Chapter 5.4, Problem 72AYU

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 $f^{-1}$.

$f\left(x\right)=\text{ ln}\left(x-3\right)$

To determine

To find:

a) The domain of $f$.

Answer to Problem 72AYU

a) The domain is $\left(3, \infty \right)$.

Explanation of Solution

Given:

$f\left(x\right)=\ln \left(x-3\right)$

Calculation:

a) The domain of $f$ consists of all $x$ for which $x-3>0$, that $x>3$. Therefore domain is $\left(3, \infty \right)$.

To determine

To find:

b) Graph $f$.

Answer to Problem 72AYU

b)

Explanation of Solution

Given:

$f\left(x\right)=\ln \left(x-3\right)$

Calculation:

b) Graph $f$.

$x$	$f\left(x\right)=\ln \left(x-3\right)$
$4$	0
$4.5$	$0.41$
5	$0.69$

To determine

To find:

c) Range and asymptotes of $f$.

Answer to Problem 72AYU

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

Asymptotes of $f$ is $x=3$

Explanation of Solution

Given:

$f\left(x\right)=\ln \left(x-3\right)$

Calculation:

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

Asymptotes of $f$ is $x=3$

To determine

To find:

d) $f^{-1}$

Answer to Problem 72AYU

d) The inverse of $f$ is $f^{-1}\left(x\right)=e^x+3$.

Explanation of Solution

Given:

$f\left(x\right)=\ln \left(x-3\right)$

Calculation:

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

Given $y=\ln \left(x-3\right)$

Solve for $y$, $x=\ln \left(y-3\right)$

Taking exponential on both sides $e^x=e^{\ln \left(y-3\right)}$

$e^x=y-3$

$y=e^x+3$

Therefore the inverse of $\text{f}$ is $f^{-1}\left(x\right)=e^x+3$.

To determine

To find:

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

Answer to Problem 72AYU

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

The range of $f^{-1}$ equal the domain of $f=\left(3,\infty \right)$

Explanation of Solution

Given:

$f\left(x\right)=\ln \left(x-3\right)$

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 $f=\left(3, \infty \right)$

To determine

To find:

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

Answer to Problem 72AYU

(f)

Explanation of Solution

Given:

$f\left(x\right)=\ln \left(x-3\right)$

Calculation:

f)

$x$	$f^{-1}\left(x\right)=e^x+3$
$0$	$4$
1	$5.72$
$-1$	$3.37$