Q 71. - Precalculus Enhanced with Graphing Utilities

Question

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 $f^{-1}$.

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

Step-by-Step Solution

Answer

Part (a) $\left(-4,\infty \right)$.

Part (b)

Part (c) Range $\left(-\infty \infty \right)$ and asymptotes of $f \mathrm{is} x=-4$.

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

Part (e) Domain $\left(-\infty ,\infty \right)$ and Range $\left(-4,\infty \right)$

Part (f) 

Part (a) Step 1. Given information.

The given function is:

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

Part (a) Step 2. Find the domain of f .

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

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 $\left(-4,\infty \right)$.

Part (b) Step 1. Graph f .

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

Part (c) Step 1. Determine the range and any asymptotes of f from the graph.

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 $x=-4$.

Part (d) Step 1. Find f - 1 .

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

For $f^{-1}$replace $f\left(x\right) \mathrm{with} y$,

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

Interchange the variables x and y,

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

Solve for y,

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

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

Part (e) Find the domain and the range of f - 1 .

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 $\left(-4,\infty \right)$.

Part (f) Graph f - 1 .

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

First, create a table of values by assigning values to x and then solve for y for each x,

Now draw the graph by plotting each point: