Chapter 5.4, Problem 80AYU

In Problems 73-88, use the given function $f$.

a. Find the domain of $f$.

b. Graph $f$.

c. From the graph, determine the range and any asymptotes of $f$.

d. Find $f^{-1}$, the inverse of $f$.

e. Find the domain and the range of $f^{-1}$.

f. Graph $f^{-1}$.

$f\left(x\right)=\log \left(-2x\right)$

To determine

To find:

a. The domain of $f$.

Answer to Problem 80AYU

a. The domain is $\left(-\infty ,0\right)$.

Explanation of Solution

Given:

$f\left(x\right)=\log \left(-2x\right)$

Calculation:

a. The domain of $f$ consists of all $x$ for which $x<0$. Therefore domain is $\left(-\infty ,0\right)$.

To determine

To find:

b. Graph $f$.

Answer to Problem 80AYU

b.

Explanation of Solution

Given:

$f\left(x\right)=\log \left(-2x\right)$

Calculation:

b. Graph $f$.

$x$	$f\left(x\right)=\log \left(-2x\right)$
$-1/2$	0
$-1$	$0.30$
$-5$	1

To determine

To find:

c. Range and asymptotes of $f$.

Answer to Problem 80AYU

c. Range of $f$ is $\left(-\infty ,\infty \right)$.

Asymptotes of $f$ is $x=0$.

Explanation of Solution

Given:

$f\left(x\right)=\log \left(-2x\right)$

Calculation:

c. Range of $f$ is $\left(-\infty ,\infty \right)$ .

Asymptotes of $f$ is $x=0$.

To determine

To find:

d. $f^{-1}$

Answer to Problem 80AYU

d. The inverse of $f$ is $f^{-1}\left(x\right)=-\frac{1}{2}{10}^x$.

Explanation of Solution

Given:

$f\left(x\right)=\log \left(-2x\right)$

Calculation:

d. To find $f^{-1}$.

Given $y=\log \left(-2x\right)$.

Solve for $y$, $x=\log \left(-2y\right)$.

Change to exponential form ${10}^x={10}^{\log \left(-2y\right)}$.

${10}^x=-2y$

$y=-\frac{1}{2}{10}^x$

Therefore the inverse of $f$ is $f^{-1}\left(x\right)=-\frac{1}{2}{10}^x$.

To determine

To find:

e. Domain and range of $f^{-1}$.

Answer to Problem 80AYU

e. The domain of $f^{-1}$ equal the range of $f=\left(-\infty ,\infty \right)$.

The range of $f^{-1}$ equal the domain of $f=\left(-\infty ,0\right)$ .

Explanation of Solution

Given:

$f\left(x\right)=\log \left(-2x\right)$

Calculation:

e. The domain of $f^{-1}$ equal the range of $f=\left(-\infty ,\infty \right)$.

The range of $f^{-1}$ equal the domain of $f=\left(-\infty ,0\right)$.

To determine

To find:

f. Graph $f^{-1}$.

Answer to Problem 80AYU

f.

Explanation of Solution

Given:

$f\left(x\right)=\log \left(-2x\right)$

Calculation:

f.

$x$	$f^{-1}\left(x\right)=-\frac{1}{2}{10}^x$
$-1$	$-0.05$
0	$-0.5$
1	$-5$