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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. 63 (page 269)

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

Short Answer

Expert verified

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

Step by step solution

01

Step 1. Given Information

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

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, $y = \frac{3 x}{x + 2}$

Interchange the variables

$x = \frac{3 y}{y + 2}$

03

Step 3. Solve for y

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

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

04

Step 4. Check the answer

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

So,

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

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

Now,

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

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

Hence our result is right.

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Most popular questions from this chapter

In Problems 61鈥�72, the function f is one-to-one. Find its inverse and check your answer.

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

Begin with the graph of $y = e^{x}$ and use transformation to graph the function. Determine the domain, range and horizontal asymptote of the function.

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

Begin with the graph of $y = e^{x}$ and use transformation to graph the function. Determine the domain, range and horizontal asymptote of the function.

$f (x) = e^{- x}$

solve each equation. Verify your results using a graphing utility.

$3^{x^{2} - 7} = 27^{2 x}$

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

True or False

To graph ${(x - 2)}^{3}$ , shift the graph of $y = x^{3}$ to the left $2$ units.

See all solutions

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