Intermediate Algebra (11th edition)

574 CHAPTER 10 Inverse, Exponential, and Logarithmic Functions

Using the Horizontal Line Test

Use the horizontal line test to determine whether each graph is the graph of a one-to-

one function.

NOW TRY EXAMPLE 2

EXERCISE 2
Use the horizontal line test to
determine whether each graph
is the graph of a one-to-one
function.

(a)

(b)

x

y

0

Finding the Equation of the Inverse of

For a one-to-one function ƒ defined by an equation , find the defining

equation of the inverse as follows.

Step 1 Interchange xand y.

Step 2 Solve for y.

Step 3 Replace ywith ƒ-^11 x 2.

y=ƒ 1 x 2

yƒ 1 x 2

(a)

Because a horizontal line intersects

the graph in more than one point (actu-

ally three points), the function is not

one-to-one.

(b)

Every horizontal line will intersect

the graph in exactly one point. This

function is one-to-one.

NOW TRY

(x 1 , y)(x 3 , y)

y = f(x)

0

x

y

(x 2 , y) y = f(x)

0

x

y

OBJECTIVE 3 Find the equation of the inverse of a function.The inverse

of a one-to-one function is found by interchanging the x- and y-values of each of its

ordered pairs. The equation of the inverse of a function defined by is found

in the same way.

y= ƒ 1 x 2

Finding Equations of Inverses

Decide whether each equation defines a one-to-one function. If so, find the equation

that defines the inverse.

(a)

The graph of is a nonvertical line, so by the horizontal line test, ƒ is

a one-to-one function. To find the inverse, let and follow the steps.

Interchange xand y. (Step 1)

Solve for y. (Step 2)

Replace ywith (Step 3)

This equation can be written as follows.

or ƒ-^11 x 2 = a-cb=ac-cb

1

2

x-

5

2

ƒ-^11 x 2 =

x

2

-

5

2

,

ƒ-^11 x 2 = ƒ-^11 x 2.

x- 5

2

y=

x- 5

2

2 y=x- 5

x= 2 y+ 5

y= 2 x+ 5

y= ƒ 1 x 2

y= 2 x+ 5

ƒ 1 x 2 = 2 x+ 5

EXAMPLE 3

x

y

0

NOW TRY ANSWERS

2. (a)one-to-one
   (b)not one-to-one