To find the average rate of change of the function [tex]\( f(x) \)[/tex] over the interval [tex]\( 4 \leq x \leq 20 \)[/tex], we can use the following formula:
[tex]\[
\text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}
\][/tex]
where [tex]\( x_1 \)[/tex] and [tex]\( x_2 \)[/tex] are the endpoints of the interval and [tex]\( f(x_1) \)[/tex] and [tex]\( f(x_2) \)[/tex] are the corresponding function values.
1. Identify the points:
- We are given the interval [tex]\( 4 \leq x \leq 20 \)[/tex].
- Let [tex]\( x_1 = 4 \)[/tex] and [tex]\( x_2 = 20 \)[/tex].
2. Find the function values at these points:
- At [tex]\( x = 4 \)[/tex], [tex]\( f(x_1) = 4 \)[/tex].
- At [tex]\( x = 20 \)[/tex], [tex]\( f(x_2) = 8 \)[/tex].
3. Substitute these values into the formula:
[tex]\[
\text{Average Rate of Change} = \frac{f(20) - f(4)}{20 - 4} = \frac{8 - 4}{20 - 4} = \frac{4}{16} = \frac{1}{4}
\][/tex]
Therefore, the average rate of change of the function [tex]\( f(x) \)[/tex] over the interval [tex]\( 4 \leq x \leq 20 \)[/tex] is [tex]\( \boxed{0.25} \)[/tex] or [tex]\( \frac{1}{4} \)[/tex] in simplest form.
