Intermediate Algebra (11th edition)

(c)

The arrowheads indicate that the

line extends indefinitely left and right,

as well as up and down. Therefore, both

the domain and the range include all real

numbers, written

(d)

The graph extends indefinitely left and

right, as well as upward. The domain is

Because there is a least y-value,

the range includes all numbers greater

than or equal to -3,written 3 - 3, q 2.

- 3,

1 - q, q 2.

1 - q, q 2.

SECTION 3.5 Introduction to Relations and Functions 185

The graphs in (a), (c), and (d) satisfy the vertical line test and represent functions.

The graph in (b) fails the vertical line test, since the same x-value corresponds to two

different y-values, and is not the graph of a function. NOW TRY

x

y

0 x

y

(^02)
–3
NOW TRY

OBJECTIVE 4 Identify functions defined by graphs and equations. Since

each value of xin a function corresponds to only one value of y, any vertical line

drawn through the graph of a function must intersect the graph in at most one point.

Vertical Line Test

If every vertical line intersects the graph of a relation in no more than one point,

then the relation is a function.

FIGURE 46illustrates the vertical line test with the graphs of two relations.

x

y

0

Not a function

The same x-value corresponds

to four different y-values.

FIGURE 46

A vertical line

intersects the graph

more than once.

Using the Vertical Line Test

Use the vertical line test to determine whether each relation graphed in Example 3

is a function. (We repeat the graphs here.)

(a) (b) (c) (d)

EXAMPLE 4

x

y

0

(4, –3)

(–1, 1)

(1, 2)

(0, –1)

Function

x

y

–4^0

–6

6

4

Not a function

x

y

0

Function

x

y

(^02)
–3
Function
NOW TRY
EXERCISE 3
Give the domain and range of
the relation.
x
y
(^0) –2
–3
NOW TRY ANSWERS
3.domain: ;
range: 3 - 2, q 2
1 - q, q 2
Function
Each x-value corresponds
to only one y-value.
y
0 x
NOW TRY
EXERCISE 4
Use the vertical line test to
determine whether the rela-
tion is a function.
x
y
0 3
3
6
6
4.not a function
Any vertical line
intersects the
graph only once.