Q. 59 - 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.

$g\left(x\right)=2\sqrt{x-2}+1$

Step-by-Step Solution

Answer

The graph of the function $g\left(x\right)=2\sqrt{x-2}+1$ is:

The domain of the function is $\{x\in \mathrm{ℝ}:x\ge 2\}$ and the range of the function is $\{y\in \mathrm{ℝ}:y\ge 1\}$

Step 1. Given

The function $g\left(x\right)=2\sqrt{x-2}+1$

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 - 2

Replace $y=\sqrt{x}$ by

              $y=\sqrt{x-2}$ so shifting the graph horizontally right by $2$ unit.

Step 4. Multiply the right side by 2

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

Step 5. Add the right side by 1

Replace $y=2\sqrt{x-2}$ by

               $y=2\sqrt{x-2}+1$ so that the graph shifted vertically upward by $1$ unit.

Step 6. Find domain and range

The domain of the function is $\{x\in \mathrm{ℝ}:x\ge 2\}$ and the range is $\{y\in \mathrm{ℝ}:y\ge 1\}$

Step 7. Verify the graph

Verify the graph by using graphing utility.

The graph of the function using graphing utility is: