Question

question
given the function defined in the table below, find the average rate of change, in simplest form, of the function over the interval 3 ≤ x ≤ 5.

x	f(x)
3	6
4	18
5	54
6	162

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, \( a = 3 \) and \( b = 5 \).

Step2: Find \( f(3) \) and \( f(5) \) from the table

From the table, when \( x = 3 \), \( f(3)=6 \); when \( x = 5 \), \( f(5) = 54 \).

Step3: Substitute into the formula

Substitute \( a = 3 \), \( b = 5 \), \( f(3)=6 \), and \( f(5)=54 \) into the formula: \( \frac{f(5)-f(3)}{5 - 3}=\frac{54 - 6}{5 - 3} \).

Step4: Simplify the expression

First, calculate the numerator: \( 54 - 6 = 48 \). Then, calculate the denominator: \( 5 - 3 = 2 \). Now, divide: \( \frac{48}{2}=24 \).

Answer:

24