Q. 61 - 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

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

Step-by-Step Solution

Answer

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

The domain of the function is the set of all positive real numbers ad the range is $\{y\in \mathrm{ℝ}:y\ge -2\}$

Step 1. Given

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

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

Step 2. Graph the basic function

Graph the basic function $y=\sqrt{x}$

Step 3. Replace x with - x

Replace $y=\sqrt{x}$ by

              $y=\sqrt{-x}$ so that it reflected by $x-$axis

Step 4. Subtract the right side by - 2

Replace $y=\sqrt{-x}$ by

               $y=\sqrt{-x}-2$, so that the graph shifted vertically downward by the factor $2$.

Step 5. Find domain and range

The domain of the function is the set of all positive real numbers.

And the range is $\{y\in \mathrm{ℝ}:y\ge -2\}$

Step 6. Verify the graph

Verify the graph by using graphing utility.

The graph of the function using graphing utility is: