To evaluate the function composition [tex]\((f \circ g)(-1)\)[/tex], we need to follow these steps:
1. Evaluate [tex]\(g(-1)\)[/tex]:
[tex]\[
g(x) = 4x^2 + 2
\][/tex]
Substitute [tex]\(x = -1\)[/tex]:
[tex]\[
g(-1) = 4(-1)^2 + 2 = 4(1) + 2 = 4 + 2 = 6
\][/tex]
Therefore, [tex]\(g(-1) = 6\)[/tex].
2. Use the result from [tex]\(g(-1)\)[/tex] to evaluate [tex]\(f(g(-1))\)[/tex]:
[tex]\[
g(-1) = 6
\][/tex]
Now, we need to evaluate [tex]\(f(6)\)[/tex]:
[tex]\[
f(x) = \frac{8}{x+2}
\][/tex]
Substitute [tex]\(x = 6\)[/tex]:
[tex]\[
f(6) = \frac{8}{6+2} = \frac{8}{8} = 1
\][/tex]
Therefore, [tex]\(f(6) = 1\)[/tex].
So, the composition [tex]\((f \circ g)(-1)\)[/tex] is:
[tex]\[
(f \circ g)(-1) = 1
\][/tex]
The correct answer is [tex]\(1\)[/tex].
