To find the value of [tex]\((g \circ h)(-3)\)[/tex], we need to evaluate the functions step by step.
First, let's understand the meaning of [tex]\((g \circ h)(x)\)[/tex]. This notation represents the composition of the functions [tex]\(g(x)\)[/tex] and [tex]\(h(x)\)[/tex], where [tex]\(g(h(x))\)[/tex] is to be evaluated. In this case, we will first evaluate [tex]\(h(x)\)[/tex] at [tex]\(-3\)[/tex], and then use this result as the input to the function [tex]\(g(x)\)[/tex].
Given the functions:
[tex]\[ g(x) = \frac{x+1}{x-2} \][/tex]
[tex]\[ h(x) = 4 - x \][/tex]
1. Evaluate [tex]\(h(-3)\)[/tex]:
[tex]\[ h(-3) = 4 - (-3) = 4 + 3 = 7 \][/tex]
2. Evaluate [tex]\(g\)[/tex] at the result of [tex]\(h(-3)\)[/tex], which is [tex]\(7\)[/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 choice from the given options is:
[tex]\[ \boxed{\frac{8}{5}} \][/tex]
