Beginning Algebra, 11th Edition

NOTE An equation in which yis squared does not usually define a function, be-

cause most x-values will lead to two y-values. This is true for any evenpower of y,

such as and so on. Similarly, an equation involving does not usually

define a function, because some x-values lead to more than one y-value.

y^2 ,y^4 ,y^6 , | y |

232 CHAPTER 3 Linear Equations and Inequalities in Two Variables; Functions

While xcan take any

real number value,

notice that yis always

greater than or equal

to 0.

NOW TRY

OBJECTIVE 4 Find domains and ranges.The set of all numbers that can be

used as replacements for xin a function is the domain of the function, and the set of

all possible values of yis the range of the function.

Finding the Domain and Range of Functions

Find the domain and range of each function.

(a)

Any number may be used for x, so the domain is the set

of all real numbers. Also, any number may be used for y,

so the range is also the set of all real numbers. As indicated

in FIGURE 44, the graph of the equation is a straight line that

extends infinitely in both directions, confirming that both

the domain and range are

(b)

Any number can be squared, so the domain is the set of all real numbers.

However, since the values of ycannot be negative, making the range the set

of all nonnegative numbers, or in interval notation. The ordered pairs shown

in the table are used to get the graph of the function in FIGURE 45.

3 0, q 2

y= x^2 ,

y=x^2

1 - q, q 2.

y= 2 x- 4

EXAMPLE 4

x

y

y = x^2

Domain

(–, )

Range

[0, )

–2 0 2

6

4

2

FIGURE 45

xy

00

11

1

24

4

39

• 3 9


• 2


• 1

(d)

Use the vertical line test. Any vertical

line will intersect the graph just once, so

this is the graph of a function.

(e)

The vertical line test shows that this graph

is not the graph of a function. A vertical

line could intersect the graph twice.

y

0 x

y

0 x

(f )

The graph of is a vertical line, so the equation does notdefine a function.

NOW TRY

x= 4

x= 4

x

y

Range

(–, )

Domain

(–, )

0 2

–4 y = 2x – 4

FIGURE 44

NOW TRY
EXERCISE 3
Determine whether each
relation is a function.

(a)

(b)

y=x- 5

NOW TRY ANSWERS

3. (a)yes (b)no

x

y

0

NOW TRY
EXERCISE 4
Find the domain and range of
the function.

y=x^2 - 2

4.domain: ; range: 1 - q, q 2 3 - 2, q 2

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