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

$\mathit{y}\mathbf{=}\mathbf{3}\sqrt{\mathbf{x}}$

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: $y=3\sqrt{x}$

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

$\begin{array}{c}\text{For\hspace{0.17em}\hspace{0.17em}}x=0,\\ y=3\sqrt{0}\\ =3\left(0\right)\\ =0\end{array}$

$\begin{array}{c}\text{For\hspace{0.17em}\hspace{0.17em}}x=1,\\ y=3\sqrt{1}\\ =3\left(1\right)\\ =3\end{array}$

$\begin{array}{c}\text{For\hspace{0.17em}\hspace{0.17em}}x=4,\\ y=3\sqrt{4}\\ =3\left(2\right)\\ =6\end{array}$

$\begin{array}{c}\text{For\hspace{0.17em}\hspace{0.17em}}x=9,\\ y=3\sqrt{9}\\ =3\left(3\right)\\ =9\end{array}$

$\begin{array}{c}\text{For\hspace{0.17em}\hspace{0.17em}}x=16,\\ y=3\sqrt{16}\\ =3\left(4\right)\\ =12\end{array}$

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

The parent function of $y=3\sqrt{x}$ is the simple square root function.

That is, $y=\sqrt{x}$

The graph of parent function $y=3\sqrt{x}$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 ‘3’, which is greater than 1.

Therefore, the graph $y=3\sqrt{x}$is a vertical stretch of $y=\sqrt{x}$.

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 $y=3\sqrt{x}$, the square root of $x$is multiplied by 3.

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 $y=3\sqrt{x}$.

$\begin{array}{c}y=3\left(\sqrt{0}\right)\\ =3\left(0\right)\\ =0\end{array}$

In $y=3\sqrt{x}$, coefficient of $\sqrt{x}$ is 3 which is positive.

Therefore, $y$takes all the values in real numbers which are greater than or equal to zero.

Therefore,$y\ge 0,\Rightarrow y\in [0,\infty )$

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