x x2

22

= + -

+

Solution


x x2

22

+

+ - gives

00

lim lim

lim

xx x

x

x x

x

22

2

1 2

1

3

x x

x

2

2

2

2

+

+ -=

+

- +

=

= -

-

- -

-

^^ ^

^

hh h

h2 y

x

x xx

y x

x

x

x

22

2

2

1

1


=+

+ -= -

=+

= -

+ -^ ^h h


Remember that x 1




1 Find(a) lim x 52

+x 4

(b) lim t 7-t 3 -

(c) lim x x2 43+ -

x 2

(d) lim xx x32

+

x 0

(e) limh

h h

2

22

-

- -

h 2

(f) lim y

y

5

1253

-

-

y 5

(g) limx

x x

12 12

+

+ +

x 1-

(h) limx

x x4

2 82

+

+ -

x 4 -

(i) limc

c

4

22

-

-

c 2

(j) limx x

x 12

-

-

x 1

(k) lim h

h h h2 73 2+ -

h 0

(l) limh

hx hx h32 2- +

h 0

(m) limh

hx h x hx h2 3 53 2 2- + -

h 0

(n) lim x c x c 3 3

-

-

x c



discontinuous

(a) 3 y x2= -

(b)1

1 y

x=

+

(c) f x x 1= -] g

(d)4

1 y

x2=

+

(e)4

1

y x2=

-



(a)3

y xx x2

= +

(b)33

y x

x x2

=+

+

(c)1

5 4 y

xx x2

=+

+ +

510 Exercises




Regions

Class Investigation


them all







EXAMPLE


Solution



y

x =3

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4







region

EXAMPLE


Solution



included


shown

y

x

1

1

-2 2 3 4

-3

-4

-5

2

3

4

5

-1

-2

-3-4

-1 y = -1



y -intercept 1-






EXAMPLES


1 y x 2$ +

Solution






y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0



2 x y 2 3 61-

Solution


region


side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g








3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g







range y y 02



range all real y

Cubic function


EXAMPLE



Solution



y minus25 minus6 1 2 3 10 29

y

x

1

1

-2 2 3 4

-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4



If you already

know the shape of


3






Domain and range


EXAMPLE


Solution




function are

f

f

1 1 2

1 21

4 4 2

64 2

66

3

3

- = - +

= - +

=

= +

= +

=

] ]

] ]

g g

g g


these points

y

x

(-1 1)

(4 66)

Range 1 66 y y










positive number

EXAMPLE


Solution


x 4So $

$


(or zero) So y 0$


59 Exercises


(a) 4 3 y x= +

(b) f x 4= -] g

(c) 3x =



(f) f x xx 12 2= - -] g

(g) 64x y 2 2+ =

(h) f t t 4

3=

-] g

(i) ( ) g 2

5zz

= +

(j) | |f x x=] g


(a) y x=

(b) 2 y x= -

(c) | |f x x2 3= -] g

(d) | | 2 y x= -

(e) f x x2 5= - +] g

(f) | | y x5= -

(g) 2 y x=

(h) y 5x= -

(i) f x xx 1

= +] g

(j)2

4 3 y x

x= -


(a) y x x 5 2= -] g


= +-

(d) g x x x164 2= -] g

(e) 49x y 2 2+ =

You may like to


by dividing by x




4 (a) Solve x1 02$-


f x x1 2= -] g


(a) 2 y x x2= - -

(b) g t t t 62= +] g



in each case


x3 3 -


x2 3 -


x2 1 - (d)


x1 5


0 4x


x3 3 -


x1 1 -


x2 3 -


x4 4 -

(j) y x x7 62= - + - in the

domain 0 7x


function y x 1

3=

+





x=] g


function

find its range(b)


plane

(a) f x x4=] g

(b) y x3= -

(c) y x 34= -

(d) 2 p x x3=] g

(e) 1 g x x3= +] g

(f) 100x y 2 2+ =

(g) 2 1 y x= +


y x 1= -









(b) f x x 22= -] g

(c)1

y x=

(d) f x x3=] g

(e) f x 3x=] g

13 (a) Solve x4 02$-


(i) 4 y x2= -

(ii) y x4

2= - -




DID YOU KNOW




a light



Limits






a a1

1

03

Z

=

=

3

3

-


lim a 0x=

x 3-



asymptote

EXAMPLES

1 Find lim x

x x5x 0

2+

Solution


which is undefined


( )

lim lim

lim

xx x

x

x x

x

5 5

5

5

x x

x

0

2

0 1

1

0

+=

+

= +

=

] g




2 Find limx

x

4

22

-

-

x 2

Solution


lim lim

lim

x

x

x x

x

x

4

2

2 2

2

21

41

2 1

1

-

-=

+ -

-

=+

=

x x2 2

x 2

^ _h i

3 Find limh

h x hx h2 72 2+ -

h 0

Solution

lim lim

lim

h

h x hx h

h

h hx x

hx x

x

2 7 2 7

2 7

7

2 2 2

2

2

+ -=

+ -

= + -

= -

h 0

h h0 0

^ h

Continuity





EXAMPLES

1 Find lim

x

x

1

12

-

-

x 1


11

y x

x2

=-

- Sketch the curve

Solution


xx2

-

- gives

00

CONTINUED




( )

lim lim

limx

x

x

x x

x11

1

1 1

1

2

x x

x

1

2

1

1

-

-=

-

+ -

= +

=

-

] ]g g

11

y xx2

=-





y xx

x

x x

x

11

1

1 1

1

2

=-

-

=

+

= +

-

-^ ^h h


2 Find limx

x x2

2x 2

2

+

+ -

-


x x2

22

= + -

+

Solution


x x2

22

+

+ - gives

00

lim lim

lim

xx x

x

x x

x

22

2

1 2

1

3

x x

x

2

2

2

2

+

+ -=

+

- +

=

= -

-

- -

-

^^ ^

^

hh h

h2 y

x

x xx

y x

x

x

x

22

2

2

1

1


=+

+ -= -

=+

= -

+ -^ ^h h


Remember that x 1




1 Find(a) lim x 52

+x 4

(b) lim t 7-t 3 -

(c) lim x x2 43+ -

x 2

(d) lim xx x32

+

x 0

(e) limh

h h

2

22

-

- -

h 2

(f) lim y

y

5

1253

-

-

y 5

(g) limx

x x

12 12

+

+ +

x 1-

(h) limx

x x4

2 82

+

+ -

x 4 -

(i) limc

c

4

22

-

-

c 2

(j) limx x

x 12

-

-

x 1

(k) lim h

h h h2 73 2+ -

h 0

(l) limh

hx hx h32 2- +

h 0

(m) limh

hx h x hx h2 3 53 2 2- + -

h 0

(n) lim x c x c 3 3

-

-

x c



discontinuous

(a) 3 y x2= -

(b)1

1 y

x=

+

(c) f x x 1= -] g

(d)4

1 y

x2=

+

(e)4

1

y x2=

-



(a)3

y xx x2

= +

(b)33

y x

x x2

=+

+

(c)1

5 4 y

xx x2

=+

+ +

510 Exercises




Regions

Class Investigation


them all







EXAMPLE


Solution



y

x =3

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4







region

EXAMPLE


Solution



included


shown

y

x

1

1

-2 2 3 4

-3

-4

-5

2

3

4

5

-1

-2

-3-4

-1 y = -1



y -intercept 1-






EXAMPLES


1 y x 2$ +

Solution






y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0



2 x y 2 3 61-

Solution


region


side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g








3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g






Domain and range


EXAMPLE


Solution




function are

f

f

1 1 2

1 21

4 4 2

64 2

66

3

3

- = - +

= - +

=

= +

= +

=

] ]

] ]

g g

g g


these points

y

x

(-1 1)

(4 66)

Range 1 66 y y










positive number

EXAMPLE


Solution


x 4So $

$


(or zero) So y 0$


59 Exercises


(a) 4 3 y x= +

(b) f x 4= -] g

(c) 3x =



(f) f x xx 12 2= - -] g

(g) 64x y 2 2+ =

(h) f t t 4

3=

-] g

(i) ( ) g 2

5zz

= +

(j) | |f x x=] g


(a) y x=

(b) 2 y x= -

(c) | |f x x2 3= -] g

(d) | | 2 y x= -

(e) f x x2 5= - +] g

(f) | | y x5= -

(g) 2 y x=

(h) y 5x= -

(i) f x xx 1

= +] g

(j)2

4 3 y x

x= -


(a) y x x 5 2= -] g


= +-

(d) g x x x164 2= -] g

(e) 49x y 2 2+ =

You may like to


by dividing by x




4 (a) Solve x1 02$-


f x x1 2= -] g


(a) 2 y x x2= - -

(b) g t t t 62= +] g



in each case


x3 3 -


x2 3 -


x2 1 - (d)


x1 5


0 4x


x3 3 -


x1 1 -


x2 3 -


x4 4 -

(j) y x x7 62= - + - in the

domain 0 7x


function y x 1

3=

+





x=] g


function

find its range(b)


plane

(a) f x x4=] g

(b) y x3= -

(c) y x 34= -

(d) 2 p x x3=] g

(e) 1 g x x3= +] g

(f) 100x y 2 2+ =

(g) 2 1 y x= +


y x 1= -









(b) f x x 22= -] g

(c)1

y x=

(d) f x x3=] g

(e) f x 3x=] g

13 (a) Solve x4 02$-


(i) 4 y x2= -

(ii) y x4

2= - -




DID YOU KNOW




a light



Limits






a a1

1

03

Z

=

=

3

3

-


lim a 0x=

x 3-



asymptote

EXAMPLES

1 Find lim x

x x5x 0

2+

Solution


which is undefined


( )

lim lim

lim

xx x

x

x x

x

5 5

5

5

x x

x

0

2

0 1

1

0

+=

+

= +

=

] g




2 Find limx

x

4

22

-

-

x 2

Solution


lim lim

lim

x

x

x x

x

x

4

2

2 2

2

21

41

2 1

1

-

-=

+ -

-

=+

=

x x2 2

x 2

^ _h i

3 Find limh

h x hx h2 72 2+ -

h 0

Solution

lim lim

lim

h

h x hx h

h

h hx x

hx x

x

2 7 2 7

2 7

7

2 2 2

2

2

+ -=

+ -

= + -

= -

h 0

h h0 0

^ h

Continuity





EXAMPLES

1 Find lim

x

x

1

12

-

-

x 1


11

y x

x2

=-

- Sketch the curve

Solution


xx2

-

- gives

00

CONTINUED




( )

lim lim

limx

x

x

x x

x11

1

1 1

1

2

x x

x

1

2

1

1

-

-=

-

+ -

= +

=

-

] ]g g

11

y xx2

=-





y xx

x

x x

x

11

1

1 1

1

2

=-

-

=

+

= +

-

-^ ^h h


2 Find limx

x x2

2x 2

2

+

+ -

-


x x2

22

= + -

+

Solution


x x2

22

+

+ - gives

00

lim lim

lim

xx x

x

x x

x

22

2

1 2

1

3

x x

x

2

2

2

2

+

+ -=

+

- +

=

= -

-

- -

-

^^ ^

^

hh h

h2 y

x

x xx

y x

x

x

x

22

2

2

1

1


=+

+ -= -

=+

= -

+ -^ ^h h


Remember that x 1




1 Find(a) lim x 52

+x 4

(b) lim t 7-t 3 -

(c) lim x x2 43+ -

x 2

(d) lim xx x32

+

x 0

(e) limh

h h

2

22

-

- -

h 2

(f) lim y

y

5

1253

-

-

y 5

(g) limx

x x

12 12

+

+ +

x 1-

(h) limx

x x4

2 82

+

+ -

x 4 -

(i) limc

c

4

22

-

-

c 2

(j) limx x

x 12

-

-

x 1

(k) lim h

h h h2 73 2+ -

h 0

(l) limh

hx hx h32 2- +

h 0

(m) limh

hx h x hx h2 3 53 2 2- + -

h 0

(n) lim x c x c 3 3

-

-

x c



discontinuous

(a) 3 y x2= -

(b)1

1 y

x=

+

(c) f x x 1= -] g

(d)4

1 y

x2=

+

(e)4

1

y x2=

-



(a)3

y xx x2

= +

(b)33

y x

x x2

=+

+

(c)1

5 4 y

xx x2

=+

+ +

510 Exercises




Regions

Class Investigation


them all







EXAMPLE


Solution



y

x =3

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4







region

EXAMPLE


Solution



included


shown

y

x

1

1

-2 2 3 4

-3

-4

-5

2

3

4

5

-1

-2

-3-4

-1 y = -1



y -intercept 1-






EXAMPLES


1 y x 2$ +

Solution






y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0



2 x y 2 3 61-

Solution


region


side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g








3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g












positive number

EXAMPLE


Solution


x 4So $

$


(or zero) So y 0$


59 Exercises


(a) 4 3 y x= +

(b) f x 4= -] g

(c) 3x =



(f) f x xx 12 2= - -] g

(g) 64x y 2 2+ =

(h) f t t 4

3=

-] g

(i) ( ) g 2

5zz

= +

(j) | |f x x=] g


(a) y x=

(b) 2 y x= -

(c) | |f x x2 3= -] g

(d) | | 2 y x= -

(e) f x x2 5= - +] g

(f) | | y x5= -

(g) 2 y x=

(h) y 5x= -

(i) f x xx 1

= +] g

(j)2

4 3 y x

x= -


(a) y x x 5 2= -] g


= +-

(d) g x x x164 2= -] g

(e) 49x y 2 2+ =

You may like to


by dividing by x




4 (a) Solve x1 02$-


f x x1 2= -] g


(a) 2 y x x2= - -

(b) g t t t 62= +] g



in each case


x3 3 -


x2 3 -


x2 1 - (d)


x1 5


0 4x


x3 3 -


x1 1 -


x2 3 -


x4 4 -

(j) y x x7 62= - + - in the

domain 0 7x


function y x 1

3=

+





x=] g


function

find its range(b)


plane

(a) f x x4=] g

(b) y x3= -

(c) y x 34= -

(d) 2 p x x3=] g

(e) 1 g x x3= +] g

(f) 100x y 2 2+ =

(g) 2 1 y x= +


y x 1= -









(b) f x x 22= -] g

(c)1

y x=

(d) f x x3=] g

(e) f x 3x=] g

13 (a) Solve x4 02$-


(i) 4 y x2= -

(ii) y x4

2= - -




DID YOU KNOW




a light



Limits






a a1

1

03

Z

=

=

3

3

-


lim a 0x=

x 3-



asymptote

EXAMPLES

1 Find lim x

x x5x 0

2+

Solution


which is undefined


( )

lim lim

lim

xx x

x

x x

x

5 5

5

5

x x

x

0

2

0 1

1

0

+=

+

= +

=

] g




2 Find limx

x

4

22

-

-

x 2

Solution


lim lim

lim

x

x

x x

x

x

4

2

2 2

2

21

41

2 1

1

-

-=

+ -

-

=+

=

x x2 2

x 2

^ _h i

3 Find limh

h x hx h2 72 2+ -

h 0

Solution

lim lim

lim

h

h x hx h

h

h hx x

hx x

x

2 7 2 7

2 7

7

2 2 2

2

2

+ -=

+ -

= + -

= -

h 0

h h0 0

^ h

Continuity





EXAMPLES

1 Find lim

x

x

1

12

-

-

x 1


11

y x

x2

=-

- Sketch the curve

Solution


xx2

-

- gives

00

CONTINUED




( )

lim lim

limx

x

x

x x

x11

1

1 1

1

2

x x

x

1

2

1

1

-

-=

-

+ -

= +

=

-

] ]g g

11

y xx2

=-





y xx

x

x x

x

11

1

1 1

1

2

=-

-

=

+

= +

-

-^ ^h h


2 Find limx

x x2

2x 2

2

+

+ -

-


x x2

22

= + -

+

Solution


x x2

22

+

+ - gives

00

lim lim

lim

xx x

x

x x

x

22

2

1 2

1

3

x x

x

2

2

2

2

+

+ -=

+

- +

=

= -

-

- -

-

^^ ^

^

hh h

h2 y

x

x xx

y x

x

x

x

22

2

2

1

1


=+

+ -= -

=+

= -

+ -^ ^h h


Remember that x 1




1 Find(a) lim x 52

+x 4

(b) lim t 7-t 3 -

(c) lim x x2 43+ -

x 2

(d) lim xx x32

+

x 0

(e) limh

h h

2

22

-

- -

h 2

(f) lim y

y

5

1253

-

-

y 5

(g) limx

x x

12 12

+

+ +

x 1-

(h) limx

x x4

2 82

+

+ -

x 4 -

(i) limc

c

4

22

-

-

c 2

(j) limx x

x 12

-

-

x 1

(k) lim h

h h h2 73 2+ -

h 0

(l) limh

hx hx h32 2- +

h 0

(m) limh

hx h x hx h2 3 53 2 2- + -

h 0

(n) lim x c x c 3 3

-

-

x c



discontinuous

(a) 3 y x2= -

(b)1

1 y

x=

+

(c) f x x 1= -] g

(d)4

1 y

x2=

+

(e)4

1

y x2=

-



(a)3

y xx x2

= +

(b)33

y x

x x2

=+

+

(c)1

5 4 y

xx x2

=+

+ +

510 Exercises




Regions

Class Investigation


them all







EXAMPLE


Solution



y

x =3

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4







region

EXAMPLE


Solution



included


shown

y

x

1

1

-2 2 3 4

-3

-4

-5

2

3

4

5

-1

-2

-3-4

-1 y = -1



y -intercept 1-






EXAMPLES


1 y x 2$ +

Solution






y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0



2 x y 2 3 61-

Solution


region


side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g








3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g






4 (a) Solve x1 02$-


f x x1 2= -] g


(a) 2 y x x2= - -

(b) g t t t 62= +] g



in each case


x3 3 -


x2 3 -


x2 1 - (d)


x1 5


0 4x


x3 3 -


x1 1 -


x2 3 -


x4 4 -

(j) y x x7 62= - + - in the

domain 0 7x


function y x 1

3=

+





x=] g


function

find its range(b)


plane

(a) f x x4=] g

(b) y x3= -

(c) y x 34= -

(d) 2 p x x3=] g

(e) 1 g x x3= +] g

(f) 100x y 2 2+ =

(g) 2 1 y x= +


y x 1= -









(b) f x x 22= -] g

(c)1

y x=

(d) f x x3=] g

(e) f x 3x=] g

13 (a) Solve x4 02$-


(i) 4 y x2= -

(ii) y x4

2= - -




DID YOU KNOW




a light



Limits






a a1

1

03

Z

=

=

3

3

-


lim a 0x=

x 3-



asymptote

EXAMPLES

1 Find lim x

x x5x 0

2+

Solution


which is undefined


( )

lim lim

lim

xx x

x

x x

x

5 5

5

5

x x

x

0

2

0 1

1

0

+=

+

= +

=

] g




2 Find limx

x

4

22

-

-

x 2

Solution


lim lim

lim

x

x

x x

x

x

4

2

2 2

2

21

41

2 1

1

-

-=

+ -

-

=+

=

x x2 2

x 2

^ _h i

3 Find limh

h x hx h2 72 2+ -

h 0

Solution

lim lim

lim

h

h x hx h

h

h hx x

hx x

x

2 7 2 7

2 7

7

2 2 2

2

2

+ -=

+ -

= + -

= -

h 0

h h0 0

^ h

Continuity





EXAMPLES

1 Find lim

x

x

1

12

-

-

x 1


11

y x

x2

=-

- Sketch the curve

Solution


xx2

-

- gives

00

CONTINUED




( )

lim lim

limx

x

x

x x

x11

1

1 1

1

2

x x

x

1

2

1

1

-

-=

-

+ -

= +

=

-

] ]g g

11

y xx2

=-





y xx

x

x x

x

11

1

1 1

1

2

=-

-

=

+

= +

-

-^ ^h h


2 Find limx

x x2

2x 2

2

+

+ -

-


x x2

22

= + -

+

Solution


x x2

22

+

+ - gives

00

lim lim

lim

xx x

x

x x

x

22

2

1 2

1

3

x x

x

2

2

2

2

+

+ -=

+

- +

=

= -

-

- -

-

^^ ^

^

hh h

h2 y

x

x xx

y x

x

x

x

22

2

2

1

1


=+

+ -= -

=+

= -

+ -^ ^h h


Remember that x 1




1 Find(a) lim x 52

+x 4

(b) lim t 7-t 3 -

(c) lim x x2 43+ -

x 2

(d) lim xx x32

+

x 0

(e) limh

h h

2

22

-

- -

h 2

(f) lim y

y

5

1253

-

-

y 5

(g) limx

x x

12 12

+

+ +

x 1-

(h) limx

x x4

2 82

+

+ -

x 4 -

(i) limc

c

4

22

-

-

c 2

(j) limx x

x 12

-

-

x 1

(k) lim h

h h h2 73 2+ -

h 0

(l) limh

hx hx h32 2- +

h 0

(m) limh

hx h x hx h2 3 53 2 2- + -

h 0

(n) lim x c x c 3 3

-

-

x c



discontinuous

(a) 3 y x2= -

(b)1

1 y

x=

+

(c) f x x 1= -] g

(d)4

1 y

x2=

+

(e)4

1

y x2=

-



(a)3

y xx x2

= +

(b)33

y x

x x2

=+

+

(c)1

5 4 y

xx x2

=+

+ +

510 Exercises




Regions

Class Investigation


them all







EXAMPLE


Solution



y

x =3

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4







region

EXAMPLE


Solution



included


shown

y

x

1

1

-2 2 3 4

-3

-4

-5

2

3

4

5

-1

-2

-3-4

-1 y = -1



y -intercept 1-






EXAMPLES


1 y x 2$ +

Solution






y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0



2 x y 2 3 61-

Solution


region


side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g








3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g






DID YOU KNOW




a light



Limits






a a1

1

03

Z

=

=

3

3

-


lim a 0x=

x 3-



asymptote

EXAMPLES

1 Find lim x

x x5x 0

2+

Solution


which is undefined


( )

lim lim

lim

xx x

x

x x

x

5 5

5

5

x x

x

0

2

0 1

1

0

+=

+

= +

=

] g




2 Find limx

x

4

22

-

-

x 2

Solution


lim lim

lim

x

x

x x

x

x

4

2

2 2

2

21

41

2 1

1

-

-=

+ -

-

=+

=

x x2 2

x 2

^ _h i

3 Find limh

h x hx h2 72 2+ -

h 0

Solution

lim lim

lim

h

h x hx h

h

h hx x

hx x

x

2 7 2 7

2 7

7

2 2 2

2

2

+ -=

+ -

= + -

= -

h 0

h h0 0

^ h

Continuity





EXAMPLES

1 Find lim

x

x

1

12

-

-

x 1


11

y x

x2

=-

- Sketch the curve

Solution


xx2

-

- gives

00

CONTINUED




( )

lim lim

limx

x

x

x x

x11

1

1 1

1

2

x x

x

1

2

1

1

-

-=

-

+ -

= +

=

-

] ]g g

11

y xx2

=-





y xx

x

x x

x

11

1

1 1

1

2

=-

-

=

+

= +

-

-^ ^h h


2 Find limx

x x2

2x 2

2

+

+ -

-


x x2

22

= + -

+

Solution


x x2

22

+

+ - gives

00

lim lim

lim

xx x

x

x x

x

22

2

1 2

1

3

x x

x

2

2

2

2

+

+ -=

+

- +

=

= -

-

- -

-

^^ ^

^

hh h

h2 y

x

x xx

y x

x

x

x

22

2

2

1

1


=+

+ -= -

=+

= -

+ -^ ^h h


Remember that x 1




1 Find(a) lim x 52

+x 4

(b) lim t 7-t 3 -

(c) lim x x2 43+ -

x 2

(d) lim xx x32

+

x 0

(e) limh

h h

2

22

-

- -

h 2

(f) lim y

y

5

1253

-

-

y 5

(g) limx

x x

12 12

+

+ +

x 1-

(h) limx

x x4

2 82

+

+ -

x 4 -

(i) limc

c

4

22

-

-

c 2

(j) limx x

x 12

-

-

x 1

(k) lim h

h h h2 73 2+ -

h 0

(l) limh

hx hx h32 2- +

h 0

(m) limh

hx h x hx h2 3 53 2 2- + -

h 0

(n) lim x c x c 3 3

-

-

x c



discontinuous

(a) 3 y x2= -

(b)1

1 y

x=

+

(c) f x x 1= -] g

(d)4

1 y

x2=

+

(e)4

1

y x2=

-



(a)3

y xx x2

= +

(b)33

y x

x x2

=+

+

(c)1

5 4 y

xx x2

=+

+ +

510 Exercises




Regions

Class Investigation


them all







EXAMPLE


Solution



y

x =3

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4







region

EXAMPLE


Solution



included


shown

y

x

1

1

-2 2 3 4

-3

-4

-5

2

3

4

5

-1

-2

-3-4

-1 y = -1



y -intercept 1-






EXAMPLES


1 y x 2$ +

Solution






y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0



2 x y 2 3 61-

Solution


region


side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g








3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g






2 Find limx

x

4

22

-

-

x 2

Solution


lim lim

lim

x

x

x x

x

x

4

2

2 2

2

21

41

2 1

1

-

-=

+ -

-

=+

=

x x2 2

x 2

^ _h i

3 Find limh

h x hx h2 72 2+ -

h 0

Solution

lim lim

lim

h

h x hx h

h

h hx x

hx x

x

2 7 2 7

2 7

7

2 2 2

2

2

+ -=

+ -

= + -

= -

h 0

h h0 0

^ h

Continuity





EXAMPLES

1 Find lim

x

x

1

12

-

-

x 1


11

y x

x2

=-

- Sketch the curve

Solution


xx2

-

- gives

00

CONTINUED




( )

lim lim

limx

x

x

x x

x11

1

1 1

1

2

x x

x

1

2

1

1

-

-=

-

+ -

= +

=

-

] ]g g

11

y xx2

=-





y xx

x

x x

x

11

1

1 1

1

2

=-

-

=

+

= +

-

-^ ^h h


2 Find limx

x x2

2x 2

2

+

+ -

-


x x2

22

= + -

+

Solution


x x2

22

+

+ - gives

00

lim lim

lim

xx x

x

x x

x

22

2

1 2

1

3

x x

x

2

2

2

2

+

+ -=

+

- +

=

= -

-

- -

-

^^ ^

^

hh h

h2 y

x

x xx

y x

x

x

x

22

2

2

1

1


=+

+ -= -

=+

= -

+ -^ ^h h


Remember that x 1




1 Find(a) lim x 52

+x 4

(b) lim t 7-t 3 -

(c) lim x x2 43+ -

x 2

(d) lim xx x32

+

x 0

(e) limh

h h

2

22

-

- -

h 2

(f) lim y

y

5

1253

-

-

y 5

(g) limx

x x

12 12

+

+ +

x 1-

(h) limx

x x4

2 82

+

+ -

x 4 -

(i) limc

c

4

22

-

-

c 2

(j) limx x

x 12

-

-

x 1

(k) lim h

h h h2 73 2+ -

h 0

(l) limh

hx hx h32 2- +

h 0

(m) limh

hx h x hx h2 3 53 2 2- + -

h 0

(n) lim x c x c 3 3

-

-

x c



discontinuous

(a) 3 y x2= -

(b)1

1 y

x=

+

(c) f x x 1= -] g

(d)4

1 y

x2=

+

(e)4

1

y x2=

-



(a)3

y xx x2

= +

(b)33

y x

x x2

=+

+

(c)1

5 4 y

xx x2

=+

+ +

510 Exercises




Regions

Class Investigation


them all







EXAMPLE


Solution



y

x =3

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4







region

EXAMPLE


Solution



included


shown

y

x

1

1

-2 2 3 4

-3

-4

-5

2

3

4

5

-1

-2

-3-4

-1 y = -1



y -intercept 1-






EXAMPLES


1 y x 2$ +

Solution






y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0



2 x y 2 3 61-

Solution


region


side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g








3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g






( )

lim lim

limx

x

x

x x

x11

1

1 1

1

2

x x

x

1

2

1

1

-

-=

-

+ -

= +

=

-

] ]g g

11

y xx2

=-





y xx

x

x x

x

11

1

1 1

1

2

=-

-

=

+

= +

-

-^ ^h h


2 Find limx

x x2

2x 2

2

+

+ -

-


x x2

22

= + -

+

Solution


x x2

22

+

+ - gives

00

lim lim

lim

xx x

x

x x

x

22

2

1 2

1

3

x x

x

2

2

2

2

+

+ -=

+

- +

=

= -

-

- -

-

^^ ^

^

hh h

h2 y

x

x xx

y x

x

x

x

22

2

2

1

1


=+

+ -= -

=+

= -

+ -^ ^h h


Remember that x 1




1 Find(a) lim x 52

+x 4

(b) lim t 7-t 3 -

(c) lim x x2 43+ -

x 2

(d) lim xx x32

+

x 0

(e) limh

h h

2

22

-

- -

h 2

(f) lim y

y

5

1253

-

-

y 5

(g) limx

x x

12 12

+

+ +

x 1-

(h) limx

x x4

2 82

+

+ -

x 4 -

(i) limc

c

4

22

-

-

c 2

(j) limx x

x 12

-

-

x 1

(k) lim h

h h h2 73 2+ -

h 0

(l) limh

hx hx h32 2- +

h 0

(m) limh

hx h x hx h2 3 53 2 2- + -

h 0

(n) lim x c x c 3 3

-

-

x c



discontinuous

(a) 3 y x2= -

(b)1

1 y

x=

+

(c) f x x 1= -] g

(d)4

1 y

x2=

+

(e)4

1

y x2=

-



(a)3

y xx x2

= +

(b)33

y x

x x2

=+

+

(c)1

5 4 y

xx x2

=+

+ +

510 Exercises




Regions

Class Investigation


them all







EXAMPLE


Solution



y

x =3

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4







region

EXAMPLE


Solution



included


shown

y

x

1

1

-2 2 3 4

-3

-4

-5

2

3

4

5

-1

-2

-3-4

-1 y = -1



y -intercept 1-






EXAMPLES


1 y x 2$ +

Solution






y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0



2 x y 2 3 61-

Solution


region


side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g








3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g






1 Find(a) lim x 52

+x 4

(b) lim t 7-t 3 -

(c) lim x x2 43+ -

x 2

(d) lim xx x32

+

x 0

(e) limh

h h

2

22

-

- -

h 2

(f) lim y

y

5

1253

-

-

y 5

(g) limx

x x

12 12

+

+ +

x 1-

(h) limx

x x4

2 82

+

+ -

x 4 -

(i) limc

c

4

22

-

-

c 2

(j) limx x

x 12

-

-

x 1

(k) lim h

h h h2 73 2+ -

h 0

(l) limh

hx hx h32 2- +

h 0

(m) limh

hx h x hx h2 3 53 2 2- + -

h 0

(n) lim x c x c 3 3

-

-

x c



discontinuous

(a) 3 y x2= -

(b)1

1 y

x=

+

(c) f x x 1= -] g

(d)4

1 y

x2=

+

(e)4

1

y x2=

-



(a)3

y xx x2

= +

(b)33

y x

x x2

=+

+

(c)1

5 4 y

xx x2

=+

+ +

510 Exercises




Regions

Class Investigation


them all







EXAMPLE


Solution



y

x =3

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4







region

EXAMPLE


Solution



included


shown

y

x

1

1

-2 2 3 4

-3

-4

-5

2

3

4

5

-1

-2

-3-4

-1 y = -1



y -intercept 1-






EXAMPLES


1 y x 2$ +

Solution






y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0



2 x y 2 3 61-

Solution


region


side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g








3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g






Regions

Class Investigation


them all







EXAMPLE


Solution



y

x =3

x

1

1

-2 2 3 4-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4







region

EXAMPLE


Solution



included


shown

y

x

1

1

-2 2 3 4

-3

-4

-5

2

3

4

5

-1

-2

-3-4

-1 y = -1



y -intercept 1-






EXAMPLES


1 y x 2$ +

Solution






y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0



2 x y 2 3 61-

Solution


region


side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g








3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g







region

EXAMPLE


Solution



included


shown

y

x

1

1

-2 2 3 4

-3

-4

-5

2

3

4

5

-1

-2

-3-4

-1 y = -1



y -intercept 1-






EXAMPLES


1 y x 2$ +

Solution






y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0



2 x y 2 3 61-

Solution


region


side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g








3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g






EXAMPLES


1 y x 2$ +

Solution






y x 2

0 0 2

0 2 (false)

$

$

$

+

+

This means that 0 0



2 x y 2 3 61-

Solution


region


side of the line


( )

x y 2 3 6

2 0 3 1 63 6 (true)

1

1

1

-

-

-] g








3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g







3 x y 12 22+

Solution



the curve


0 0 1

0 1 (false)

2 2

2 2

2

2

2

+

+


4 y x2$




2 x -3 y = 6

CONTINUED





3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g







3 1

y x

3 1

(true)

2

22

2

$





EXAMPLE


Solution



y = x 2





algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g







algebraically


EXAMPLES


x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6


y 6


described as x 41


y 6 and x 41

CONTINUED




2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g






2 y

x

2

2

-2

-2

Solution






equation eg (0 0)

x y

0 0

0

4

LHS

RHS

2 2

2 2

1

= +

= +

=

] g


511 Exercises


(a) x 2

(b) x 12

(c) y 0$

(d) y 51

(e) y x 1 +

(f) y x2 3$ -

(g) x y 12+

(h) 3 6 0x y 1- -

(i) x y 2 2 0$+ -

(j) x2 1 01-





each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g







each region

(a) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(b) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

(c) y

x

1

1

-2 2 3 4-1

-3

-4

2

3

4

5

-1

-2

-3-4

6

y = x + 1

(d) y

1

1

-2 2 3 4 5-1

-3

-4

-5

2

3

4

5

-1

-2

-3-4

y = x 2 - 4

(e) y

x

1

1

2

3

y = 2 x


(a) ndash y x 122

(b) x y 92 2+

(c) x y 12 2$+

(d) y x2

(e) y x31





= +











5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g






5 Shade the region

(a) x y 2 42 2- +] g

(b) x y 1 2 12 2- + -] ^g h

(c) 1x y 2 92 22+ + -] ^g h


regions

(a) x 3 y 1$ -

(b) x y x3 32$ - -

(c) y 1 y x3 5$ -

(d) y x y x1 32 + -

(e) y x y 1 92 2 +

(f) x x y 1 42 22 1- +

(g) y y x4 2 $

(h) x y y x2 3 31 2-

(i) y x y 0 12 2 $+

(j) x y 1 21 - -









the curve(d)2

y x= the x -axis


the curve(e)2

1 y

x=

+ the


2x =


the intersection of

(a) x y 2 511 and y x2

(b) x y y x3 1 21 $ - -

(c) y x y x x y 1 2 1 2 3 6 - + -

(d) x y x y 3 2 92 2

$ $- + (e) | |x y y x2 31 $



are both positive

Application




EXAMPLE











SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g






SOLUTION

Profit P $25 $21

Machine A 2 3 12

Machine B 2 8

X Y

X Y

X Y

+

+

+

=





(0 4) $25 0 $21 4 $84(4 0) $25 4 $21 0 $100(3 2) $25 3 $21 2 $117

P

P

P

= + =

= + =

= + =

] ]] ]] ]g gg gg g





Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g






Test Yourself 5

1 If f x x x3 42= - -] g find


2 Sketch each graph

(a) 3 4 y x x2= - -

(b) f x x3=] g

(c) 1x y

2 2+ =

(d) 1 y x2

= -

(e) 1 y x2= - -

(f)2

y x=

(g) 2 5 10 0x y - + =

(h) | 2 | y x= +


in question 2

4 If f xx x

x x

2 1

3 1

if

if

21

$=

-

] g

find f f f 5 0 1- +] ] ]g g g

5 Given f x

x

x x

x x

3 3

1 3

2 1

if

if

if

2

2

1

=

-

] g

find






2 3 6 0 2x y xand $+ - -


x y 2+


(a)

(b)

(c)


the curve3

2 y

x=

-


2 y

x=

-






(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g








(ii) | |x 1 31+

(iii) | |x 1 32+


(a) f 2] g (b) x when f x 7=] g



(a) 2 5 20 0x y - + =

(b) 5 14 y x x2= - -



(a) 1 y x2= -

(b) 1 y x= +

(c) y x3=

(d) y x4=

(e) 2 y x=

17 Find

(a) lim xx x32 3x 3

2

-

- -

(b) limx x

x

5

2x 0 2

+

(c) limx

x

1

1x 1 2

3

-

+

-

(d) limh

xh h2 3h 0

2+


same number plane


2 4 y x= -


20 Show that

(a) f x x x3 14 2= + -] g is even




and f b 7=] g


x3 0 -

3 Sketch the region

y x4 2

2 - in the first

quadrant


5 f x

x x

x

x x

2 3 2

1 2 2

2

when

when

when2

2

1

=

+

-

-

] gZ

[

]]

]]


curve


1

1 y

x2=

-


2 4 0x y $+ -




f x 0=] g


12

+





11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g






11 If h t t t

t t

1 1

1 1

if

if

2

2

2

=-

-] g )


sketch the curve








answers)

16 (a) Show that3

2 72

31

xx

x+

+= +

+


3

2 7 y xx=

+

+


32 7

y x

x=

+

+

17 Sketch 2 y 1x=

-

18 Sketch| |

y x

x2

=


xf x2 6= -] g


1 y x2

=

-

21 Sketch f xx

11

2= -] g





Date post:	03-Jun-2018
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Math In Focus Year 11 2 unit Ch5functions and Graphs

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