Q. 62 - Precalculus Enhanced with Graphing Utilities

Question

In Problems $39-68$, graph each function using the techniques of shifting, compressing, stretching, and/or reflecting. Start with the graph of the basic function (for example, $y=x^2$) and show all stages. Be sure to show at least three key points. Find the domain and the range of each function.Verify your results using a graphing utility

$h\left(x\right)=\frac{4}{x}+2$

Step-by-Step Solution

Answer

The graph of the function $h\left(x\right)=\frac{4}{x}+2$ is:

The domain of the function is $\{x\in \mathrm{ℝ}:x\ne 0\}$ and the range is $\{y\in \mathrm{ℝ}:y\ne 2\}$

Step 1. Given

The function $h\left(x\right)=\frac{4}{x}+2$

To graph the function and to find its domain and range.

Step 2. Graph the basic function

Graph the basic function $h\left(x\right)=\frac{1}{x}$

Step 3. Multiply the right side by 4

Replace $y=\frac{1}{x}$ by

              $y=\frac{4}{x}$, so that the graph stretched vertically by the factor $4$

Step 4. Add the right side by 2

Replace $y=\frac{4}{x}$ by

               $y=\frac{4}{x}+2$, so that the graph shifted vertically upward by the factor $2$.

Step 5. Find domain and range

The domain of the function is $\{x\in \mathrm{ℝ}:x\ne 0\}$.

An the range of the function is $\{y\in \mathrm{ℝ}:y\ne 2\}$

Step 6. Verify the graph

Verify the graph by using graphing utility.

The graph of the function using graphing utility is: