To find the average rate of change of the function over the interval [tex]\(3 \leq x \leq 6\)[/tex], we can follow these steps:
1. Identify the given points on the function that correspond to the interval boundaries. From the table, these points are:
- When [tex]\(x = 3\)[/tex], [tex]\(f(x) = 43\)[/tex]
- When [tex]\(x = 6\)[/tex], [tex]\(f(x) = 13\)[/tex]
2. Recall the formula for the average rate of change of a function over an interval [tex]\([x_1, x_2]\)[/tex]:
[tex]\[
\text{Average rate of change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}
\][/tex]
3. Substitute the given points into the formula. Here, [tex]\(x_1 = 3\)[/tex] and [tex]\(x_2 = 6\)[/tex]:
[tex]\[
\text{Average rate of change} = \frac{f(6) - f(3)}{6 - 3}
\][/tex]
4. Substitute the function values [tex]\(f(3) = 43\)[/tex] and [tex]\(f(6) = 13\)[/tex]:
[tex]\[
\text{Average rate of change} = \frac{13 - 43}{6 - 3}
\][/tex]
5. Perform the calculations:
[tex]\[
\text{Average rate of change} = \frac{13 - 43}{6 - 3} = \frac{-30}{3} = -10
\][/tex]
Thus, the average rate of change of the function over the interval [tex]\(3 \leq x \leq 6\)[/tex] is [tex]\(\boxed{-10.0}\)[/tex].
