Chapter 3.10, Problem 25WE

To determine

To graph:the relation $\left\{\left(x,y\right)\text{: }y=-\left|x\right|\right\}$ and whether the given relation is a function or not.

Answer to Problem 25WE

The relation $\left\{\left(x,y\right)\text{: }y=-\left|x\right|\right\}$ is a function.

Explanation of Solution

Given information:

The relation $\left\{\left(x,y\right)\text{: }y=-\left|x\right|\right\}$ .

Result used:

Vertical line test:

A relation is a function if and only if no vertical line intersects its graph more than once.

Calculation:

It is given that the relation $y=-\left|x\right|$ .

Substitute $x=-2$ in $y=-\left|x\right|$ and find the value of y ,

  $\begin{array}{l}y=-\left|-2\right|\\ y=-2\end{array}$

Thus, the ordered pair is $\left(-2,-2\right)$ .

Substitute $x=-1$ in $y=-\left|x\right|$ and find the value of y ,

  $\begin{array}{l}y=-\left|-1\right|\\ y=-1\end{array}$

Thus, the ordered pair is $\left(-1,-1\right)$ .

Substitute $x=0$ in $y=-\left|x\right|$ and find the value of y ,

  $\begin{array}{l}y=-\left|0\right|\\ y=0\end{array}$

Thus, the ordered pair is $\left(0,0\right)$ .

Substitute $x=1$ in $y=-\left|x\right|$ and find the value of y ,

  $\begin{array}{l}y=-\left|1\right|\\ y=-1\end{array}$

Thus, the ordered pair is $\left(1,-1\right)$ .

Substitute $x=2$ in $y=-\left|x\right|$ and find the value of y ,

  $\begin{array}{l}y=-\left|2\right|\\ y=-2\end{array}$

Thus, the ordered pair is $\left(2,-2\right)$ .

Therefore, the ordered pairs $\left(-2,-2\right)$ , $\left(-1,-1\right)$ , $\left(0,0\right)$ , $\left(1,-1\right)$ , and $\left(2,-2\right)$ .

Graph:

Plot the ordered pair and connect the points by straight line as shown in Figure 1,

  

From Figure 1, it is observed that every vertical line of any point of x intersects exactly one on the graph.

That is, the graph satisfies the vertical line test.

Therefore, the relation $\left\{\left(x,y\right)\text{: }y=-\left|x\right|\right\}$ is a function.