To find the value of [tex]\((g \circ h)(-3)\)[/tex], we need to evaluate the function [tex]\(h\)[/tex] at [tex]\(-3\)[/tex] first, and then substitute this result into the function [tex]\(g\)[/tex]. Let's go through the process step-by-step:
1. Evaluate [tex]\(h(-3)\)[/tex]:
Given the function [tex]\(h(x) = 4 - x\)[/tex], we substitute [tex]\(-3\)[/tex] for [tex]\(x\)[/tex]:
[tex]\[
h(-3) = 4 - (-3) = 4 + 3 = 7.
\][/tex]
So, [tex]\(h(-3) = 7\)[/tex].
2. Evaluate [tex]\(g(h(-3))\)[/tex]:
Next, we take the result from the first step, which is [tex]\(7\)[/tex], and substitute it into the function [tex]\(g(x)\)[/tex]. The function [tex]\(g(x)\)[/tex] is given by [tex]\(\frac{x+1}{x-2}\)[/tex]. Now we need to evaluate [tex]\(g(7)\)[/tex]:
[tex]\[
g(7) = \frac{7+1}{7-2} = \frac{8}{5}.
\][/tex]
Thus, the value of [tex]\((g \circ h)(-3)\)[/tex] is [tex]\(\frac{8}{5}\)[/tex].
Therefore, the correct answer is:
[tex]\[
\boxed{\frac{8}{5}}
\][/tex]
