given the function defined in the table below, find the average of the function over the interval 4 ≤ x ≤ 8…

Explanation:

Step1: Recall average - value formula

The average value of a function $y = f(x)$ over the interval $[a,b]$ is given by $\frac{1}{b - a}\int_{a}^{b}f(x)dx$. For a discrete - valued function, the average value over the interval $[a,b]$ with $x$ values $x_1,x_2,\cdots,x_n$ in the interval is $\frac{\sum_{i}f(x_i)}{n}$, where $n$ is the number of data points in the interval. Here, $a = 4$, $b = 8$, and the $x$ values in the interval $[4,8]$ are $x_1=4$, $x_2 = 6$, $x_3=8$, so $n = 3$.

Step2: Calculate the sum of $f(x)$ values

We have $f(4)=53$, $f(6)=33$, $f(8)=13$. The sum $\sum_{i = 1}^{3}f(x_i)=53 + 33+13=99$.

Step3: Calculate the average value

The average value of the function over the interval $[4,8]$ is $\frac{\sum_{i = 1}^{3}f(x_i)}{3}=\frac{99}{3}=33$.

Answer:

33