Question

given the function defined in the table below, find the average rate of change, in simplest form, of the function over the interval $2\leq x\leq 3$.

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, \(a = 2\) and \(b=3\).

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

From the table, when \(x = 2\), \(f(2)=17\) (so \(f(a)=17\)) and when \(x = 3\), \(f(3)=19\) (so \(f(b)=19\)).

Step3: Substitute into the formula

Substitute \(a = 2\), \(b = 3\), \(f(a)=17\), and \(f(b)=19\) into \(\frac{f(b)-f(a)}{b - a}\). We get \(\frac{19 - 17}{3-2}\).

Step4: Simplify the expression

\(\frac{19 - 17}{3 - 2}=\frac{2}{1}=2\).

Answer:

\(2\)