Question

given the function defined in the table below, find the average rate of change, in simplest form, of the function over the interval ( 1 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, \(a = 1\) and \(b=3\).

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

From the table, when \(x = 1\), \(f(1)=78\) (so \(f(a)=78\)), and when \(x = 3\), \(f(3)=66\) (so \(f(b)=66\)).

Step3: Substitute into the formula

Substitute \(a = 1\), \(b = 3\), \(f(a)=78\), and \(f(b)=66\) into \(\frac{f(b)-f(a)}{b - a}\). We get \(\frac{66 - 78}{3-1}\).

Step4: Simplify the expression

First, calculate the numerator: \(66-78=-12\). Then calculate the denominator: \(3 - 1=2\). So \(\frac{-12}{2}=-6\).

Answer:

\(-6\)