To evaluate the function composition [tex]\((f \circ g)(-64)\)[/tex], we need to follow these steps:
1. First, evaluate [tex]\(g\)[/tex] at [tex]\(-64\)[/tex]:
[tex]\[
g(x) = \sqrt[3]{x} + 1
\][/tex]
Substitute [tex]\(-64\)[/tex] for [tex]\(x\)[/tex]:
[tex]\[
g(-64) = \sqrt[3]{-64} + 1
\][/tex]
2. Calculate [tex]\(\sqrt[3]{-64}\)[/tex]:
[tex]\[
\sqrt[3]{-64} = -4
\][/tex]
Therefore, substituting this value back into the function [tex]\(g\)[/tex]:
[tex]\[
g(-64) = -4 + 1 = -3
\][/tex]
So, [tex]\(g(-64) = -3\)[/tex].
3. Next, use this result to evaluate [tex]\(f\)[/tex] at [tex]\(-3\)[/tex]:
[tex]\[
f(x) = (x + 1)^3
\][/tex]
Substitute [tex]\(-3\)[/tex] for [tex]\(x\)[/tex]:
[tex]\[
f(-3) = (-3 + 1)^3 = (-2)^3 = -8
\][/tex]
So, [tex]\(f(g(-64)) = f(-3) = -8\)[/tex].
After evaluating these steps, we obtain [tex]\((f \circ g)(-64) = -8\)[/tex].
