To find the average rate of change of the function [tex]\( g(x) = \frac{5}{x - 1} + 2 \)[/tex] over the interval [tex]\([-4, 3]\)[/tex], we follow these steps:
1. Evaluate the function at the endpoints of the interval:
- For [tex]\( x = -4 \)[/tex]:
[tex]\[
g(-4) = \frac{5}{-4 - 1} + 2 = \frac{5}{-5} + 2 = -1 + 2 = 1
\][/tex]
- For [tex]\( x = 3 \)[/tex]:
[tex]\[
g(3) = \frac{5}{3 - 1} + 2 = \frac{5}{2} + 2 = 2.5 + 2 = 4.5
\][/tex]
2. Calculate the average rate of change using the formula:
[tex]\[
\text{Average Rate of Change} = \frac{g(3) - g(-4)}{3 - (-4)}
\][/tex]
3. Substitute the evaluated values and solve:
[tex]\[
\text{Average Rate of Change} = \frac{4.5 - 1}{3 - (-4)} = \frac{4.5 - 1}{3 + 4} = \frac{3.5}{7} = 0.5
\][/tex]
Thus, the average rate of change of [tex]\( g \)[/tex] over the interval [tex]\([-4, 3]\)[/tex] is [tex]\(0.5\)[/tex]. Hence, the correct answer is:
[tex]\[
\boxed{\frac{1}{2}}
\][/tex]
