Intermediate Algebra (11th edition)

For example, suppose the point shown in

FIGURE 2belongs to a one-to-one function ƒ. Then the

point belongs to. The line segment connecting

and is perpendicular to, and cut in half by,

the line The points and are “mirror

images” of each other with respect to

We can find the graph of from the graph of ƒby

locating the mirror image of each point in ƒwith

respect to the line

Graphing the Inverse

Graph the inverses of the functions ƒ (shown in blue) in FIGURE 3.

In FIGURE 3the graphs of two functions ƒ are shown in blue. Their inverses are

shown in red. In each case, the graph of is a reflection of the graph of ƒ with

respect to the line y= x.

ƒ-^1

EXAMPLE 4

yx.

ƒ^1

y= x.

y= x. 1 a, b 2 1 b, a 2

1 a, b 2 1 b, a 2

1 b, a 2 ƒ-^1

1 a, b 2

576 CHAPTER 10 Inverse, Exponential, and Logarithmic Functions

NOW TRY
EXERCISE 4
Use the given graph to graph
the inverse of ƒ.

x

y f

0

2

7

–3

–2

NOW TRY ANSWER
4.

x

y

0

–3

–2 7

f –1

x

y

0

f–1

f

y = x

x

y

0

y = x

f –1

f

FIGURE 3

In Example 3we showed that the inverse of the one-to-one function defined by

is given by If we use a square viewing window of a

graphing calculator and graph

and

we can see how this reflection appears on the screen. See FIGURE 4.

y 2 =ƒ-^11 x 2 = y 3 = x,

x- 5

2

y 1 = ƒ 1 x 2 = 2 x+ 5, ,

ƒ 1 x 2 = 2 x+ 5 ƒ-^11 x 2 = x 2 -^5.

CONNECTIONS

y 2 = f –1(x)

y 1 = f(x)

10

–10

–15 15

FIGURE 4

For Discussion or Writing

Some graphing calculators have the capability to “draw” the inverse of a function.

Use a graphing calculator to draw the graphs of and its inverse in a

square viewing window.

ƒ 1 x 2 =x^3 + 2

x

y

b

a

ab

(b, a)

(a, b)

y = x

0

FIGURE 2

NOW TRY