Chapter 5: Q. 97 (page 271)

Given
$f (x) = \frac{a x + b}{c x + d}$
find $f^{- 1}$ $(x)$ . If $c \neq 0$ , under what conditions on $a, b, c,$ and
$d$ is $f = f^{- 1} (x)$ ?

Short Answer

Expert verified

Part (a). The inverse of the function is

$f^{- 1} (x) = \frac{- d x + b}{c x - a}$ .

Part (b). If $f^{- 1} (x) = f (x)$ then $a = - d$ .

Step by step solution

Step 1. Find f-1x

The given function is $f (x) = \frac{a x + b}{c x + d}$ .

Substituting $f (x) = y$ and replace $y$ by $x$

localid="1646328397323" $x = \frac{a y + b}{c y + d}$

Finding the inverse, we get:

$c x y + d x = a y + b c x y - a y = b - d x y (c x - a) = b - d x y = \frac{b - d x}{c x - a}$

Replace $y = f^{- 1} (x)$ , we get:

localid="1646639581462" $f^{- 1} (x) = \frac{- d x + b}{c x - a}$

Step 2. Equate fx=f-1x

Equating $f^{- 1} (x) = f (x)$ , we get:

$\frac{a x + b}{c x + d} = \frac{- d x + b}{c x - a}$

From above equation, we can observe that, $a = - d$

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Given
$f (x) = \frac{a x + b}{c x + d}$
find $f^{- 1}$ $(x)$ . If $c \neq 0$ , under what conditions on $a, b, c,$ and
$d$ is $f = f^{- 1} (x)$ ?

Short Answer

Step by step solution

Step 1. Find f-1x

Step 2. Equate fx=f-1x

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Givenfx=ax+bcx+dfind f-1x. If c≠0, under what conditions on a,b,c,anddisf=f-1x?

Short Answer

Step by step solution

Step 1. Find f-1x

Step 2. Equate fx=f-1x

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Pure Maths

Decision Maths

Mechanics Maths

Applied Mathematics

Logic and Functions

Study anywhere. Anytime. Across all devices.

Given
$f (x) = \frac{a x + b}{c x + d}$
find $f^{- 1}$ $(x)$ . If $c \neq 0$ , under what conditions on $a, b, c,$ and
$d$ is $f = f^{- 1} (x)$ ?