Chapter 3.3, Problem 21PPS

To determine

To find whether the given function is linear.

Answer to Problem 21PPS

The function is a linear function.

Explanation of Solution

Given:

A table is given as

Concept Used:

A function is linear if the rate of change is constant.

  $\text{Rate of change}=\frac{\text{change in }y\text{coordinate}}{\text{change in }x\text{coordinate}}$

Calculation:

Consider the given coordinates in the table.

$\left(-7,11\right),\left(-5,14\right),\left(-3,17\right),\left(-1,20\right)$ , and $\left(0,23\right)$

In order to evaluate the rate of change

  $\begin{array}{c}\text{Rate of change}=\frac{\text{change in }y\text{coordinate}}{\text{change in }x\text{coordinate}}\\ =\frac{14-11}{-5+7}\\ =\frac{3}{2}\\ \text{Rate of change}=\frac{\text{change in }y\text{coordinate}}{\text{change in }x\text{coordinate}}\\ =\frac{17-14}{-3+5}\\ =\frac{3}{2}\end{array}$

And

  $\begin{array}{c}\text{Rate of change}=\frac{\text{change in }y\text{coordinate}}{\text{change in }x\text{coordinate}}\\ =\frac{20-17}{3-1}\\ =\frac{3}{2}\\ \text{Rate of change}=\frac{\text{change in }y\text{coordinate}}{\text{change in }x\text{coordinate}}\\ =\frac{23-20}{3-1}\\ =\frac{3}{2}\end{array}$

Thus, rate of change is constant and it is $\frac{3}{2}$ .

Therefore, the function is a linear function.