FunctionsLesson 2

Warm Up 1. Write an equation of the line that passes through the points (-2, 1) and (3, 2). 2. Find the gradient of the line that is perpendicular to the line 4x – 7y = 12.

3. Write the equation of the vertical line that passes through the point (3, 2).

075 yx

4

7m

3x

Relation Relation – pairs of quantities that are related

to each other

Example: The area A of a circle is related to its radius r by the formula

.2rA

Function There are different kinds of relations.

When a relation matches each item from one set with exactly one item from a different set the relation is called a function.

Definition of a Function A function is a relationship between two

variables such that each value of the first variable is paired with exactly one value of the second variable.

The domain is the set of permitted x values.

The range is the set of found values of y. These will be called images.

Let’s take a look at the function that relates the time of day to the temperature.

Rules to be a Function

Is it a Function? For each x, there is

only one value of y.

Therefore, it IS a function.

Domain, x Range, y

1 -3.6

2 -3.6

3 4.2

4 4.2

5 10.7

6 12.1

52 52

Is it a function? Three different y-

values (7, 8, and 10) are paired with one x-value.

Therefore, it is NOT a function

Domain, x Range, y

3 7

3 8

3 10

4 42

10 34

11 18

52 52

Function? Is it a function? Name the domain and range.

YES. For every x-value, there is only one value of y.

Domain: (3, 4, 5, 7, 8) Range: (-5, -8, 6, 10, 2)

{(3, -5), (4, -8), (5, 6), (7, 10), (8, 2)}

Function? Is it a function? State the domain and range.

No. The x-value of 5 is paired with two different y-values.

Domain: (5, 6, 3, 4, 12) Range: (8, 7, -1, 2, 9, -2)

{(5, 8), (6, 7), (3, -1), (4, 2), (5, 9), (12, -2)

Function? Is it a function? Name the domain and range.

Yes. For every x-value, there is only one value of y.

Domain: (-2, 4, 3, 7, 9, 2) Range: (3, 6, 1, -3, 8)

{(-2, 3), (4, 6), (3, 1), (7, 6), (9, -3), (2, 8)}

Function?

YES

Vertical Line Test Used to determine if a graph is a function.

If a vertical line intersects the graph at more than one point, then the graph is NOT a function.

NOT a function

IS a function

You Try…...

You Try….

You Try: Is it a Function? YES

You Try…Is it a function? YES.

You Try…Is it a Function? NO.

Is it a function? Give the domain and range.

4,4:

2,4:

Range

Domain

FUNCTION

Give the Domain and Range.

2:

1:

yRange

xDomain

30:

22:

yRange

xDomain

IB Notation….

When a function is defined for all real values, we write the domain of f as

Functional Notation

We have seen an equation written in the form y = some expression in x.

Another way of writing this is to use functional notation.

For Example, you could write y = x²

as f(x) = x².

Functional Notation

f(x) = 3x + 5

Find:

( 2)f (0)f (5)f

1

56

523

5

50

503

20

515

553

Functional Notation

Find:

( 3)f (0)f (4)f

2( ) 3 2f x x x

32

230

2327

2333 2

2

200

2003 2

46

2448

24163

2443 2

Functional Notation

Find:

( )f m

2( ) 2f x x x

3( )f m

22 mm

85

23933

2333

233

2

2

2

mm

mmmm

mmm

mm

Let’s look at Functions Graphically

Find: 2 4( ) ( )f g

( )f x ( )g x

Find: 1 0( ) ( )f g

( )f x ( )g x

Find: 2 1( ) ( )f g

( )f x ( )g x

Find: 5 0( ) ( )f g

( )f x ( )g x

Find: 4 1( ) ( )f g

( )f x ( )g x

Find: 4 2( ) ( )f g

( )f x ( )g x

Find: 2 0( ) ( )f g

( )f x ( )g x

Find: 5 3( ) ( )g f

( )f x ( )g x

Piecewise-Defined Function

A piecewise-defined function is a function that is defined by two or more equations over a specified domain.

The absolute value function

can be written as a piecewise-defined function.

The basic characteristics of the absolute value function are summarized on the next page.

xxf

Example

Evaluate the function when x = -1 and 0.

Domain of a Function

The domain of a function can be implied by the expression used to define the function

The implied domain is the set of all real numbers for which the expression is defined.

For example,

The function has an implied

domain that consists of all real x other than

x = ±2

The domain excludes x-values that result in division by zero.

Another common type of implied domain is that used to avoid even roots of negative numbers.

EX:

is defined only for

The domain excludes x-values that result in even roots of negative numbers.

.0x