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 ).

Explanation:

Step1: Recall the formula for average rate of change

The formula for the average rate of change of a function \(y = f(x)\) over the interval \([a,b]\) is \(\frac{f(b)-f(a)}{b - a}\). Here, \(a = 3\) and \(b=5\).

Step2: Identify \(f(a)\) and \(f(b)\)

From the table, when \(x = 3\), \(f(3)=3\) (so \(f(a)=3\)) and when \(x = 5\), \(f(5)=27\) (so \(f(b)=27\)).

Step3: Substitute into the formula

Substitute \(a = 3\), \(b = 5\), \(f(a)=3\), and \(f(b)=27\) into the formula \(\frac{f(b)-f(a)}{b - a}\). We get \(\frac{27-3}{5 - 3}\).

Step4: Simplify the expression

First, calculate the numerator: \(27-3=24\). Then calculate the denominator: \(5 - 3=2\). So \(\frac{24}{2}=12\).

Answer:

\(12\)