which of the following functions have an average rate of change equal to 0 on the interval from x = -2 to x…

Explanation:

Step1: Recall average - rate - of - change formula

The average rate of change of a function $y = f(x)$ over the interval $[a,b]$ is $\frac{f(b)-f(a)}{b - a}$. Here, $a=-2$ and $b = 2$.

Step2: Calculate for $f(x)=7 - 3x$

$f(2)=7-3\times2=7 - 6=1$, $f(-2)=7-3\times(-2)=7 + 6 = 13$. Then $\frac{f(2)-f(-2)}{2-(-2)}=\frac{1 - 13}{4}=\frac{-12}{4}=-3\neq0$.

Step3: Calculate for $f(x)=7x$

$f(2)=7\times2 = 14$, $f(-2)=7\times(-2)=-14$. Then $\frac{f(2)-f(-2)}{2-(-2)}=\frac{14-(-14)}{4}=\frac{28}{4}=7\neq0$.

Step4: Calculate for $f(x)=7 - 3x^{2}$

$f(2)=7-3\times2^{2}=7 - 12=-5$, $f(-2)=7-3\times(-2)^{2}=7 - 12=-5$. Then $\frac{f(2)-f(-2)}{2-(-2)}=\frac{-5-(-5)}{4}=\frac{-5 + 5}{4}=0$.

Step5: Calculate for $f(x)=7$

$f(2)=7$, $f(-2)=7$. Then $\frac{f(2)-f(-2)}{2-(-2)}=\frac{7 - 7}{4}=0$.

Answer:

$f(x)=7 - 3x^{2}$, $f(x)=7$