In Problems 71– 86, use the given function $f$ to:

(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 $f^{-1}$.

$f\left(x\right)=\ln \left(x+4\right)$

The given function is:

$f\left(x\right)=\ln \left(x+4\right)$

$f\left(x\right)=\ln \left(x+4\right)$

The domain of $f$ consists of all x for which $\left(x+4\right)>0$or $x>-4$.

Therefore, the domain of the given function is$\left(-4,\infty \right)$.

Sketch the graph of $f\left(x\right)=\ln (x+4)$:

We can see from the graph that the function's value varies from negative infinity to positive infinity.

Therefore, the range of the function is $\left(-\infty ,\infty \right)$.

The vertical asymptote of the given function is $x=-4$because the curve approaches but will never touch the vertical line$x=-4$.

$f\left(x\right)=\ln \left(x+4\right)$

$y=\ln \left(x+4\right)$

Interchange the variables xandy,

$x=\ln \left(y+4\right)$

Solve for y,

$$\begin{gathered} e^x=y+4 \\ {e}^x-4=y \\ y=e^x-4 \end{gathered}$$

Therefore, the inverse of$f\mathrm{is}{f}^{-1}\left(x\right)=e^x-4$.

We know that the domain of a function $f\left(x\right)$is the range of its inverse.

In the same way, the range of a function $f\left(x\right)$is the range of its inverse.

Therefore, the domain of$f^{-1}\mathrm{is}\left(-\infty ,\infty \right)$and its range is$\left(-4,\infty \right)$.

$f^{-1}\left(x\right)=e^x-4$

First, create a table of values by assigning values to x and then solve fory for eachx,

Now draw the graph by plotting each point: