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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. 68 (page 269)

In Problems 61鈥�72, the function f is one-to-one. Find its inverse and check your answer.
$f (x) = \frac{2 x - 3}{x + 4}$

Short Answer

Expert verified

The inverse off is $\frac{3 + 4 x}{2 - x} .$

Step by step solution

01

Step 1. Given Information

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

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

Interchange the variables

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

03

Step 3. Solve for y

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

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

04

Step 4. Check the answer

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

So,

$f^{- 1} (f (x)) = f^{- 1} (\frac{2 x - 3}{x + 4}) = \frac{4 (\frac{2 x - 3}{x + 4}) + 3}{2 - (\frac{2 x - 3}{x + 4})} = \frac{11 x}{11} = x$

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

Now,

$f (f^{- 1} (x)) = f (\frac{4 x + 3}{2 - x}) = \frac{2 (\frac{4 x + 3}{2 - x}) - 3}{(\frac{4 x + 3}{2 - x}) + 4} = \frac{11 x}{11} = x$

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

Hence our result is right.

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Most popular questions from this chapter

A function $y = f (x)$ is increasing on the interval $(0, 5) .$ What conclusions can you draw about the graph of $y = f^{- 1} (x)$ ?

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

True or False

The domain of the exponential function

$f (x) = a^{x}$ where $a > 0, a 鈮� 1$ is the set of all real numbers.

A function $f$ has an inverse function. If the graph of $f$ lies in quadrant II, in which quadrant does the graph of $f^{- 1}$ lie?

If $5^{- x} = 3$ , what does $5^{3 x}$ equal?

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.

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