Question

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

x	f(x)
3	57
5	33
7	17
9	9
11	9

answer

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function \( f(x) \) over the interval \( [a, b] \) is given by \( \frac{f(b) - f(a)}{b - a} \). Here, \( a = 5 \) and \( b = 7 \).

Step2: Find \( f(5) \) and \( f(7) \) from the table

From the table, when \( x = 5 \), \( f(5) = 33 \), and when \( x = 7 \), \( f(7) = 17 \).

Step3: Substitute into the formula

Substitute \( a = 5 \), \( b = 7 \), \( f(5) = 33 \), and \( f(7) = 17 \) into the formula:
\[
\frac{f(7) - f(5)}{7 - 5} = \frac{17 - 33}{7 - 5}
\]

Step4: Simplify the numerator and denominator

First, calculate the numerator: \( 17 - 33 = -16 \).
Then, calculate the denominator: \( 7 - 5 = 2 \).
So, we have \( \frac{-16}{2} \).

Step5: Simplify the fraction

\( \frac{-16}{2} = -8 \).

Answer:

\( -8 \)