James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 1.5 Inverse Functions and Logarithms 57
A
f
x
f–!
This definition says that if f maps x into y, then f 21 maps y back into x. (If f were
not one-to-one, then f 21 would not be uniquely defined.) The arrow diagram in Figure 5
indicates that f 21 reverses the effect of f. Note that
B
FIGURE 5
y
domain of f 21 − range of f
range of f 21 − domain of f
For example, the inverse function of f sxd − x 3 is f 21 sxd − x 1y3 because if y − x 3 ,
then
f 21 syd − f 21 sx 3 d − sx 3 d 1y3 − x
CAUTION Do not mistake the 21 in f 21 for an exponent. Thus
f 21 sxd does not mean
1
f sxd
The reciprocal 1yf sxd could, however, be written as f f sxdg 21 .
A
1
3
8
B
5
7
_10
ExamplE 3 If f s1d − 5, f s3d − 7, and f s8d − 210, find f 21 s7d, f 21 s5d, and
f 21 s210d.
SOLUTION From the definition of f 21 we have
f 21 s7d − 3 because f s3d − 7
f
f 21 s5d − 1 because f s1d − 5
A
B
f 21 s210d − 8 because f s8d − 210
1
3
5
7
The diagram in Figure 6 makes it clear how f 21 reverses the effect of f in this case.
■
8
f –!
_10
The letter x is traditionally used as the independent variable, so when we concentrate
on f 21 rather than on f, we usually reverse the roles of x and y in Definition 2 and write
FIGURE 6
The inverse function reverses inputs
and outputs.
3 f 21 sxd − y &? f syd − x
By substituting for y in Definition 2 and substituting for x in (3), we get the following
cancellation equations:
4
f 21 s f sxdd − x
f s f 21 sxdd − x
for every x in A
for every x in B
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.