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≤x≤1. the average rate of change is (a)
x | 2 | 3 | 4 | 5 | 6
f(x) | 12 | 31 | 22 | 15 | 10

Explanation:

Step1: Identify the interval

The interval is from \( x = 3 \) to \( x = 6 \). So, \( x_1 = 3 \), \( f(x_1) = 31 \); \( x_2 = 6 \), \( f(x_2) = 10 \).

Step2: Apply the average rate of change formula

The formula for the average rate of change of a function \( f(x) \) over the interval \([x_1, x_2]\) is \(\frac{f(x_2) - f(x_1)}{x_2 - x_1}\).

Substitute the values: \(\frac{10 - 31}{6 - 3}\)

Step3: Calculate the numerator and denominator

Numerator: \( 10 - 31 = -21 \)

Denominator: \( 6 - 3 = 3 \)

Step4: Simplify the fraction

\(\frac{-21}{3} = -7\)

Answer:

-7