Chapter 3.10, Problem 16WE

To determine

To find:the domain of the given relation and draw the function also whether the relation is a function or not.

Answer to Problem 16WE

The domain of relation is $\left\{-2,\text{ 2}\right\}$ and the relation is not a function.

Explanation of Solution

Given information:

The relation is $\left\{\left(x,y\right):\left|x\right|=2\text{ and }\left|y\right|\le 2\right\}$ where x and y are integer.

Definition used:

A function is relation in which different ordered pairs has different first coordinates.

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 is $\left\{\left(x,y\right):\left|x\right|=2\text{ and }\left|y\right|\le 2\right\}$ where x and y are integer.

Note that, the modulus of x defines as $\left|x\right|=\left\{\begin{array}{l}x\text{ if }x\ge 0\\ -x\text{ if }x<0\end{array}\right\}$

Since $\left|x\right|=2$ , the values of x are $-2\text{ and 2}$ .

The value of $\left|y\right|\le 2$ is computed by using modulus definition.

  $\left|y\right|\le 2\Rightarrow -2\le y\le 2\text{ where }y\text{ is integers}$

That is, the values of y are $-2,-1,\text{ 0, 1, and 2}$ .

The relation is a set of ordered pairs as listed below,

{ $\left(-2,-2\right),\left(-2,-1\right),\left(-2,0\right)$ , $\left(-2,1\right)$ , $\left(-2,2\right)$ , $\left(2,-2\right),\left(2,-1\right),\left(2,0\right),\left(2,1\right),\left(2,2\right)$ }

Note that, the domain of the relation is a set of first coordinates in the ordered pairs.

Therefore, the domain of relation is $\left\{-2,\text{ 2}\right\}$ .

A function is relation in which different ordered pairs has different first coordinates.

Here, the different ordered pairs $\left(-2,-2\right),\left(-2,-1\right),\left(-2,0\right),\left(-2,1\right)$ and $\left(-2,-2\right)$ has a same first coordinate.

Therefore, the given relation $\left\{\left(x,y\right):\left|x\right|=2\text{ and }\left|y\right|\le 2\right\}$ is not a function.

Graph:

The graph of the relation $\left\{\left(x,y\right):\left|x\right|=2\text{ and }\left|y\right|\le 2\right\}$ as shown in Figure 1,

  

From Figure 1, it is observed that the vertical line passing through 2 intersects more than one points so it cannot satisfy the vertical line test.

Therefore, the relation is not function.