Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 5: Q. 97 (page 271)

Given
$f (x) = \frac{a x + b}{c x + d}$
find $f^{- 1}$ $(x)$ . If $c 鈮� 0$ , 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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91影视

Given
$f (x) = \frac{a x + b}{c x + d}$
find $f^{- 1}$ $(x)$ . If $c 鈮� 0$ , 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

One App. One Place for Learning.

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91影视

Givenfx=ax+bcx+dfind f-1x. If c鈮�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

Geometry

Applied Mathematics

Probability and Statistics

Logic and Functions

Study anywhere. Anytime. Across all devices.

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