If [tex]\( g(x) = \frac{x+1}{x-2} \)[/tex] and [tex]\( h(x) = 4 - x \)[/tex], what is the value of [tex]\( (g \circ h)(-3) \)[/tex]?

A. [tex]\(\frac{8}{5}\)[/tex]
B. [tex]\(\frac{5}{2}\)[/tex]
C. [tex]\(\frac{15}{2}\)[/tex]
D. [tex]\(\frac{18}{5}\)[/tex]



Answer :

To determine the value of [tex]\((g \circ h)(-3)\)[/tex], which is the same as finding [tex]\(g(h(-3))\)[/tex], we need to perform the following steps:

1. Evaluate [tex]\(h(-3)\)[/tex]:
[tex]\[ h(x) = 4 - x \][/tex]
Substitute [tex]\(x = -3\)[/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]\(g(7)\)[/tex]:
[tex]\[ g(x) = \frac{x + 1}{x - 2} \][/tex]
Substitute [tex]\(x = 7\)[/tex]:
[tex]\[ g(7) = \frac{7 + 1}{7 - 2} = \frac{8}{5} \][/tex]

Therefore, [tex]\((g \circ h)(-3) = g(h(-3)) = g(7) = \frac{8}{5}\)[/tex].

So, the value of [tex]\((g \circ h)(-3)\)[/tex] is:
[tex]\[ \boxed{\frac{8}{5}} \][/tex]