Graph the function, and compare to the parent graph. State the domain and range.

$\mathit{g}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}\mathbf{4}\sqrt{\mathbf{x}}$

Parent graph:The simplest form of the given function is called the parent function of that function and the graph of the parent function is called parent graph.

Domain: The set of all possible values for which given function defined is called domain.

Range: The set of all possible values of the given function is called range.

The given function is:$g\left(x\right)=4\sqrt{x}$

In order to graph a function, find few co-ordinates by substituting values of ‘x’ and find finding the respective values of ‘g(x)’.

$\begin{array}{l}\text{For\hspace{0.17em}\hspace{0.17em}}x=0,\\ g\left(0\right)=4\sqrt{0}\\ =4\left(0\right)\\ =0\end{array}$

$\begin{array}{l}\text{For\hspace{0.17em}\hspace{0.17em}}x=1,\\ g\left(1\right)=4\sqrt{1}\\ =4\left(1\right)\\ =4\end{array}$

$\begin{array}{l}\text{For\hspace{0.17em}\hspace{0.17em}}x=4,\\ g\left(4\right)=4\sqrt{4}\\ =4\left(2\right)\\ =8\end{array}$

$\begin{array}{l}\text{For\hspace{0.17em}\hspace{0.17em}}x=9,\\ g\left(9\right)=4\sqrt{9}\\ =4\left(3\right)\\ =12\end{array}$

$\begin{array}{l}\text{For\hspace{0.17em}\hspace{0.17em}}x=16,\\ g\left(x\right)=4\sqrt{16}\\ =4\left(4\right)\\ =16\end{array}$

Plot these co-ordinates on a coordinate plane and join those points to get the required graph.

The parent function of$g\left(x\right)=4\sqrt{x}$ is the simple square root function.

That is, $g\left(x\right)=\sqrt{x}$

The graph of parent function $g\left(x\right)=\sqrt{x}$is given below.

Note: Since the parent function is just used for comparison, it is graphed using graphing calculator.

The parent function is multiplied by ‘4’, which is greater than 1.

Therefore, the graph$g\left(x\right)=4\sqrt{x}$ is a vertical stretch of $g\left(x\right)=\sqrt{x}$.

Since ‘$x$’ is inside the root, the values inside the root must be positive.

Therefore, values of $x$is all positive real numbers including zero.

That is, $x\ge 0,\Rightarrow x\in [0,\infty )$.

Therefore, domain:$[0,\infty )$

In $g\left(x\right)=4\sqrt{x}$, the square root of x is multiplied by 4.

As square root is always positive, the least value it takes is zero.

Find the starting value of the function by substituting$x=0$ in $g\left(x\right)=4\sqrt{x}$.

$\begin{array}{l}g\left(x\right)=4\left(\sqrt{0}\right)\\ =4\left(0\right)\\ =0\end{array}$

In $g\left(x\right)=4\sqrt{x}$, coefficient of $\sqrt{x}$is 4 which is positive.

Therefore, $g\left(x\right)$takes all the values in real numbers which are greater than or equal to zero.

Therefore, $g\left(x\right)\ge 0,\Rightarrow g\left(x\right)\in [0,\infty )$

Therefore, Range: $[0,\infty )$