Chapter 5.3, Problem 30AYU

To determine

To find: The function is linear, exponential or neither.

Answer to Problem 30AYU

Neither linear function nor exponential function.

Explanation of Solution

Given:

$x$	$-1$	$0$	$1$	$2$	$3$
$f\left(x\right)$	$6$	$1$	$0$	$3$	$10$

$x$	$g\left(x\right)$	Average rate of change	Ratio of consecutive outputs
$-1$	$6$
		$\frac{\Delta y}{\Delta x}=\frac{1-6}{0-\left(-1\right)}=\frac{-5}{1}=-5$	$\frac{1}{6}=0.167$
$0$	$1$
		$\frac{\Delta y}{\Delta x}=\frac{0-1}{1-0}=\frac{-1}{1}=-1$	$\frac{0}{1}=0$
$1$	$0$
		$\frac{\Delta y}{\Delta x}=\frac{3-0}{2-1}=\frac{3}{1}=3$	$\frac{3}{0}=\infty$
$2$	$3$
		$\frac{\Delta y}{\Delta x}= \frac{10-3}{3-2}= \frac{7}{1}=7$	$\frac{10}{3}=3.33$
$3$	$10$

From the above table, the average rate of change from $-1$ to 0 is $-5$, and the average rate of change from 0 to 1 is $-1$. Because the average rate of change is not constant, the function is not a linear function. The ratio of consecutive output from $-1$ to 0 is $\text{0}\text{.167}$, and the ratio consecutive output from 0 to 1 is 0. Because the consecutive output from is not a constant, the function is not an exponential function. Therefore the given function is neither linear nor exponential.