To find the average rate of change of a function over a given interval [tex]\([x_1, x_2]\)[/tex], we follow these steps:
1. Identify the interval:
The interval provided is from [tex]\(x = 3\)[/tex] to [tex]\(x = 5\)[/tex].
2. Find the corresponding [tex]\(y\)[/tex]-values for these [tex]\(x\)[/tex]-values from the table:
- When [tex]\(x = 3\)[/tex], [tex]\(y = 8\)[/tex]
- When [tex]\(x = 5\)[/tex], [tex]\(y = 32\)[/tex]
3. Use the formula for the average rate of change:
[tex]\[
\text{Average rate of change} = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Here, [tex]\((x_1, y_1) = (3, 8)\)[/tex] and [tex]\((x_2, y_2) = (5, 32)\)[/tex].
4. Substitute the values into the formula:
[tex]\[
\text{Average rate of change} = \frac{32 - 8}{5 - 3}
\][/tex]
5. Calculate the differences:
[tex]\[
32 - 8 = 24
\][/tex]
[tex]\[
5 - 3 = 2
\][/tex]
6. Divide the differences to find the average rate of change:
[tex]\[
\text{Average rate of change} = \frac{24}{2} = 12
\][/tex]
Thus, the average rate of change for the function over the interval from [tex]\(x = 3\)[/tex] to [tex]\(x = 5\)[/tex] is 12.
So, the correct answer is:
A. 12
