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]
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]
