Beginning Algebra, 11th Edition

SECTION 3.6 Introduction to Functions^231

Determining Whether Relations Define Functions

Determine whether each relation graphed or defined here is a function.

EXAMPLE 3

OBJECTIVE 3 Decide whether an equation defines a function.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.

In FIGURE 43(a), the indicated x-value leads to two y-values, so this graph is not

the graph of a function. A vertical line can be drawn that intersects the graph in more

than one point.

x

y

0

(x, y 1 )

(x, y 2 )

Not a function

x

y

(x, y 1 )^0

A function

x

y

0

(x, y 1 )

A function

(a) (b) (c) FIGURE 43

By contrast, in FIGURE 43(b)and FIGURE 43(c)any vertical line will intersect each

graph in no more than one point, so these graphs are graphs of functions. This idea

leads to the vertical line testfor a function.

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.

As FIGURE 43(b)suggests, any nonvertical line is the graph of a function. Thus,

any linear equation of the form defines a function.(Recall that a ver-

tical line has undefined slope.)

ymxb

(a)

Because there are two ordered pairs with

first component , as shown in red,

this is not the graph of a function.

- 4

(b)

Every first component is matched with

one and only one second component, and

as a result, no vertical line intersects the

graph in more than one point. Therefore,

this is the graph of a function.

y 4

–4 4 –4

0 x

y