Chapter 5.4, Problem 83AYU

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 $f^{-1}$.

$f(x)\text{ = }{e}^{x\text{ + 2}}-\text{ 3}$

To determine

To find:

a. The domain of $f$.

Answer to Problem 83AYU

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

Explanation of Solution

Given:

$f\left(x\right)=e^{x+2}-3$

Calculation:

a. The domain of $f$ consists of all the values of $x$. Therefore domain is $\left(-\infty ,\infty \right)$.

To determine

To find:

b. Graph $f$.

Answer to Problem 83AYU

b.

Explanation of Solution

Given:

$f\left(x\right)=e^{x+2}-3$

Calculation:

b. Graph $f$.

$x$	$f\left(x\right)=e^{x+2}-3$
$-1$	$-0.28$
0	$4.39$
1	$17.08$

To determine

To find:

c. Range and asymptotes of $f$.

Answer to Problem 83AYU

c. Range of $f$ is $\left(-3,\infty \right)$.

Asymptotes of $f$ is $y=-3$.

Explanation of Solution

Given:

$f\left(x\right)=e^{x+2}-3$

Calculation:

c. Range of $f$ is $\left(-3,\infty \right)$.

Asymptotes of $f$ is $y=-3$.

To determine

To find:

d. $f^{-1}$

Answer to Problem 83AYU

d. The inverse of $f$ is $f^{-1}\left(x\right)=\ln \left(3+x\right)-2$.

Explanation of Solution

Given:

$f\left(x\right)=e^{x+2}-3$

Calculation:

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

Given $y=e^{x+2}-3$.

Solve for $y$, $x=e^{y+2}-3$.

$3+x=e^{y+2}$

Taking logarithm on both sides with base $e$.

$\ln \left(3+x\right)=\ln e^{y+2}$

$\ln \left(3+x\right)=y+2$

$y=\ln \left(3+x\right)-2$

Therefore the inverse of $f$ is $f^{-1}\left(x\right)=\ln \left(3+x\right)-2$.

To determine

To find:

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

Answer to Problem 83AYU

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

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

Explanation of Solution

Given:

$f\left(x\right)=e^{x+2}-3$

Calculation:

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

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

To determine

To find:

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

Answer to Problem 83AYU

f.

Explanation of Solution

Given:

$f\left(x\right)=e^{x+2}-3$

Calculation:

f.

$x$	$f^{-1}\left(x\right)=\ln \left(3+x\right)-2$
$-2$	$-2$
0	$-0.90$
1	$-0.61$