Chapter 2.5, Problem 48AYU

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.

$g\left(x\right)\text{ = }\frac{1}{2}\sqrt{x}$

To determine

To graph: The function $f\left(x\right)=\frac{1}{2}\sqrt{x}$, 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 48AYU

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

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

Explanation of Solution

Given:

$f\left(x\right)=\frac{1}{2}\sqrt{x}$

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 $f\left(x\right)=\frac{1}{2}\sqrt{x}$, multiply each $y\text{-coordinate}$ of the graph of $f\left(x\right)=\sqrt{x}$, by $\frac{1}{2}$, that it is vertically stretched by the factor of $\frac{1}{2}$.

$f\left(x\right)=\frac{1}{2}\sqrt{x}$     multiply by $\frac{1}{2}$; vertical stretch by a factor of $\frac{1}{2}$.

Interpretation:

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

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