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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: $y=\frac{1}{2}\sqrt{x}−1$

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

$\begin{array}{c}\text{¹ó´Ç°ù â¶Ä‰}x=0,\\ y=\frac{1}{2}\sqrt{0}−1\\ =\frac{1}{2}\left(0\right)−1\\ =−1\end{array}$

$\begin{array}{c}\text{¹ó´Ç°ù â¶Ä‰}x=1,\\ y=\frac{1}{2}\sqrt{1}−1\\ =\frac{1}{2}\left(1\right)−1\\ =0.5−1\\ =−0.5\end{array}$

$\begin{array}{c}\text{¹ó´Ç°ù â¶Ä‰}x=4,\\ y=\frac{1}{2}\sqrt{4}−1\\ =\frac{1}{2}\left(2\right)−1\\ =1−1\\ =0\end{array}$

$\begin{array}{c}\text{¹ó´Ç°ù â¶Ä‰}x=9,\\ y=\frac{1}{2}\sqrt{9}−1\\ =\frac{1}{2}\left(3\right)−1\\ =1.5−1\\ =0.5\end{array}$

$\begin{array}{c}\text{¹ó´Ç°ù â¶Ä‰}x=16,\\ y=\frac{1}{2}\sqrt{16}−1\\ =\frac{1}{2}\left(4\right)−1\\ =2−1\\ =1\end{array}$

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

The parent function of$y=\frac{1}{2}\sqrt{x}−1$ is the simplest square root function.

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

Using graphing calculator, the graph of parent function$y=\sqrt{x}$ 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$\frac{1}{2}$ (a value less than 1 and greater than zero) and is subtracted is ‘1’. So the graph$y=\frac{1}{2}\sqrt{x}−1$ is a vertical compression of$y=\sqrt{x}$ and is translated(shifted) downward by 1 units from the origin, on comparing with the parent graph $y=\sqrt{x}$.

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

Therefore, domain: $[0,∞)$

In $y=\frac{1}{2}\sqrt{x}−1$, the square root of$x$is multiplied by$\frac{1}{2}$ and is subtracted by ‘1’.

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 $y=\frac{1}{2}\sqrt{x}−1$

$\begin{array}{c}y=\frac{1}{2}\left(0\right)−1\\ =0−1\\ =−1\end{array}$

In $y=\frac{1}{2}\sqrt{x}−1$, coefficient of $\sqrt{x}$is 1, which is positive.

Therefore $y$takes all the values of real numbers which are greater

That is,$yâ‰\yen −1,\text{ â¶Ä‰â¶Ä‰}⇒y∈[−1,∞)$

Therefore, Range:$[−1,∞)$