Consider functions [tex]f[/tex] and [tex]g[/tex].

[tex]\[
\begin{array}{l}
f(x)=(x+1)^3 \\
g(x)=\sqrt[3]{x}+1
\end{array}
\][/tex]

Evaluate the function composition.

[tex]\[
(f \circ g)(-64)= \boxed{\square}
\][/tex]



Answer :

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]