Chapter 2.5, Problem 41AYU

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.

$g\left(x\right)=\sqrt{3x}$

To determine

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

Answer to Problem 41AYU

Solution:

The graph of $g(x)=\sqrt{3x}$ is:

The domain and range of the function $y=\sqrt{3x}$ is $\{x/x\ge 0\}$

Explanation of Solution

Given Information:

The function is $g(x)=\sqrt{3x}$

Explanation:

To graph the function $g(x)=\sqrt{3x}$, first graph the square root function $y=\sqrt{x}$

Now, replace $x$ by $3x$ in the function $y=\sqrt{x}$

$\Rightarrow y=\sqrt{3x}$

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 $3x$ results in horizontal compression by 3 units

Therefore, multiply each $x-$ coordinate of $y=\sqrt{x}$ by $\frac{1}{3}$

Hence, the graph of $y=\sqrt{3x}$ is as follows:

From the above graph, the domain and range of the function $y=\sqrt{x}$ is $\{x/x\ge 0\}$

Also, the domain and range of the function $y=\sqrt{3x}$ is $\{x/x\ge 0\}$