Let's find the composition of functions given [tex]\( f(x) = \sqrt[3]{x-4} \)[/tex] and [tex]\( g(x) = x^3 + 4 \)[/tex].
### (a) [tex]\( f \circ g \)[/tex]
The composition [tex]\( f \circ g \)[/tex] is defined as [tex]\( f(g(x)) \)[/tex].
1. Start with [tex]\( g(x) = x^3 + 4 \)[/tex].
2. Substitute [tex]\( g(x) \)[/tex] into [tex]\( f \)[/tex]:
[tex]\[
f(g(x)) = f(x^3 + 4)
\][/tex]
3. Now, use the definition of [tex]\( f(x) \)[/tex] which is [tex]\( f(x) = \sqrt[3]{x - 4} \)[/tex]:
[tex]\[
f(x^3 + 4) = \sqrt[3]{(x^3 + 4) - 4}
\][/tex]
4. Simplify the expression inside the cube root:
[tex]\[
f(x^3 + 4) = \sqrt[3]{x^3}
\][/tex]
5. The cube root of [tex]\( x^3 \)[/tex] is [tex]\( x \)[/tex]:
[tex]\[
f(x^3 + 4) = x
\][/tex]
So,
[tex]\[
f \circ g = x
\][/tex]
### Testing [tex]\( f \circ g \)[/tex] at [tex]\( x = 2 \)[/tex]:
[tex]\[
f \circ g (2) = 2
\][/tex]
Thus,
[tex]\[
f \circ g (2) = 2.0
\][/tex]
### (b) [tex]\( g \circ f \)[/tex]
The composition [tex]\( g \circ f \)[/tex] is defined as [tex]\( g(f(x)) \)[/tex].
1. Start with [tex]\( f(x) = \sqrt[3]{x-4} \)[/tex].
2. Substitute [tex]\( f(x) \)[/tex] into [tex]\( g \)[/tex]:
[tex]\[
g(f(x)) = g(\sqrt[3]{x-4})
\][/tex]
3. Now, use the definition of [tex]\( g(x) \)[/tex] which is [tex]\( g(x) = x^3 + 4 \)[/tex]:
[tex]\[
g(\sqrt[3]{x-4}) = (\sqrt[3]{x-4})^3 + 4
\][/tex]
4. Simplify the expression:
[tex]\[
(\sqrt[3]{x-4})^3 = x - 4
\][/tex]
5. Thus,
[tex]\[
g(\sqrt[3]{x-4}) = (x - 4) + 4
\][/tex]
6. Simplify the expression:
[tex]\[
g(\sqrt[3]{x-4}) = x
\][/tex]
So,
[tex]\[
g \circ f = x
\][/tex]
### Testing [tex]\( g \circ f \)[/tex] at [tex]\( x = 2 \)[/tex]:
[tex]\[
g \circ f (2) = g(\sqrt[3]{2 - 4}) = g(\sqrt[3]{-2})
\][/tex]
[tex]\[
g(\sqrt[3]{-2}) = (\sqrt[3]{-2})^3 + 4 = -2 + 4 = 2
\][/tex]
[tex]\[
g \circ f (2) = 2 + 6.661338147750939 \times 10^{-16} \imath
\][/tex]
Thus,
[tex]\[
g \circ f (2) = 2 + 6.661338147750939 \times 10^{-16} \imath
\][/tex]
### (c) [tex]\( g \circ g \)[/tex]
The composition [tex]\( g \circ g \)[/tex] is defined as [tex]\( g(g(x)) \)[/tex].
1. Start with [tex]\( g(x) = x^3 + 4 \)[/tex].
2. Substitute [tex]\( g(x) \)[/tex] into itself:
[tex]\[
g(g(x)) = g(x^3 + 4)
\][/tex]
3. Now, use the definition of [tex]\( g(x) \)[/tex]:
[tex]\[
g(x^3 + 4) = (x^3 + 4)^3 + 4
\][/tex]
4. Simplify inside the expression:
When evaluated at [tex]\( x = 2 \)[/tex]:
[tex]\[
g(x) = 2^3 + 4 = 8 + 4 = 12
\][/tex]
[tex]\[
g(g(2)) = g(12) = 12^3 + 4 = 1728 + 4 = 1732
\][/tex]
So,
[tex]\[
g \circ g = 1732
\][/tex]
Therefore, the compositions are:
[tex]\[
\begin{aligned}
(a) & \quad f \circ g (2) = 2.0 \\
(b) & \quad g \circ f (2) = 2 + 6.661338147750939 \times 10^{-16} \imath \\
(c) & \quad g \circ g (2) = 1732 \\
\end{aligned}
\][/tex]
