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

$\mathit{h}\left(x\right)\mathbf{=}\sqrt{\mathbf{x}\mathbf{+}\mathbf{3}}$

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: $h\left(x\right)=\sqrt{x+3}$

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

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

$\begin{array}{c}\text{For\hspace{0.17em}\hspace{0.17em}}x=-2,\\ h\left(-2\right)=\sqrt{-2+3}\\ =\sqrt{1}\\ =1\end{array}$

$\begin{array}{c}\text{For\hspace{0.17em}\hspace{0.17em}}x=1,\\ h\left(1\right)=\sqrt{1+3}\\ =\sqrt{4}\\ =2\end{array}$

$\begin{array}{c}\text{For\hspace{0.17em}\hspace{0.17em}}x=6,\\ h\left(6\right)=\sqrt{6+3}\\ =\sqrt{9}\\ =3\end{array}$

$\begin{array}{c}\text{For\hspace{0.17em}\hspace{0.17em}}x=13,\\ h\left(13\right)=\sqrt{13+3}\\ =\sqrt{16}\\ =4\end{array}$

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

The parent function of$h\left(x\right)=\sqrt{x+3}$ is the simplest square root function.

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

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

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

The value 3 is added inside square root of parent function $h\left(x\right)=\sqrt{x}$. So the graph $h\left(x\right)=\sqrt{x+3}$ is the translation of the parent graph $h\left(x\right)=\sqrt{x}$.

Since 3 is added inside the root, the graph is translated to the left by 3 units.

Since the values inside the root must be positive.

$x+3\ge 0$

Adding ‘$-3$’ on both the sides.

$\begin{array}{c}x+3-3\ge 0-3\\ x+0\ge -3\\ x\ge -3\\ x=0\end{array}$

$x\ge -3\Rightarrow x\in [-3,\infty )$

Therefore, domain:$[-3,\infty )$

The h(x)values are square root of $x+3$. As square roots are positive, $h\left(x\right)$takes all the positive real values including zero.

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

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