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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 5: Q. 65 (page 269)

In Problems 61鈥�72, the function f is one-to-one. Find its inverse and check your answer.
$f (x) = \frac{2 x}{3 x - 1}$

Short Answer

Expert verified

The inverse off is $\frac{x}{3 x - 2} .$

Step by step solution

01

Step 1. Given Information

The given function is $f (x) = \frac{2 x}{3 x - 1}$

We have to find its inverse and check the answer.

02

Step 2. Replace f(x) with y and interchange the variables x and y

So, $y = \frac{2 x}{3 x - 1}$

Interchange the variables

$x = \frac{2 y}{3 y - 1}$

03

Step 3. Solve for y

$x = \frac{2 y}{3 y - 1} x (3 y - 1) = 2 y 3 x y - x = 2 y 3 x y - 2 y = x y (3 x - 2) = x y = \frac{x}{3 x - 2}$

The inverse is $f^{- 1} (x) = \frac{x}{3 x - 2} .$

04

Step 4. Check the answer

We can check the answers by $f^{- 1} (f (x)) a n d f (f^{- 1} (x)) .$

So,

$f^{- 1} (f (x)) = f^{- 1} (\frac{2 x}{3 x - 1}) = \frac{(\frac{2 x}{3 x - 1})}{3 (\frac{2 x}{3 x - 1}) - 2} = \frac{2 x}{2} = x$

Thus, $f^{- 1} (f (x)) = x$

Now,

$f (f^{- 1} (x)) = f (\frac{x}{3 x - 2}) = \frac{2 (\frac{x}{3 x - 2})}{3 (\frac{x}{3 x - 2}) - 1} = \frac{2 x}{2} = x$

Thus, $f (f^{- 1} (x)) = x$

Hence our result is right.

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Most popular questions from this chapter

Find the inverse of the linear function $f (x) = m x + b m 鈮� 0$

The graph of an exponential function is given. Match each graph to one of the following functions.

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Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red.

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