Chapter 3.10, Problem 19WE

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 19WE

The domain of relation is $\left\{-2,-1\text{, 1, 2}\right\}$ and the relation is not a function.

Explanation of Solution

Given information:

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

The term $\left|x\right|y=2$ can be written as $\left|x\right|=\frac{2}{y}$ .

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

By using the definition of the modulus, $x=\frac{2}{y}\text{ or }x=-\frac{2}{y}$ .

If the value of $x=\frac{2}{y}$ ,

The value of x and y are integers, so the value of y cannot be 0, more than 2 and less than

  $-2$ .

Thus, the only possible values of y are $-2,-1,\text{ 1,}\text{and 2}$ .

The ordered pairs of the elements are $\left(-2,-1\right),\left(-1,-2\right),\left(1,2\right),\left(2,1\right)$ such that $x=\frac{2}{y}$ .

If the value of $x=-\frac{2}{y}$ , the only possible values of y are $-2,-1,\text{ 1,}\text{and 2}$ .

The ordered pairs of the elements are $\left(-2,1\right),\left(-1,2\right),\left(1,-2\right),\left(2,-1\right)$ such that $x=-\frac{2}{y}$ .

Thus, the relation is a set of ordered pairs as listed below,

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

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

Therefore, the domain of relation is $\left\{-2,-1\text{, 1, 2}\right\}$ .

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

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

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

Graph:

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

  

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

Therefore, the relation is not function.