Q.11 The function fis one-to-one. Fin... [FREE SOLUTION]

91影视

Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 5: Q.11 (page 348)

The function $f$ is one-to-one. Find the inverse of each function and check your answer. State the
domain and the range of $f$ and $f^{- 1}$ .

Short Answer

Expert verified

The inverse is

$f^{- 1} (x) = \frac{x + 1}{x}$ .

Step by step solution

Step 1: Concept used

At first in $y = f (x)$ , interchange the variables $x$ and $y$ , to obtain $x = f (y)$ .

Then solve the equation for $y$ in terms of $x$ .

After that, check the result by showing that,

$f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$

Interchange the variables x and y

Given that, $f (x) = \frac{1}{x - 1}$

Replacing $f (x)$ with $y$ and interchange the variables $x$ and $y$ in

$y = \frac{1}{x - 1}$

to obtain

role="math" localid="1646377359268" $x = \frac{1}{y - 1}$

Solve for y in terms of x

$x = \frac{1}{y - 1}$

Multiplying both sides by $(y - 1)$ ,

$x (y - 1) = 1$

Applying distribution property,

$x y - x = 1$

Adding $x$ with both sides,

$x y = x + 1$

Divide by role="math" localid="1646377835824" $x$ ,

$y = \frac{x + 1}{x}$

The inverse is

$f^{- 1} (x) = \frac{x + 1}{x}$

Check result

$f^{- 1} (f (x)) = f^{- 1} (\frac{1}{x - 1})$

$= \frac{\frac{1}{x - 1} + 1}{\frac{1}{x - 1}}$

$= \frac{1 + (x - 1)}{1}$

$= x$

$f (f^{- 1} (x)) = f (\frac{x + 1}{x})$

$= \frac{1}{\frac{x + 1}{x} - 1}$

$= \frac{1}{\frac{x + 1 - x}{x}}$

$= x$

Thus, the result is checked.

Statement of domain and range

$f^{- 1} (f (x)) = x$ , where $x$ is in domain of $f$ and

$D o m a i n o f f = R a n g e o f f^{- 1}$

$f (f^{- 1} (x)) = x$ , where $x$ is in domain of $f^{- 1}$ and

$R a n g e o f f = D o m a i n o f f^{- 1}$ .

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91影视

The function $f$ is one-to-one. Find the inverse of each function and check your answer. State the
domain and the range of $f$ and $f^{- 1}$ .

Short Answer

Step by step solution

Step 1: Concept used

Interchange the variables x and y

Solve for y in terms of x

Check result

Statement of domain and range

One App. One Place for Learning.

Most popular questions from this chapter

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91影视

The function fis one-to-one. Find the inverse of each function and check your answer. State thedomain and the range offandf-1.

Short Answer

Step by step solution

Step 1: Concept used

Interchange the variables x and y

Solve for y in terms of x

Check result

Statement of domain and range

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Discrete Mathematics

Decision Maths

Statistics

Probability and Statistics

Logic and Functions

Study anywhere. Anytime. Across all devices.

The function $f$ is one-to-one. Find the inverse of each function and check your answer. State the
domain and the range of $f$ and $f^{- 1}$ .