what is the average rate of change of the following function on the interval 0, 7? f(x) = { -4 -∞ < x ≤ -4…

Explanation:

Step1: Recall average - rate - of - change formula

The average rate of change of a function $y = f(x)$ on the interval $[a,b]$ is $\frac{f(b)-f(a)}{b - a}$. Here, $a = 0$ and $b = 7$.

Step2: Find $f(0)$ and $f(7)$

Since $0$ satisfies $- 4<0<3$, then $f(0)=0$ according to the piece - wise function $f(x)=x$ for $-4 < x<3$. Since $7$ satisfies $3\leqslant7<10$, then $f(7) = 3$ according to the piece - wise function $f(x)=3$ for $3\leqslant x<10$.

Step3: Calculate the average rate of change

Substitute $a = 0$, $b = 7$, $f(0)=0$, and $f(7)=3$ into the formula $\frac{f(b)-f(a)}{b - a}$. We get $\frac{f(7)-f(0)}{7 - 0}=\frac{3-0}{7}=\frac{3}{7}$.

Answer:

$\frac{3}{7}$