Chapter 9.3, Problem 33PPS

To determine

To graph the function and state the domain, range, and the asymptotes.

Explanation of Solution

Given information:

The given function is:

  $f\left(x\right)=\frac{2}{x-4}+3$

Graph:

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

  $a=2$ : The graph is expanded.

  $h=4$ : The graph is translated $4\text{ units}$ right. There is an asymptote at $x=4$ .

  $k=3$ : The graph is translated $3\text{ units}$ up. There is an asymptote at $f\left(x\right)=3$ .

The graph of the function is:

  

Interpretation:

The asymptotes are $x=4\text{ and }f\left(x\right)=3$ .

The domain of the graph is all real numbers not equal to $4$ and the range is all real numbers except $3$ .