Chapter 1, Problem 37RE

(a)

To determine

To find: The domain of the function $y=\{\begin{array}{l}\sqrt{-x},-4\le x\le 0\\ \sqrt{x},0<x\le 4\end{array}$ .

Answer to Problem 37RE

The domain of the $y=\{\begin{array}{l}\sqrt{-x},-4\le x\le 0\\ \sqrt{x},0<x\le 4\end{array}$ function is the interval $\left[-4,4\right]$ .

Explanation of Solution

Given information:

The given function is $y=\{\begin{array}{l}\sqrt{-x},-4\le x\le 0\\ \sqrt{x},0<x\le 4\end{array}$ .

Calculation:

The function $y=\{\begin{array}{l}\sqrt{-x},-4\le x\le 0\\ \sqrt{x},0<x\le 4\end{array}$ is defined for all the real values of $x$ from $-4$ to $4$ .

Therefore, the domain of the $y=\{\begin{array}{l}\sqrt{-x},-4\le x\le 0\\ \sqrt{x},0<x\le 4\end{array}$ function is the interval $\left[-4,4\right]$ .

(b)

To determine

To find: The range of the function $y=\{\begin{array}{l}\sqrt{-x},-4\le x\le 0\\ \sqrt{x},0<x\le 4\end{array}$ .

Answer to Problem 37RE

The range of the $y=\{\begin{array}{l}\sqrt{-x},-4\le x\le 0\\ \sqrt{x},0<x\le 4\end{array}$ function is the interval $\left[0,2\right]$ .

Explanation of Solution

Given information:

The given function is $y=\{\begin{array}{l}\sqrt{-x},-4\le x\le 0\\ \sqrt{x},0<x\le 4\end{array}$ .

Calculation:

The value of the function $y=\{\begin{array}{l}\sqrt{-x},-4\le x\le 0\\ \sqrt{x},0<x\le 4\end{array}$ lies between $0$ to $2$ for all the real values of $x$ in the domain.

Therefore, the range of the $y=\{\begin{array}{l}\sqrt{-x},-4\le x\le 0\\ \sqrt{x},0<x\le 4\end{array}$ function is the interval $\left[0,2\right]$ .

(c)

To determine

To graph: The function $y=\{\begin{array}{l}\sqrt{-x},-4\le x\le 0\\ \sqrt{x},0<x\le 4\end{array}$ .

Explanation of Solution

Given information:

The given functionis $y=\{\begin{array}{l}\sqrt{-x},-4\le x\le 0\\ \sqrt{x},0<x\le 4\end{array}$ .

Graph:

To graph a function $y=\{\begin{array}{l}\sqrt{-x},-4\le x\le 0\\ \sqrt{x},0<x\le 4\end{array}$ , follow the steps using graphing calculator.

First press “ON” button on graphical calculator, press $\boxed{\text{Y}=}$ key and enter right hand side of the equation $y=\sqrt{-x}$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{\text{2nd}}\boxed{x^2}\boxed{(-)}\boxed{\text{X}}\boxed{)}\boxed{÷}\boxed{(}\boxed{\text{X}}\boxed{2\text{nd}}\boxed{\text{MATH}}\boxed{4}\boxed{(-)}\boxed{4}\boxed{2\text{nd}}\boxed{\text{MATH}}\boxed{\to }\boxed{1}\boxed{\text{X}}\boxed{2\text{nd}}\boxed{\text{MATH}}\boxed{6}\boxed{0}\boxed{)}$ . Now, press the $\boxed{\text{ENTER}}$ key and enter right hand side of the equation $y=\sqrt{x}$ after the symbol ${\text{Y}}_2$ . Enter the keystrokes $\boxed{\text{2nd}}\boxed{x^2}\boxed{\text{X}}\boxed{)}\boxed{÷}\boxed{(}\boxed{\text{X}}\boxed{2\text{nd}}\boxed{\text{MATH}}\boxed{3}\boxed{0}\boxed{2\text{nd}}\boxed{\text{MATH}}\boxed{\to }\boxed{1}\boxed{\text{X}}\boxed{2\text{nd}}\boxed{\text{MATH}}\boxed{6}\boxed{4}\boxed{)}$ .

The display will show the equations,

  $\begin{array}{l}{\text{Y}}_{\text{1}}=\sqrt{-\text{X}}/\left(\text{X}\ge -4\text{and X}\le \text{0}\right)\\ {\text{Y}}_2=\sqrt{\text{X}}/\left(\text{X}>0\text{and X}\le 4\right)\end{array}$

Now, press the $\boxed{\text{GRAPH}}$ key and $\boxed{\text{TRACE}}$ key to produce the graph of both the equations as shown below.

  

Figure (1)

Interpretation: From the above graph it can be observed that the graph the function $y=\{\begin{array}{l}\sqrt{-x},-4\le x\le 0\\ \sqrt{x},0<x\le 4\end{array}$ is symmetric about the y -axis.