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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: $y=\sqrt{x}+1.5$

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

role="math" $\begin{array}{c}\text{¹ó´Ç°ù â¶Ä‰}x=0,\\ y=\sqrt{0}+1.5\\ =0+1.5\\ =1.5\end{array}$

role="math" localid="1648140062818" $\begin{array}{c}\text{¹ó´Ç°ù â¶Ä‰}x=1,\\ y=\sqrt{1}+1.5\\ =1+1.5\\ =2.5\end{array}$

$\begin{array}{c}\text{¹ó´Ç°ù â¶Ä‰}x=4,\\ y=\sqrt{4}+1.5\\ =2+1.5\\ =3.5\end{array}$

$\begin{array}{c}\text{¹ó´Ç°ù â¶Ä‰}x=9,\\ y=\sqrt{9}+1.5\\ =3+1.5\\ =4.5\end{array}$

$\begin{array}{c}\text{¹ó´Ç°ù â¶Ä‰}x=16,\\ y=\sqrt{16}+1.5\\ =4+1.5\\ =5.5\end{array}$

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

The parent function$y=\sqrt{x}+1.5$ is the simplest square root function.

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

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 graph$y=\sqrt{x}+1.5$ is obtained by adding ‘1.5’ to the parent function.

Therefore, the graph$y=\sqrt{x}+3$is translated(shifted) upward by 1.5 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$x$are all positive real numbers including zero.

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

Therefore, domain:$[0,∞)$

In $y=\sqrt{x}+1.5$, the square root of xis added by ‘1.5’.

As the square root is always positive, the least value it takes is zero.

Find the starting value of $y$by substituting $x=0$in $y=\sqrt{x}+1.5$.

$\begin{array}{c}y=\sqrt{0}+1.5\\ =0+1.5\\ =1.5\end{array}$

Also, in $y=\sqrt{x}+1.5$, coefficient of $\sqrt{x}$is 1, which is positive.

Therefore,$y$ takes all the real values greater than or equal to ‘1.5’.

That is,$yâ‰\yen 1.5,\text{ â¶Ä‰â¶Ä‰}⇒y∈[1.5,∞)$

Therefore, Range:$[1.5,∞)$