Answer :
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].
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].
