Chapter 9.4, Problem 53STP

To determine

To graph each function and state the domain and range

Answer to Problem 53STP

The graph is plotted and the domain and range of the function is obtained.

The domain of the function is $\left\{x|x\ne -6\right\}$ and the range of the function is $\left\{f\left(x\right)|f\left(x\right)\ne 1\right\}$

Explanation of Solution

Given information:

The function is $f\left(x\right)=\frac{1}{x+6}+1$

The general equation for the transformation of reciprocal function is,

  $f\left(x\right)=\frac{a}{x-h}+k$

So, here $a=1$ that means the graph is expanded, $h=-6$ that means the graph is translated 6 units left and there is an asymptote at $x=-6$ . And $k=1$ that means the graph is translated 1 unit up and there is asymptote at $f\left(x\right)=1$

The graph can be represented as,

  

Therefore, the domain of the function is $\left\{x|x\ne -6\right\}$ and the range of the function is $\left\{f\left(x\right)|f\left(x\right)\ne 1\right\}$