Chapter 5.4, Problem 74AYU

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 $f^{-1}$.

$f\left(x\right)=-\ln \left(-x\right)$

To determine

To find:

a) The domain of $f$.

Answer to Problem 74AYU

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

Explanation of Solution

Given:

$f\left(x\right)=-\ln \left(-x\right)$

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 74AYU

b)

Explanation of Solution

Given:

$f\left(x\right)=-\ln \left(-x\right)$

Calculation:

b) Graph $f$.

$x$	$f\left(x\right)=-\ln \left(-x\right)$
$-0.5$	$0.69$
$-1$	0
$-2$	$0.69$

To determine

To find:

c) Range and asymptotes of $f$.

Answer to Problem 74AYU

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

Asymptotes of $f$ is $x=0$

Explanation of Solution

Given:

$f\left(x\right)=-\ln \left(-x\right)$

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 74AYU

d) The inverse of $f$ is $f^{-1}\left(x\right)=-e^{-x}$

Explanation of Solution

Given:

$f\left(x\right)=-\ln \left(-x\right)$

Calculation:

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

Given $y=-\ln \left(-x\right)$

Solve for $y$, $-x=\ln \left(-y\right)$

Taking exponential on both sides $e^{-x}=e^{\ln \left(-y\right)}$

$e^{-x}=-y$

$y=-e^{-x}$

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

To determine

To find:

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

Answer to Problem 74AYU

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:

$f\left(x\right)=-\ln \left(-x\right)$

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 74AYU

f)

Explanation of Solution

Given:

$f\left(x\right)=-\ln \left(-x\right)$

Calculation:

f)

$x$	$f^{-1}\left(x\right)=-e^{-x}$
$0$	$-1$
1	$-0.36$
$-1$	$-2.71$