To determine the average rate of change of the function [tex]\( g(x) = \frac{5}{x-1} + 2 \)[/tex] over the interval [tex]\([-4, 3]\)[/tex], follow these steps:
1. Evaluate the function [tex]\( g(x) \)[/tex] at the endpoints of the interval:
- At [tex]\( x = -4 \)[/tex]:
[tex]\[
g(-4) = \frac{5}{-4 - 1} + 2 = \frac{5}{-5} + 2 = -1 + 2 = 1
\][/tex]
- At [tex]\( x = 3 \)[/tex]:
[tex]\[
g(3) = \frac{5}{3 - 1} + 2 = \frac{5}{2} + 2 = 2.5 + 2 = 4.5
\][/tex]
2. Compute the average rate of change:
The formula for the average rate of change of a function [tex]\( g \)[/tex] over the interval [tex]\([a, b]\)[/tex] is given by:
[tex]\[
\text{Average Rate of Change} = \frac{g(b) - g(a)}{b - a}
\][/tex]
Here, [tex]\( a = -4 \)[/tex] and [tex]\( b = 3 \)[/tex]:
[tex]\[
\text{Average Rate of Change} = \frac{g(3) - g(-4)}{3 - (-4)} = \frac{4.5 - 1}{3 + 4} = \frac{3.5}{7} = 0.5
\][/tex]
3. Select the correct answer:
The average rate of change of [tex]\( g(x) \)[/tex] over the interval [tex]\([-4, 3]\)[/tex] is [tex]\( 0.5 \)[/tex].
So, the correct answer is:
[tex]\[
\boxed{\frac{1}{2}}
\][/tex]
