Chapter 2.5, Problem 42AYU

In Problems $37-60,$ graph each function using the techniques of shifting, compressing, stretching, and/or reflecting. Start with the graph of the basic function $\left(forexample,y=x^2\right)$ and show all the steps. Be sure to show at least three key points. Find the domain and the range of each function.

$g\left(x\right)=\sqrt[3]{\frac{1}{2}x}$

To determine

The domain and range of the function $g(x)=\sqrt[3]{\frac{1}{2}x}$ and graph the function $g(x)=\sqrt[3]{\frac{1}{2}x}$ using techniques of shifting, compressing, stretching and/or reflecting.

Answer to Problem 42AYU

Solution:

The graph of $g(x)=\sqrt[3]{\frac{1}{2}x}$ is:

The domain and range of the function $g(x)=\sqrt[3]{\frac{1}{2}x}$ is $(-\infty ,\infty )$.

Explanation of Solution

Given Information:

The function is $g(x)=\sqrt[3]{\frac{1}{2}x}$

Explanation:

Step 1: The cube root function $y=\sqrt[3]{x}$

Step 2: Replace $x$ by $\frac{1}{2}x$ in the function $y=\sqrt{x}$

$g(x)=\sqrt[3]{\frac{1}{2}x}$.

The graph of function $y=f(hx)$ is obtained from the graph of $y=f(x)$ by multiplying each $x-$ coordinate of $y=f(x)$ by $\frac{1}{h}$

Replacing $x$ by $\frac{1}{2}x$ results in horizontal compression by $\frac{1}{2}$ units

Therefore, multiply each $x-$ coordinate of $y=\sqrt{x}$ by 2.

Hence, the graph of $g(x)=\sqrt[3]{\frac{1}{2}x}$ is as follows:

The domain and range of the function $y=\sqrt[3]{x}$ is $(-\infty ,\infty )$

Also, the domain and range of the function $g(x)=\sqrt[3]{\frac{1}{2}x}$ is $(-\infty ,\infty )$