SECTION 1.2 Basics of Functions and Their Graphs

154

Chapter 1 Functions and Graphs

SECTION 1.2

Objectives

??? Find the domain and

range of a relation.

??? Determine whether a

???

???

???

???

???

???

???

relation is a function.

Determine whether an

equation represents a

function.

Evaluate a function.

Graph functions by

plotting points.

Use the vertical line test

to identify functions.

Obtain information

about a function from

its graph.

Identify the domain and

range of a function from

its graph.

Identify intercepts from

a function¡¯s graph.

Basics of Functions and Their Graphs

Magni?ed 6000 times, this color-scanned image

shows a T-lymphocyte blood cell (green)

infected with the HIV virus (red). Depletion of

the number of T cells causes destruction of the

immune system.

The average number of T cells in

a person with HIV is a function

of time after infection. In this

section, you will be introduced

to the basics of functions and their

graphs. We will analyze the graph

of a function using an example that

illustrates the progression of HIV

and T cell count. Much of our work

in this course will be devoted to the

important topic of functions and how

they model your world.

Relations

Forbes magazine published a list of the highest-paid TV celebrities between June 2010

and June 2011. The results are shown in Figure 1.13.

Highest Paid TV Celebrities between June 2010 and June 2011

440

400

Find the domain and range

of a relation.

Earnings (millions of dollars)

???

360

320

$315 million

280

240

200

160

120

$80 million

80

$80 million

$55 million

$51 million

Ellen

DeGeneres

Ryan

Seacrest

40

Oprah

Winfrey

Simon

Cowell

Dr. Phil

McGraw

Celebrity

FIGURE 1.13

Source: Forbes

The graph indicates a correspondence between a TV celebrity and that person¡¯s

earnings, in millions of dollars. We can write this correspondence using a set of

ordered pairs:

{(Winfrey, 315), (Cowell, 80), (McGraw, 80), (DeGeneres, 55), (Seacrest, 51)}.

These braces indicate we are representing a set.

Section 1.2 Basics of Functions and Their Graphs

155

The mathematical term for a set of ordered pairs is a relation.

De?nition of a Relation

A relation is any set of ordered pairs. The set of all ?rst components of the ordered

pairs is called the domain of the relation and the set of all second components is

called the range of the relation.

EXAMPLE 1

Finding the Domain and Range of a Relation

Find the domain and range of the relation:

{(Winfrey, 315), (Cowell, 80), (McGraw, 80), (DeGeneres, 55), (Seacrest, 51)}.

SOLUTION

The domain is the set of all ?rst components. Thus, the domain is

{Winfrey, Cowell, McGraw, DeGeneres, Seacrest}.

The range is the set of all second components. Thus, the range is

{315, 80, 55, 51}.

Although Cowell and McGraw both

earned $80 million, it is not

necessary to list 80 twice.

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Check Point 1 Find the domain and range of the relation:

{(0, 9.1), (10, 6.7), (20, 10.7), (30, 13.2), (40, 21.2)}.

As you worked Check Point 1, did you wonder if there was a rule that assigned the

¡°inputs¡± in the domain to the ¡°outputs¡± in the range? For example, for the ordered

pair (30, 13.2), how does the output 13.2 depend on the input 30? The ordered pair

is based on the data in Figure 1.14(a), which shows the percentage of ?rst-year U.S.

college students claiming no religious af?liation.

Percentage of First-Year United States College

Students Claiming No Religious Affiliation

Women

Men

21.2

21%

16.9

18%

12%

9%

27%

25.1

24%

14.0 13.2

11.9

9.7

9.1

10.7

6.7

6%

3%

1970

1980

1990

Year

2000

2010

FIGURE 1.14(a) Data for women and men

Source: John Macionis, Sociology, Fourteenth Edition, Pearson, 2012.

Percentage Claiming No

Religious Affiliation

Percentage Claiming No

Religious Affiliation

27%

15%

Percentage of First-Year United

States College Women Claiming

No Religious Affiliation

y

(40, 21.2)

24%

21%

18%

(30, 13.2)

15%

12%

(0, 9.1)

9%

(20, 10.7)

6%

(10, 6.7)

3%

5

10

15

20

25

30

35

40

Years after 1970

FIGURE 1.14(b) Visually representing the relation for the

women¡¯s data

x

156

Chapter 1 Functions and Graphs

In Figure 1.14(b), we used the data for college women to create the following

ordered pairs:

Percentage of First-Year

United States College

Women Claiming

y

No Religious Affiliation

Percentage Claiming No

Religious Affiliation

27%

24%

21%

18%

12%

Consider, for example, the ordered pair (30, 13.2).

(30, 13.2)

15%

(30, 13.2)

(0, 9.1)

9%

(20, 10.7)

6%

30 years after 1970,

or in 2000,

(10, 6.7)

3%

5 10 15 20 25 30 35 40

Years after 1970

Determine whether a relation is

a function.

Table 1.1 Highest-Paid

TV Celebrities

Celebrity

Earnings

(millions of dollars)

Winfrey

315

Cowell

80

McGraw

80

DeGeneres

55

Seacrest

51

13.2% of first-year college women

claimed no religious affiliation.

x

The ?ve points in Figure 1.14(b) visually represent the relation formed from the

women¡¯s data. Another way to visually represent the relation is as follows:

FIGURE 1.14(b) (repeated)

Visually representing the relation for the

women¡¯s data

???

percentage of first@year college

women claiming no religious ¡Ý.

affiliation

? years after 1970,

(40, 21.2)

0

10

20

30

40

9.1

6.7

10.7

13.2

21.2

Domain

Range

Functions

Table 1.1, based on our earlier discussion, shows the highest-paid TV celebrities and

their earnings between June 2010 and June 2011, in millions of dollars. We¡¯ve used

this information to de?ne two relations.

Figure 1.15(a) shows a correspondence between celebrities and their earnings.

Figure 1.15(b) shows a correspondence between earnings and celebrities.

Winfrey

Cowell

McGraw

DeGeneres

Seacrest

315

80

55

51

315

80

55

51

Winfrey

Cowell

McGraw

DeGeneres

Seacrest

Domain

Range

Domain

Range

FIGURE 1.15(a) Celebrities correspond

to earnings.

FIGURE 1.15(b) Earnings correspond to

celebrities.

A relation in which each member of the domain corresponds to exactly one

member of the range is a function. Can you see that the relation in Figure 1.15(a) is a

function? Each celebrity in the domain corresponds to exactly one earnings amount

in the range: If we know the celebrity, we can be sure of his or her earnings. Notice

that more than one element in the domain can correspond to the same element in

the range: Cowell and McGraw both earned $80 million.

Is the relation in Figure 1.15(b) a function? Does each member of the domain

correspond to precisely one member of the range? This relation is not a function

because there is a member of the domain that corresponds to two different members

of the range:

(80, Cowell)

(80, McGraw).

The member of the domain 80 corresponds to both Cowell and McGraw in the range.

If we know that earnings are $80 million, we cannot be sure of the celebrity. Because

a function is a relation in which no two ordered pairs have the same ?rst component

and different second components, the ordered pairs (80, Cowell) and (80, McGraw)

are not ordered pairs of a function.

Section 1.2 Basics of Functions and Their Graphs

157

Same first component

(80, Cowell)

(80, McGraw)

Different second components

De?nition of a Function

A function is a correspondence from a ?rst set, called the domain, to a second set,

called the range, such that each element in the domain corresponds to exactly one

element in the range.

In Check Point 1, we considered a relation that gave a correspondence between

years after 1970 and the percentage of ?rst-year college women claiming no religious

af?liation. Can you see that this relation is a function?

Each element in the domain

{(0, 9.1), (10, 6.7), (20, 10.7), (30, 13.2), (40, 21.2)}

corresponds to exactly one element in the range.

However, Example 2 illustrates that not every correspondence between sets is a

function.

EXAMPLE 2

Determining Whether a Relation Is a Function

Determine whether each relation is a function:

a. {(1, 6), (2, 6), (3, 8), (4, 9)}

b. {(6, 1), (6, 2), (8, 3), (9, 4)}.

SOLUTION

1

2

3

4

6

8

9

Domain

Range

FIGURE 1.16(a)

6

8

9

1

2

3

4

Domain

Range

FIGURE 1.16(b)

GREAT QUESTION!

If I reverse a function¡¯s

components, will this new

relation be a function?

If a relation is a function,

reversing the components in each

of its ordered pairs may result in a

relation that is not a function.

We begin by making a ?gure for each relation that shows the domain and the range

(Figure 1.16).

a. Figure 1.16(a) shows that every element in the domain corresponds to exactly

one element in the range. The element 1 in the domain corresponds to the

element 6 in the range. Furthermore, 2 corresponds to 6, 3 corresponds to 8,

and 4 corresponds to 9. No two ordered pairs in the given relation have the same

?rst component and different second components. Thus, the relation is a function.

b. Figure 1.16(b) shows that 6 corresponds to both 1 and 2. If any element in

the domain corresponds to more than one element in the range, the relation

is not a function. This relation is not a function; two ordered pairs have the

same ?rst component and different second components.

Same first component

(6, 1)

(6, 2)

Different second components

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Look at Figure 1.16(a) again. The fact that 1 and 2 in the domain correspond to

the same number, 6, in the range does not violate the de?nition of a function. A

function can have two different ?rst components with the same second component.

By contrast, a relation is not a function when two different ordered pairs have

the same ?rst component and different second components. Thus, the relation in

Figure 1.16(b) is not a function.

158

Chapter 1 Functions and Graphs

Check Point 2 Determine whether each relation is a function:

a. {(1, 2), (3, 4), (5, 6), (5, 8)}

???

Determine whether an equation

represents a function.

b. {(1, 2), (3, 4), (6, 5), (8, 5)}.

Functions as Equations

Functions are usually given in terms of equations rather than as sets of ordered pairs.

For example, here is an equation that models the percentage of ?rst-year college

women claiming no religious af?liation as a function of time:

y = 0.014x2 - 0.24x + 8.8.

The variable x represents the number of years after 1970. The variable y represents

the percentage of ?rst-year college women claiming no religious af?liation. The

variable y is a function of the variable x. For each value of x, there is one and only

one value of y. The variable x is called the independent variable because it can be

assigned any value from the domain. Thus, x can be assigned any nonnegative integer

representing the number of years after 1970. The variable y is called the dependent

variable because its value depends on x. The percentage claiming no religious

af?liation depends on the number of years after 1970. The value of the dependent

variable, y, is calculated after selecting a value for the independent variable, x.

We have seen that not every set of ordered pairs de?nes a function. Similarly, not

all equations with the variables x and y de?ne functions. If an equation is solved for y

and more than one value of y can be obtained for a given x, then the equation does

not de?ne y as a function of x.

EXAMPLE 3

Determining Whether an Equation

Represents a Function

Determine whether each equation de?nes y as a function of x:

a. x2 + y = 4

b. x2 + y2 = 4.

SOLUTION

Solve each equation for y in terms of x. If two or more values of y can be obtained

for a given x, the equation is not a function.

a.

x2 + y = 4

x2 + y - x2 = 4 - x2

y = 4 - x2

This is the given equation.

Solve for y by subtracting x2 from both sides.

Simplify.

From this last equation we can see that for each value of x, there is one

and only one value of y. For example, if x = 1, then y = 4 - 12 = 3. The

equation de?nes y as a function of x.

b.

x2 + y2 = 4

x2 + y2 - x2 = 4 - x2

y2 = 4 - x2

This is the given equation.

Isolate y2 by subtracting x2 from both sides.

Simplify.

y = { 24 - x2 Apply the square root property: If u2 = d,

then u = { 1d .

The { in this last equation shows that for certain values of x (all values

between -2 and 2), there are two values of y. For example, if x = 1, then

y = { 24 - 12 = { 23. For this reason, the equation does not de?ne y

as a function of x.

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