https://file.aiquickdraw.com/mathbot/file/17260024920398rwyjntj.jpg

To solve this problem, we need to calculate the average rate of change of the function \( f(x) \) over the interval \( [5, 7] \). 

The formula for the average rate of change of a function \( f(x) \) over an interval \( [x_1, x_2] \) is:

\[
\text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}
\]

From the table in the image:
- \( x_1 = 5 \) and \( f(5) = 6 \)
- \( x_2 = 7 \) and \( f(7) = 8 \)

Now applying the formula:

\[
\text{Average Rate of Change} = \frac{f(7) - f(5)}{7 - 5} = \frac{8 - 6}{7 - 5} = \frac{2}{2} = 1
\]

Thus, the average rate of change is **1**.

Do you need further details or have any questions? Here are 5 related questions for practice:

1. What is the average rate of change of the function over the interval \( [1, 3] \)?
2. How do you interpret the average rate of change geometrically?
3. Can the average rate of change be negative? If so, when does it happen?
4. How does the average rate of change relate to the slope of a secant line?
5. If the function was \( f(x) = x^2 \), how would you calculate the average rate of change over the same interval?

**Tip:** The average rate of change is equivalent to the slope of the line connecting two points on the graph of the function over the given interval.

Given the function defined in the table, find the average rate of change, in simplest form, of the function over the interval 5 ≤ x ≤ 7.

Find the average rate of change of the function f(x) over the interval 5 ≤ x ≤ 7, using the formula for rate of change. Suitable for students in Grades 9-11, this problem highlights core concepts such as functions, slope, and algebra.

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