Chapter 1, Problem 38RE

(a)

To determine

To find: The domain of the function $y=\{\begin{array}{l}-x-2,-2\le x\le -1\\ x,-1<x\le 1\\ -x+2,1<x\le 2\end{array}$ .

Answer to Problem 38RE

The domain of the function $y=\{\begin{array}{l}-x-2,-2\le x\le -1\\ x,-1<x\le 1\\ -x+2,1<x\le 2\end{array}$ is the interval $\left[-2,2\right]$ .

Explanation of Solution

Given information:

The given function is $y=\{\begin{array}{l}-x-2,-2\le x\le -1\\ x,-1<x\le 1\\ -x+2,1<x\le 2\end{array}$ .

Calculation:

The function $y=\{\begin{array}{l}-x-2,-2\le x\le -1\\ x,-1<x\le 1\\ -x+2,1<x\le 2\end{array}$ is defined for the real values of $x$ between $-2$ to $2$ .

Therefore, the domain of the function $y=\{\begin{array}{l}-x-2,-2\le x\le -1\\ x,-1<x\le 1\\ -x+2,1<x\le 2\end{array}$ is the interval $\left[-2,2\right]$ .

(b)

To determine

To find: The range of the function $y=\{\begin{array}{l}-x-2,-2\le x\le -1\\ x,-1<x\le 1\\ -x+2,1<x\le 2\end{array}$ .

Answer to Problem 38RE

The range of the function $y=\{\begin{array}{l}-x-2,-2\le x\le -1\\ x,-1<x\le 1\\ -x+2,1<x\le 2\end{array}$ is the interval $\left[-1,1\right]$ .

Explanation of Solution

Given information:

The given function is $y=\{\begin{array}{l}-x-2,-2\le x\le -1\\ x,-1<x\le 1\\ -x+2,1<x\le 2\end{array}$ .

Calculation:

For $y=-x-2$ on the interval $\left[-2,-1\right]$ , if $x=-2$ , then $y=0$ and if $x=-1$ , then $y=-1$ . So, this portion of the graph has range of $\left[1,0\right]$ .

For $y=x$ on the interval $\left(-1,1\right]$ , if $x=-1$ , then $y=-1$ and if $x=1$ , then $y=1$ . So, the range of this portion is $\left(-1,1\right]$ .

For $y=-x+2$ on the interval $\left(1,2\right]$ , if $x=1$ , then $y=1$ and if $x=2$ , then $y=0$ . So, this portion has a range of $\left[0,1\right)$ .

Therefore, thefunction of the range is $\left[-1,1\right]$ .

(c)

To determine

To graph: The function $y=\{\begin{array}{l}-x-2,-2\le x\le -1\\ x,-1<x\le 1\\ -x+2,1<x\le 2\end{array}$ .

Explanation of Solution

Given information:

The given functionis $y=\{\begin{array}{l}-x-2,-2\le x\le -1\\ x,-1<x\le 1\\ -x+2,1<x\le 2\end{array}$ .

Graph:

To graph a function $y=\{\begin{array}{l}-x-2,-2\le x\le -1\\ x,-1<x\le 1\\ -x+2,1<x\le 2\end{array}$ , 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 $y=-x-2$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes:

  $\begin{array}{l}\boxed{(}\boxed{(-)}\boxed{\text{X}}\boxed{-}\boxed{2}\boxed{)}\boxed{÷}\boxed{(}\boxed{\text{X}}\boxed{2\text{nd}}\boxed{\text{MATH}}\boxed{4}\boxed{(-)}\boxed{2}\boxed{2\text{nd}}\\ \boxed{\text{MATH}}\boxed{\to }\boxed{1}\boxed{\text{X}}\boxed{2\text{nd}}\boxed{\text{MATH}}\boxed{6}\boxed{(-)}\boxed{1}\boxed{)}\end{array}$

Press the $\boxed{\text{ENTER}}$ key and enter right hand side of the equation $y=x$ after the symbol ${\text{Y}}_2$ . Enter the keystrokes:

  $\begin{array}{l}\boxed{(}\boxed{\text{X}}\boxed{)}\boxed{÷}\boxed{(}\boxed{\text{X}}\boxed{2\text{nd}}\boxed{\text{MATH}}\boxed{3}\boxed{(-)}\boxed{1}\boxed{2\text{nd}}\\ \boxed{\text{MATH}}\boxed{\to }\boxed{1}\boxed{\text{X}}\boxed{2\text{nd}}\boxed{\text{MATH}}\boxed{6}\boxed{1}\boxed{)}\end{array}$

Now, press the $\boxed{\text{ENTER}}$ key and enter right hand side of the equation $y=-x+2$ after the symbol ${\text{Y}}_2$ . Enter the keystrokes:

  $\begin{array}{l}\boxed{(}\boxed{(-)}\boxed{\text{X}}\boxed{+}\boxed{2}\boxed{)}\boxed{÷}\boxed{(}\boxed{\text{X}}\boxed{2\text{nd}}\boxed{\text{MATH}}\boxed{3}\boxed{1}\boxed{2\text{nd}}\\ \boxed{\text{MATH}}\boxed{\to }\boxed{1}\boxed{\text{X}}\boxed{2\text{nd}}\boxed{\text{MATH}}\boxed{6}\boxed{2}\boxed{)}\end{array}$

The display will show the equations,

  $\begin{array}{l}{\text{Y}}_{\text{1}}=\left(-\text{X}-2\right)/\left(\text{X}\ge -2\text{and}\text{X}\le -1\right)\\ {\text{Y}}_2=\left(\text{X}\right)/\left(\text{X}>-1\text{and}\text{X}\le 1\right)\\ {\text{Y}}_2=\left(-\text{X}+2\right)/\left(\text{X}>1\text{and}\text{X}\le 2\right)\end{array}$

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 given function is symmetric about the origin.