Chapter 8.3, Problem 41HP

To determine

To write two different reciprocal functions with graphs having the same vertical and horizontal asymptotes and graph the functions.

Explanation of Solution

Given information:

The given statement is:

“Write two different reciprocal functions with graphs having the same vertical and horizontal asymptotes. Graph the functions.”

Graph:

The two functions having the same vertical and horizontal asymptotes are:

  $f\left(x\right)=\frac{3}{x+1}+2\text{ and }g\left(x\right)=\frac{5}{x+1}+2$

For the function $f\left(x\right)=\frac{3}{x+1}+2$ :

This represents a transformation of the graph of $f\left(x\right)=\frac{1}{x}$ .

  $a=3$ : The graph is expanded.

  $h=-1$ : The graph is translated $1\text{ unit}$ left. There is an asymptote at $x=-1$ .

  $k=2$ : The graph is translated $\text{2 units}$ up. There is an asymptote at $f\left(x\right)=2$ .

For the function $g\left(x\right)=\frac{5}{x+1}+2$ :

This represents a transformation of the graph of $f\left(x\right)=\frac{1}{x}$ .

  $a=5$ : The graph is expanded.

  $h=-1$ : The graph is translated $1\text{ unit}$ left. There is an asymptote at $x=-1$ .

  $k=2$ : The graph is translated $\text{2 units}$ up. There is an asymptote at $f\left(x\right)=2$ .

The graph of the functions are: