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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 60. (page 269)

The function $f (x) = \frac{4}{x + 2}$ is one-to-one. Find its inverse and check the answer. Graph $f, f^{- 1}$ and $y = x$ on the same coordinate axes.

Short Answer

Expert verified

Inverse of the function $f (x) = \frac{4}{x + 2}$ is $f^{- 1} (x) = \frac{4 - 2 x}{x}$ . Graph is shown below.

Step by step solution

01

Step 1. Given information.

Given a one-to-one function $f (x) = \frac{4}{x + 2}$ .

02

Step 2. Find the inverse of the function f and verify.

To find the explicit form of the inverse solve $x = \frac{4}{y + 2}$ for y.

$\begin{array}{rcl} x & = & \frac{4}{y + 2} \\ x y + 2 x & = & 4 \\ x y & = & 4 - 2 x \\ y & = & \frac{4 - 2 x}{x} \end{array}$

It follows that $y = \frac{4 - 2 x}{x}$ .

Note that if $f (g (x)) = x$ and $g (f (x)) = x$ , then fand g are inverse of one another.

Prove $f (f^{- 1} (x)) = x$ as follows.

$\begin{array}{rcl} f (f^{- 1} (x)) & = & f (\frac{4 - 2 x}{x}) \\ = & \frac{4}{(\frac{4 - 2 x}{x}) + 2} \\ = & \frac{4 x}{4 - 2 x + 2 x} \\ = & x \end{array}$

Prove $f^{- 1} (f (x)) = x$ as follows.

$\begin{array}{rcl} f^{- 1} (f (x)) & = & f^{- 1} (\frac{4}{x + 2}) \\ = & \frac{4 - 2 (\frac{4}{x + 2})}{(\frac{4}{x + 2})} \\ = & \frac{4 x + 8 - 8}{4} \\ = & x \end{array}$

Therefore, inverse of f is $f^{- 1} (x) = \frac{4 - 2 x}{x}$ .

03

Step 3. Graph f,f-1 and y=x on the same coordinate axes.

Graph $f (x) = \frac{4}{x + 2}, f^{- 1} (x) = \frac{4 - 2 x}{x}$ and $y = x$ as follows.

Note the symmetry of the graphs with respect to the line $y = x$ .

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Most popular questions from this chapter

Solve the equation $2^{- x} = 16$

and verify your results using a graphing utility

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