Chapter 9, Problem 29SGR

To determine

To state: Graph the function and state domain and range.

Answer to Problem 29SGR

Domain is $\left\{x=|x\ne -4\right\}$ and range is $\left\{f\left(x\right)|f\left(x\right)\ne -8\right\}$ .

Explanation of Solution

Given information:

The given function is $f\left(x\right)=-\frac{4}{x+4}-8$ .

Calculation:

For the function $f\left(x\right)=\frac{a}{x-h}+k$ .

The vertical asymptotes is at $x=h$ . And

The horizontal asymptotes is at $y=k$ .

The domain of the function is all real number except the value at the vertical asymptotes and the range of the function is all real number except the value at the horizontal asymptotes.

The graph of the function $f\left(x\right)=-\frac{4}{x+4}-8$ is drawn below:

  

From the above graph:

  $a=-4$ : The graph is expanded and reflected through the x-axis.

  $h=-4$ : The graph is translated 4 units left. There is an vertical asymptotes at $x=-4$ , and

  $k=-8$ : The graph is translated 8 units down. There is a horizontal asymptotes at $f\left(x\right)=-8$ .

Therefore,

Domain: $\left\{x=|x\ne -4\right\}$ and

Range : $\left\{f\left(x\right)|f\left(x\right)\ne -8\right\}$

Hence,

Domain is $\left\{x=|x\ne -4\right\}$ and range is $\left\{f\left(x\right)|f\left(x\right)\ne -8\right\}$ .