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.

$g\left(x\right)=4\sqrt{2-x}$

Short answer

The graph of the function $g\left(x\right)=4\sqrt{2-x}$is:

The domain of the function is $\{x\in \mathrm{ℝ}:x\le 2\}$and the range is the set of all non-negative real numbers.

Step-by-step solution

Step 1. Given

The function$g\left(x\right)=4\sqrt{2-x}$

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

Step 2. Graph the basic function

Graph the basic function $g\left(x\right)=\sqrt{x}$

Step 3. Replace x with 2-x

Replace $y=\sqrt{x}$by

$y=\sqrt{2-x}$so that it reflected by $x-$axis and the graph shift horizontally right to $2$unit.

Step 4. Multiply the right side by 4

Multiply the right side of the graph by $4$, so that the graph stretched by the factor of $4$.

Step 5. Find domain and range

The domain of the function is $\{x\in \mathrm{ℝ}:x\le 2\}$.

And the range of the function is the set of all non-negative real numbers.

Step 6. Verify the graph

Verify the graph by using graphing utility.

The graph of the function using graphing utility is: