how do the average rates of change for the pair of functions compare over the given interval? f(x)=9x²…

Explanation:

Step1: Recall average - rate - of - change formula

The average rate of change of a function $y = h(x)$ over the interval $[a,b]$ is $\frac{h(b)-h(a)}{b - a}$. Here, $a=-6$ and $b = - 4$.

Step2: Calculate average rate of change of $f(x)$

For $f(x)=9x^{2}$, $f(-4)=9\times(-4)^{2}=9\times16 = 144$, $f(-6)=9\times(-6)^{2}=9\times36 = 324$. Then the average rate of change of $f(x)$ is $\frac{f(-4)-f(-6)}{-4-(-6)}=\frac{144 - 324}{2}=\frac{-180}{2}=-90$.

Step3: Calculate average rate of change of $g(x)$

For $g(x)=36x^{2}$, $g(-4)=36\times(-4)^{2}=36\times16 = 576$, $g(-6)=36\times(-6)^{2}=36\times36 = 1296$. Then the average rate of change of $g(x)$ is $\frac{g(-4)-g(-6)}{-4-(-6)}=\frac{576 - 1296}{2}=\frac{-720}{2}=-360$.

Step4: Find the ratio of average rates of change

To find how many times the average rate of change of $g(x)$ is that of $f(x)$, we calculate $\frac{\text{Average rate of change of }g(x)}{\text{Average rate of change of }f(x)}=\frac{-360}{-90}=4$.

Answer:

The average rate of change of $f(x)$ over $-6\leq x\leq - 4$ is $-90$. The average rate of change of $g(x)$ over $-6\leq x\leq - 4$ is $-360$. The average rate of change of $g(x)$ is $4$ times that of $f(x)$.