Select the correct answer.

Consider function [tex] g(x)=\frac{5}{x-1}+2 [/tex].

What is the average rate of change of function [tex] g [/tex] over the interval [tex] [-4,3] [/tex]?

A. [tex] \frac{1}{2} [/tex]
B. [tex] -\frac{1}{2} [/tex]
C. 2
D. [tex] -\frac{7}{2} [/tex]



Answer :

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]