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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: $g\left(x\right)=\sqrt{x}−4$

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

$\begin{array}{c}\text{¹ó´Ç°ù â¶Ä‰}x=0,\\ g\left(0\right)=\sqrt{0}−4\\ =0−4\\ =−4\end{array}$

$\begin{array}{c}\text{¹ó´Ç°ù â¶Ä‰}x=1,\\ g\left(1\right)=\sqrt{1}−4\\ =1−4\\ =−3\end{array}$

$\begin{array}{c}\text{¹ó´Ç°ù â¶Ä‰}x=4,\\ g\left(4\right)=\sqrt{4}−2\\ =2−4\\ =−2\end{array}$

$\begin{array}{c}\text{¹ó´Ç°ù â¶Ä‰}x=9,\\ g\left(9\right)=\sqrt{9}−4\\ =3−4\\ =−1\end{array}$

$\begin{array}{c}\text{¹ó´Ç°ù â¶Ä‰}x=16,\\ g\left(x\right)=\sqrt{16}−4\\ =4−4\\ =0\end{array}$

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

The parent function of$g\left(x\right)=\sqrt{x}−4$ 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 graph$g\left(x\right)=\sqrt{x}−4$ is obtained by parent function is subtracted by ‘4’.

Therefore, the graph$g\left(x\right)=\sqrt{x}−4$ is translated(shifted) downward by 4 units from the origin, on comparing with the parent graph $g\left(x\right)=\sqrt{x}$.

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, $xâ‰\yen 0,\text{ â¶Ä‰â¶Ä‰}⇒x∈[0,∞)$.

Therefore, domain:$[0,∞)$

In$g\left(x\right)=\sqrt{x}−4$ the square root of xis subtracted 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 $g\left(x\right)=\sqrt{x}−4$.

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

Also, in $g\left(x\right)=\sqrt{x}−4$, coefficient of $\sqrt{x}$is 1, which is positive.

Therefore, $g\left(x\right)$takes all the real values greater than or equal to ‘$-4$’.

That is,$g\left(x\right)â‰\yen −4,\text{ â¶Ä‰â¶Ä‰}⇒g\left(x\right)∈[−4,∞)$

Therefore, Range:$[−4,∞)$