given the function defined in the table below, find the average rate of change, in simplest form, of the…

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 = 5) and (b=7).

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

From the table, when (x = 5), (f(5)=36) and when (x = 7), (f(7)=324).

Step3: Substitute into the formula

Substitute (a = 5), (b = 7), (f(a)=36), and (f(b)=324) into the formula (\frac{f(b)-f(a)}{b - a}). We get (\frac{324 - 36}{7-5}).

Step4: Simplify the expression

First, calculate the numerator: (324-36=288). Then, calculate the denominator: (7 - 5=2). So, (\frac{288}{2}=144).

Answer:

(144)