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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 the parent graph.

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

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

The given function is:$y=\sqrt{x+1}$

To graph a function, find a few coordinates by substituting values of ‘$x$’ and find finding the respective values of ‘$y$’.

$\begin{array}{c}\text{¹ó´Ç°ù â¶Ä‰}x=−1,\\ y=\sqrt{−1+1}\\ =\sqrt{0}\\ =0\end{array}$

$\begin{array}{c}\text{¹ó´Ç°ù â¶Ä‰}x=0,\\ y=\sqrt{0+1}\\ =\sqrt{1}\\ =1\end{array}$

$\begin{array}{c}\text{¹ó´Ç°ù â¶Ä‰}x=3,\\ y=\sqrt{3+1}\\ =\sqrt{4}\\ =2\end{array}$

$\begin{array}{c}\text{¹ó´Ç°ù â¶Ä‰}x=8,\\ y=\sqrt{8+1}\\ =\sqrt{9}\\ =3\end{array}$

$\begin{array}{c}\text{¹ó´Ç°ù â¶Ä‰}x=15,\\ y=\sqrt{15+1}\\ =\sqrt{16}\\ =4\end{array}$

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

The parent function $y=\sqrt{x+1}$is the simplest square root function.

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

The graph of the parent function $y=\sqrt{x}$is given below.

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

The value 1 is added inside the square root of the parent function $y=\sqrt{x}$. So the graph $y=\sqrt{x+1}$is the translation of the parent graph $y=\sqrt{x}$.

Since 1 is added inside the root, the graph is translated to the left by 1 unit.

Since the values inside the root must be positive.

$x+1â‰\yen 0$

Adding ‘$-1$’ on both sides.

$\begin{array}{c}x+1−1â‰\yen 0−1\\ x+0â‰\yen −1\\ xâ‰\yen −1\end{array}$

$xâ‰\yen −1⇒x∈[−1,∞)$

Therefore, domain: $[−1,∞)$

The values of $y$ are the square root of $x+1$. As square roots are positive,$y$takes all the positive real values including zero.

Therefore,$yâ‰\yen 0,\text{ â¶Ä‰â¶Ä‰}⇒y∈[0,∞)$

Therefore, Range:$[0,∞)$