Chapter 5: Q. 70 (page 269)

the function f is one-to-one. Find its inverse and check your answer.
$f (x) = \frac{- 3 x - 4}{x - 2}$

Short Answer

Expert verified

The inverse of the function $f (x) = \frac{- 3 x - 4}{x - 2}$ is $f^{- 1} (x) = \frac{2 x - 4}{x + 3} x \neq - 3$

Step by step solution

Step 1. Given data

The given function is $f (x) = \frac{- 3 x - 4}{x - 2}$

Step 2. interchanging variables

Replace $f (x)$ with y and interchange x and y

$f (x) = \frac{- 3 x - 4}{x - 2} y = \frac{- 3 x - 4}{x - 2} x = \frac{- 3 y - 4}{y - 2}$

Step 3. The inverse of the function

Solve the equation for y

$x = \frac{- 3 y - 4}{y - 2} x (y - 2) = - 3 y - 4 x y - 2 x = - 3 y - 4 x y + 3 y = 2 x - 4 y (x + 3) = 2 x - 4 y = \frac{2 x - 4}{x + 3}$

replace $y$ with $f^{- 1} (x)$

role="math" localid="1646160883591" $f^{- 1} (x) = \frac{2 x - 4}{x + 3} x \neq - 3$

Step 4. Verification

Determine $f (f^{- 1} (x))$

$f (f^{- 1} (x)) = f (\frac{2 x - 4}{x + 3}) f (f^{- 1} (x)) = \frac{- 3 (\frac{2 x - 4}{x + 3}) - 4}{(\frac{2 x - 4}{x + 3}) - 2} f (f^{- 1} (x)) = \frac{- 3 (2 x - 4) - 4 (x + 3)}{(2 x - 4) - 2 (x + 3)} f (f^{- 1} (x)) = \frac{- 6 x + 12 - 4 x - 12}{2 x - 4 - 2 x - 6} f (f^{- 1} (x)) = \frac{- 10 x}{- 10} f (f^{- 1} (x)) = x$

determine $f^{- 1} (f (x))$

$f^{- 1} (f (x)) = f^{- 1} (\frac{- 3 x - 4}{x - 2}) f^{- 1} (f (x)) = \frac{2 (\frac{- 3 x - 4}{x - 2}) - 4}{(\frac{- 3 x - 4}{x - 2}) + 3} f^{- 1} (f (x)) = \frac{2 (- 3 x - 4) - 4 (x - 2)}{(- 3 x - 4) + 3 (x - 2)} f^{- 1} (f (x)) = \frac{- 6 x - 8 - 4 x + 8}{- 3 x - 4 + 3 x - 6} f^{- 1} (f (x)) = \frac{- 10 x}{- 10} f^{- 1} (f (x)) = x$

localid="1646178687892" $f (f^{- 1} (x)) = x a n d f^{- 1} (f (x)) = x$ so inverse function is correct

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

the function f is one-to-one. Find its inverse and check your answer.
$f (x) = \frac{- 3 x - 4}{x - 2}$

Short Answer

Step by step solution

Step 1. Given data

Step 2. interchanging variables

Step 3. The inverse of the function

Step 4. Verification

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Decision Maths

Applied Mathematics

Geometry

Discrete Mathematics

Probability and Statistics

Study anywhere. Anytime. Across all devices.

the function f is one-to-one. Find its inverse and check your answer.f(x)=-3x-4x-2

Short Answer

Step by step solution

Step 1. Given data

Step 2. interchanging variables

Step 3. The inverse of the function

Step 4. Verification

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Decision Maths

Applied Mathematics

Geometry

Discrete Mathematics

Probability and Statistics

Study anywhere. Anytime. Across all devices.

the function f is one-to-one. Find its inverse and check your answer.
$f (x) = \frac{- 3 x - 4}{x - 2}$