To find the average rate of change [tex]\(R\)[/tex] of the function [tex]\(f\)[/tex] between [tex]\(a\)[/tex] and [tex]\(b\)[/tex], we use the formula:
[tex]\[
R = \frac{f(b) - f(a)}{b - a}
\][/tex]
Given that [tex]\(f(2) = 5\)[/tex] and [tex]\(f(5) = -3\)[/tex], we can substitute these values into the formula. Here, [tex]\(a = 2\)[/tex] and [tex]\(b = 5\)[/tex].
Step-by-step, let's proceed as follows:
1. Substitute [tex]\(f(b) = -3\)[/tex] and [tex]\(f(a) = 5\)[/tex]:
[tex]\[
R = \frac{-3 - 5}{5 - 2}
\][/tex]
2. Simplify the expression in the numerator:
[tex]\[
-3 - 5 = -8
\][/tex]
So the formula becomes:
[tex]\[
R = \frac{-8}{5 - 2}
\][/tex]
3. Simplify the denominator:
[tex]\[
5 - 2 = 3
\][/tex]
Hence, the formula now is:
[tex]\[
R = \frac{-8}{3}
\][/tex]
The value of [tex]\(R\)[/tex] is:
[tex]\[
R = -\frac{8}{3}
\][/tex]
Therefore, the correct answer is:
A) [tex]\(-\frac{8}{3}\)[/tex]
