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Q. 3 ACYP

Expert-verified
Found in: Page 606

### Algebra 1

Book edition Student Edition
Author(s) Carter, Cuevas, Day, Holiday, Luchin
Pages 801 pages
ISBN 9780078884801

# Graph the function, and compare to the parent graph. State the domain and range.$\mathbit{g}\left(x\right)\mathbf{=}\sqrt{\mathbf{x}}\mathbf{-}\mathbf{4}$

The domain of the given function is $\mathbit{x}\mathbf{\in }\mathbf{\left[}\mathbf{0}\mathbf{,}\mathbf{\infty }\mathbf{\right)}$ and the range is $\mathbit{g}\left(x\right)\mathbf{\in }\mathbf{\left[}\mathbf{0}\mathbf{,}\mathbf{\infty }\mathbf{\right)}$

See the step by step solution

## Step 1. State the concept of parent graph.

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.

## Step 2. State the concept of domain and range.

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.

## Step 3. Graph the function.

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 ‘$g\left(x\right)$'.

$\begin{array}{c}\text{For\hspace{0.17em}\hspace{0.17em}}x=0,\\ g\left(0\right)=\sqrt{0}-4\\ =0-4\\ =-4\end{array}$

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

$\begin{array}{c}\text{For\hspace{0.17em}\hspace{0.17em}}x=4,\\ g\left(4\right)=\sqrt{4}-2\\ =2-4\\ =-2\end{array}$

$\begin{array}{c}\text{For\hspace{0.17em}\hspace{0.17em}}x=9,\\ g\left(9\right)=\sqrt{9}-4\\ =3-4\\ =-1\end{array}$

$\begin{array}{c}\text{For\hspace{0.17em}\hspace{0.17em}}x=16,\\ g\left(x\right)=\sqrt{16}-4\\ =4-4\\ =0\end{array}$

Values of ‘$\mathbit{x}$Values of ‘$\mathbit{g}\mathbf{\left(}\mathbit{x}\mathbf{\right)}$$\left(x,y\right)$
0$-4$$\left(0,-4\right)$
1$-3$$\left(1,-3\right)$
4$-2$$\left(4,-2\right)$
9$-1$$\left(9,-1\right)$
160$\left(16,0\right)$

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

## Step 4. Comparison with the parent 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 $g\left(x\right)=\sqrt{x}$.

## Step 5. State the domain and range.

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, $x\ge 0,\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}⇒x\in \left[0,\infty \right)$.

Therefore, domain: $\left[0,\infty \right)$

In $g\left(x\right)=\sqrt{x}-4$ the square root of x is 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, $g\left(x\right)\ge -4,\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}⇒g\left(x\right)\in \left[-4,\infty \right)$

Therefore, Range: $\left[-4,\infty \right)$