Binder Chapter 2

7/30/2019 Binder Chapter 2

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Copyright 2008 by Farimah Fazeli

College Algebra

Section 2.1 Functions and Graphs

LetXandYbe two nonempty sets. A function from Xinto Yis a relationthat associates with each element ofXexactly one element ofY.

In other words, a function is a set of ordered pairs in which no twoordered pairs have the same first coordinate and different secondcoordinate.

Domain

Range

The following guidelines can help determine whether a relation is a function.1. Each element in the domain must be matched with exactly one element of the range.2. Some elements in the range may not be matched with any element in the domain.3.Two or more elements of the domain may be matched with the same element of the range.

So how can we tell if a graph represents a function? Each ____ will correspond with exactly one ____.An easy way to check is to use the Vertical Line Test.

Theorem:

In other words, if any vertical line intersects a graph at more than one point, the graph is not the graph ofa function. Now we can play a quick game of Function? Not a Function?


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Section 2.1 Functions and Graphs Page 2


Example1 : Use the graph of the function fto answer parts (a)-(n).

(a)Find ( )0f and ( )6f .

(b)Find ( )2f and ( )2f .

(c)Is ( )3f positive or negative?

(d)Is ( )

1f

positive or negative?

(e)For what numbers x is ( ) 0f x = ?

(f) For what numbers x is ( ) 0f x < ?

(g)What is the domain off? (k) How often does the line 1y = intersect the graph?

(h)What is the range off? (l) How often does the line 1x = intersect the graph?

(i) What are the x-intercepts? (m) For what values ofx does ( ) 3f x = ?

(j) What is the y-intercept? (n) For what values ofx does ( ) 2f x = ?

-6 8


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Example2: Determine if the equation1

32

y x= defines y as a function ofx.

Example3: Determine if the equation 22 1x y= + defines y as a function ofx.

One idea for explaining the mechanics of functions is the idea of a Function Machine.

1. It accepts numbers from the domain of the function.2. For each input, there is exactly one output (which may be repeated for different inputs).

Example4: For the function f defined by ( ) 23 2f x x x= + , evaluate:

(a) ( )3f

(b) ( ) ( )3f x f+

(c) ( )f x

(d) ( )f x


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(e) ( )3f x +

(f)( ) ( )

, 0f x h f x

hh

+

( The difference quotient)

Summary Important facts about functions.

For each x in the domain off, there is exactly one image f(x) in the range; however, an elementin the range can result from more than one x in the domain.

fis the symbol that we use to denote the function. It is symbolic of the equation that we use toget from an x in the domain to f(x) in the range.

Ify = f(x), then x is called the independent variable or argument off, andy is called the dependentvariable or the value offat x.

Unless otherwise stated, the domain of a function fis the largest set of real numbers for which the value

( )f x is a real number. Sometimes its helpful to ask yourself, What values of the independent variablemake sense in the given equation?

Example5: Find the domain and range of each of the following functions:

(a) ( )2

4

2 3

xf x

x x

+=

(b) ( )2

9g x x=

(c) ( ) 3 2h x x=


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College Algebra

Section 2.2 Graph of Relations and Function


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Section 2.2 Graph of Relations and Function Page 2


Another basic function is the greatest integer function.

* Youll also see ( ) f x x=

The domain of the greatest integer function is all real numbersand the range is all integers.

The y-intercept is 0 and the x-intercept is [ )0,1 .The function is neither even nor odd and has a discontinuity at every integer.


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Section 2.2 Graph of Relations and Function Page 3


Example: If ( ) int2

xf x

=

, find ( )1.2f , ( )1.99f , ( )2.2f , ( )1.8f

Sometimes a function is defined differently on different parts of its domain. When functions are definedby more than one equation, they are calledpiecewise-defined functions.

Example: Graph ( )

3 , 3 0

3, 0

, 0

x x

f x x

x x

+

Find the domain, intercepts, and, based onthe graph, the range.

Example: An economy car rented in Florida from National Car Rental on a weekly basis costs $95 perweek. Extra days cost $24 per day until the day rate exceeds the weekly rate, in which casethe weekly rate applies. Also, any part of a day used counts as a full day. Find the cost Cofrenting an economy car as a piecewise-defined function of the numberx of days used, where7 14x . Graph this function.


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College Algebra

Section 2.3 Families of Functions, Transformations, and Symmetry

Sometimes we are asked to graph a function that is almost like one that we already know from ourlibrary of functions. In this section, well look at some of these functions and develop techniques for

graphing them. Collectively, these techniques are referred to as transformations.All the transformations of a function form a family of functions.RigidNonrigid;Well start by graphing functions using vertical andhorizontal shifts.

On the same screen, graph the following functions:

Next, on the same screen, graph the functions:


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Section 2.3 Families of Functions, Transformations, and Symmetry Page 2


Example1 : Start with the graph of the basic function and use shifts to graph ( ) ( )3

2 3f x x= + .

Basic function:

Vertical Shift:

Horizontal shift:

Does the order of the vertical and horizontal shift matter?

Example: Write the function whose graph is the graph of 2y x= , but is shifted 5 units to the right and 7

units up.

We can also transform graphs by stretching orcompressing. Stretching and compressing can be donein either the vertical or horizontal direction.

On the same screen, graph the following functions:


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Now in the horizontal directionOn the same screen, graph the functions:

Example2 : Write the equation of the function whose graph is the graph of 2y x= , but is:

(a)vertically compressed by a factor of1

3.

(b)horizontally compressed by a factor of 4.

If you were to apply both vertical and horizontalstretches/compressions to a graph, would the order in

which those transformations are made matter?

The last types of transformations we will study are reflections about the x-axis and y-axis.

First well look at a reflection about the x-axis. On the same screen, graph the functions:

( )

21

2 22

4

4 4

Y x

Y x x

=

= = +


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Next, a reflection about the y-axisOn the same screen, graph the functions:

( ) ( )

4

14 4

2

Y x x

Y x x x x

= +

= + =

Example 3 : Write the equation of the function whose graph is the graph of ( )2

3 4y x= , but is:

(a)reflected about the x-axis.

(b)reflected about the y-axis.

(c)reflected about both the x- andy- axis.

When you apply reflections in both the x- andy-axes, would the order in which thosetransformations are made matter?


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When discussing the symmetry of the graphs of functions, it can get long and tedious to use the phrasesymmetric with respect to the y-axis or symmetry with respect to the origin. Instead, well use thewords even andoddto describe the symmetry that exists for the graph of a function.

A function is even if and only if its graph is symmetric with respect to the y-axis.

A function is odd if and only if its graph is symmetric with respect to the origin.

Do the following graphs appear to be even, odd, or neither?

Verify algebraically if the following graphs are even, odd, or neither.

For aneven

function, for every point(x, y) on the graph, the point (-x, y) isalso on the graph.

So for anodd function, for everypoint (x, y) on the graph, the point (-x,-y) is also on the graph.

( ) 32h x x x= ( ) 35 1g x x= ( ) 23 24 += xxxf

( ) 4 23 2f x x x = + ( )35 1g x x = ( )

32h x x x = +


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College Algebra

Section 2.4 Operations with Functions

Functions, like numbers, can be added, subtracted, multiplied, and divided.

Iffandg are functions:

The sum f g+ is the function defined by ( ) ( ) ( ) ( )f g x f x g x+ = +

The difference f g is the function defined by ( )( ) ( ) ( )f g x f x g x =

The product f g is the function defined by ( )( ) ( ) ( )f g x f x g x =

The quotientf

gis the function defined by ( )

( )

( )( ), 0

f xfx g x

g g x

=

Example 1: For the functions ( ) 22 3f x x= + and ( ) 34 1g x x= + find the following:

(a) ( )( )f g x+ =

(b) ( ) ( )f g x =

(c) ( ) ( )f g x =

(d) ( )f

xg

=

Example 2: Ex 14 pg 235


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Section 2.4 Operations with Functions Page 2


Composition of Functions

We can evaluate composite functions either by hand or using a calculator.

Example 3: Suppose that ( ) 22 3f x x= + and ( ) 34 1g x x= + . Find:

(a) ( )( )1f g

(b) ( )( )1g f

(c) ( ) ( )2f f

(d) ( )( )1g g

Not surprisingly, the domain of a composite function f g depends on both fandg.


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Section 2.4 Operations with Functions Page 3


This means that the domain of f g is a subset of the domain ofg.

(And the range of f g is a subset of the range off.)

Example 4: Suppose that ( )1

f xx

= and ( ) 1g x x= . Find the composites and state their domains.

(a) f g

(b) f f

The composite functions f g and g f are usually different.

Guess what? We can also undo, or decompose, a composite function. These decompositions are notalways unique but there is usually one that is more natural than the others.

Example 5: Find the functions fandg such that f g H= if ( ) 21H x x= .

Example 6: If u is the sum of t and 9, and v is u divided by 3, then write v as a function of t.

Example 7: 2.4 ex 100

Example 8: 2.4ex 102


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College Algebra

Section 2.5 One-to-One Functions; Inverse Functions

Sometimes its easier to look at what a 1-1 (one-to-one) function is not:

A function is NOT 1-1 if two different inputscorrespond to the same output.

The vertical line test tells us if a _________________ is a __________________ .

The horizontal line test tells us if a _________________ is __________________ .


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Section 2.5 One-to-One Functions; Inverse Functions Page 2


For each function, use the graph to determine whether the function is one-to-one.

If a function, f, is 1-1 then it has an inverse, 1f .

Example: Find the inverse of the 1-1 function ( ) ( ) ( ) ( ){ }1,2 , 2,8 , 3,18 , 4,32

To verify that one function is the inverse of another, we need to check that

( )( )1f f x x = where x is in the domain offand

( )( )1

f f x x

= where x is in the domain of1

f

.

Example: Verify that ( ) 2 6f x x= + and ( )1

32

g x x= are inverses of each other.

Using your calculator, graph ( )1Y f x= , ( )2Y g x= , and 3Y x= . Zoom square.


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What do you notice?

Example: Verify that the inverse of ( )5

2 3

xf x

x

=

+

is ( )3 5

1 2

xg x

x

+=

both algebraically and

graphically.

To find the inverse of a function by Switch and Solve Method (pg 243)1.2.3.4.5.Example: The function ( )

2 3, 2

2

xf x x

x

+=

+

, is one-to-one. Find its inverse and check the result.

In the above example, compare the horizontal and vertical asymptotes offand 1f . What do you notice?

Surprised?


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Example: Find the inverse of ( )2

23 , 0

3xf x x

x+= > . Check your answer and state the domain offand

find its range using 1f .


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College Algebra

Section 2.6 Constructing Functions with Variation

Variation is how one quantity varies in relation to another. Quantities may vary directly, inversely, orjointly.

Example 1: the cost of a smoothie is directly proportional to its size. If a 12-ounce smoothie is $3.60,them what is the cost of a 16-ounce smoothie?

Now well take a look at problems that use eitherinverse variation orjoint (combined) variation.

Example 2: The time required to rake the grounds at Rockwood Manor varies inversely with the numberof rakers. If 4 rakers can complete the job in 12 hours, then how long would it take for 6 rakers tocomplete the job?


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Section 2.5 Linear Functions and Models Page 2

Joint and Combined variation

Example 3: The cost of a fence that is 200 t long and 5 feet high is $3000. If the cost varies jointly withthe length and height, then what is the cost of a fence that is 25 feet long and 6 feet high?

Example 4: The cost of carpeting a room varies jointly as the length and the width. Julie advertises aprice of $263.40 to carpet a 9-ft by 12-ft room with Dupont Stainmaster. Julie gave a customer a priceof $482.90 for carpeting a room with a width of 4yd with that same carpeting, but she has forgotten thelength of the room. What is the length of the room?

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