Chapter 2.5, Problem 60AYU

In Problems 39-62, graph each function using the techniques of shifting, compressing, stretching, and/or reflecting. Stan with the graph of the basic function (for example, $y\text{ = }{x}^2$) and show all stages. Be sure to show at least three key points. Find the domain and the range of each function.

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

To determine

To graph: The function $f\left(x\right)=4\sqrt{2-x}$, 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.

Answer to Problem 60AYU

Domain of the function $f\left(x\right)=4\sqrt{2-x}$ is $\left[2,\infty \right)$.

Range of the function $f\left(x\right)=4\sqrt{2-x}$ is $\left[0,\infty \right)$.

Explanation of Solution

Given:

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

Graph:

Now use the following steps to obtain the graph of $f\left(x\right)$.

Step 1: The function $f\left(x\right)=\sqrt{x}$ is the square root function.

$f\left(x\right)=\sqrt{x}$     square root function

Step 2: To obtain the graph of $f\left(x\right)=\sqrt{x+2}$, replace $x$ by $x+2$ from each $y\text{-coordinate}$ on the graph of $f\left(x\right)=\sqrt{x}$, that it is shifted left 2 units.

$f\left(x\right)=\sqrt{x+2}$     replace $x$ by $x+2$; horizontal shift left 2 units.

Step 3: To obtain the graph of $f\left(x\right)=\sqrt{2-x}$, replace $x$ by $-x$ of the graph of $f\left(x\right)=\sqrt{x+2}$, that it is reflection about the $y\text{-axis}$.

$f\left(x\right)=\sqrt{2-x}$     replace $x$ by $-x$; reflection about $y\text{-axis}$.

Step 4: To obtain the graph of $f\left(x\right)=4\sqrt{2-x}$, multiply each $y\text{-coordinate}$ of the graph of $f\left(x\right)=\sqrt{2-x}$, by 4, that it is vertically stretched by the factor of 4.

$f\left(x\right)=4\sqrt{2-x}$     multiply by 4, vertically stretched by a factor of 4

Interpretation:

Domain of the function $f\left(x\right)=4\sqrt{2-x}$ is $\left[2,\infty \right)$.

Range of the function $f\left(x\right)=4\sqrt{2-x}$ is $\left[0,\infty \right)$.