To evaluate the function composition [tex]\((f \circ g)(-64)\)[/tex], we need to follow these steps:
1. Evaluate the inner function [tex]\(g(x)\)[/tex] at [tex]\(x = -64\)[/tex]:
[tex]\[
g(x) = \sqrt[3]{x} + 1
\][/tex]
Plugging in [tex]\(x = -64\)[/tex]:
[tex]\[
g(-64) = \sqrt[3]{-64} + 1
\][/tex]
The cube root of [tex]\(-64\)[/tex] is a complex number. In this case:
[tex]\[
\sqrt[3]{-64} = 3 + 3.464101615137754j \quad (\text{where } j \text{ is the imaginary unit})
\][/tex]
Hence,
[tex]\[
g(-64) = 3 + 3.464101615137754j
\][/tex]
2. Next, evaluate the outer function [tex]\(f(x)\)[/tex] using the result from step 1:
[tex]\[
f(x) = (x + 1)^3
\][/tex]
Plugging in [tex]\(x = 3 + 3.464101615137754j\)[/tex]:
[tex]\[
f(3 + 3.464101615137754j) = \left((3 + 3.464101615137754j) + 1\right)^3
\][/tex]
Simplifying within the parenthesis:
[tex]\[
3 + 3.464101615137754j + 1 = 4 + 3.464101615137754j
\][/tex]
Now we need to cube this complex number:
[tex]\[
(4 + 3.464101615137754j)^3 = -79.99999999999994 + 124.70765814495915j
\][/tex]
Hence, the value of [tex]\((f \circ g)(-64)\)[/tex] is:
[tex]\[
(f \circ g)(-64) = -79.99999999999994 + 124.70765814495915j
\][/tex]
