Chapter 3 Section 6 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.

Chapter Chapter 33Section Section 66

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley


Introduction to Functions

11

44

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3.63.63.63.6Understand the definition of a relation.Understand the definition of a function.Decide whether an equation defines a function.Find domains and ranges.Use function notation. Apply the function concept in an application.


Objective 11

Understand the definition of a relation.

Slide 3.6 - 3


Understand the definition of a relation.

In an ordered pair (x, y), x and y are called the components of the ordered pair.

Slide 3.6 - 4

Any set of ordered pairs is called a relation.

The set of all first components of the ordered pairs of a relation is the domain of the relation, and the set of all second components of the ordered pairs is the range of the relation.


EXAMPLE 1

Solution:

Identifying Domains and Ranges of Relations Defined by Ordered Pairs

Use ordered pairs to define the relation.

4, , 6, , 7, , 3,A B B C

Slide 3.6 - 5


Objective 22

Understand the definition of a function.

Slide 3.6 - 6


Understand the definition of a function.A very important type of relation called a function.

A function is a set of ordered pairs in which each first component corresponds to exactly one second component.

Slide 3.6 - 7

By definition, the relation in the following order pairs is not a function, because the same first component, 3, corresponds to more then one second component.

5 63 3 3, , , ,7 3,8If the ordered pairs from this example were interchanged, giving the

relation

the result would be a function. In that case, each domain (first component) corresponds to exactly one range element (second component).

5 6 7, , , ,3 3 3 8,3 ,


EXAMPLE 2

Solution: function

Determining Whether Relations Are Functions

Determine whether each relation is a function.

2,8 , 1,1 , 0,0 , 1,1 , 2,8 ,

Slide 3.6 - 8

5,2 , 5,1 , 5,0

Solution: not a function


Objective 33

Decide whether an equation defines a function.

Slide 3.6 - 9


Decide whether an equation defines a function.

Slide 3.6 - 10

Given the graph of an equation, the definition of a function can be used to decide whether or not the graph represents a function. By the definition of a function , each x-value must lead to exactly one y-value.

The way to determine if a graph is a function is the vertical line test. If a vertical line intersects a graph in more than one point, then the graph is not the graph of a function.

Any nonvertical line is the graph of a function. For this reason, any linear equation of the form y = mx + b defines a function. (Recall that a vertical line has an undefined slope.)


EXAMPLE 3Deciding Whether Relations Define Functions

Determine whether each relation is a function.

Slide 3.6 - 11

Solution: functionSolution: not a function


Objective 44

Find domains and ranges.

Slide 3.6 - 12


By the definitions of domain and range given for

relations, the set of all numbers that can be used as

replacements for x in a function is the domain of the

function. The set of all possible values of y is the range

of the function.

Slide 3.6 - 13

Find domains and ranges.


EXAMPLE 4

Find the domain and range of the function y = x2 + 4.

Solution:

Domain:

Finding the Domain and Range of Functions

,

Slide 3.6 - 14

Range: 4,


Objective 55

Use function notation.

Slide 3.6 - 15


Use function notation.

The letters f, g, and h are commonly used to name functions. For example, the function y = 3x + 5 may be written

where f (x) is read “f of x.” The notation f (x) is another way of writing y in a function. For the function defined by f (x) = 3x + 5, if x = 7, then

Read this result, f (7) = 26, as “f of 7 equals 26.” The notation f (7) means the values of y when x is 7. The statement f (7) = 26 says that the value of y = 26 when x is 7. It also indicates that the point (7,26) lies on the graph of f.

Slide 3.6 - 16

3 5,f x x

7 73 5f

21 5 26

The notation f(x) does not mean f times x; f(x) means the value of x for the function f. It represents the y –value that corresponds to x.


Function NotationIn the notation f(x),

f is the name of the function,

x is the domain value,

and f(x) is the range value y for the domain value x.

Slide 3.6 - 17


EXAMPLE 5

Solution:

Using Function Notation

For the function f (x) = 6x − 2, find f (−1).

6 21 1f

6 21f

1 8f

Slide 3.6 - 18


Objective 66

Apply the function concept in an application.

Slide 3.6 - 19


EXAMPLE 6Applying the Function Concept to Population

The number of U.S. students ages 3 – 21served by educational programs for students with disabilities for selected years are given in the table.

Slide 3.6 - 20

a) Write a set of ordered pairs that defines a function f for these data.

b) Give the domain and range of f.

c) Find f (1998).

d) In which year did the number of students equal 5.7 million? That is, for what value of x does f (x) = 5.7million?


EXAMPLE 6Applying the Function Concept to Population (cont’d)

Solution:

The number of U.S. students ages 3 – 21served by educational programs for students with disabilities for selected years are given in the table.

1994 1996 1998

2

{ , ,

000

4.9 5.2 5.5

5. 2

, ,

0

, ,

027, }9, ,5.

f

1994 1996

1998

domain : { , ,

, , 2000 2002}

5.5 million

2000Slide 3.6 - 21

4.9 5.range : { , , 2 5.5 5., 7,5.9}

a)

b)

c)

d)

Chapter 3 Section 6 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.

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