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 leq x leq 5 ). the table has ( x ) values: 1, 3, 5, 7 and ( f(x) ) values: 3, 5, 15, 33.

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

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

From the table, when \( x = 3 \), \( f(3)=5 \); when \( x = 5 \), \( f(5)=15 \).

Step3: Substitute into the formula

Substitute \( a = 3 \), \( b = 5 \), \( f(3)=5 \), \( f(5)=15 \) into the formula: \(\frac{f(5)-f(3)}{5 - 3}=\frac{15 - 5}{5 - 3}\).

Step4: Simplify the expression

First, calculate the numerator: \( 15 - 5 = 10 \). Then, calculate the denominator: \( 5 - 3 = 2 \). So, \(\frac{10}{2}=5\).

Answer:

5