To solve the problem, we need to find the compositions of functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex]. The functions are given as:
[tex]\[ f(x) = \sqrt[3]{x - 9} \][/tex]
[tex]\[ g(x) = x^3 + 9 \][/tex]
Let's evaluate each composition step-by-step.
### (a) [tex]\( f \circ g \)[/tex]
The composition [tex]\( f \circ g \)[/tex] means [tex]\( f(g(x)) \)[/tex].
1. First, find [tex]\( g(x) \)[/tex]:
[tex]\[ g(x) = x^3 + 9 \][/tex]
2. Next, substitute [tex]\( g(x) \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[ f(g(x)) = f(x^3 + 9) \][/tex]
3. Use the definition of [tex]\( f(x) \)[/tex]:
[tex]\[ f(x^3 + 9) = \sqrt[3]{(x^3 + 9) - 9} \][/tex]
[tex]\[ f(x^3 + 9) = \sqrt[3]{x^3} \][/tex]
[tex]\[ f(x^3 + 9) = x \][/tex]
Therefore, [tex]\( f \circ g(x) = x \)[/tex]. For example, for [tex]\( x = 2 \)[/tex]:
[tex]\[ f \circ g(2) = 2 \][/tex]
### (b) [tex]\( g \circ f \)[/tex]
The composition [tex]\( g \circ f \)[/tex] means [tex]\( g(f(x)) \)[/tex].
1. First, find [tex]\( f(x) \)[/tex]:
[tex]\[ f(x) = \sqrt[3]{x - 9} \][/tex]
2. Next, substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex]:
[tex]\[ g(f(x)) = g(\sqrt[3]{x - 9}) \][/tex]
3. Use the definition of [tex]\( g(x) \)[/tex]:
[tex]\[ g(\sqrt[3]{x - 9}) = (\sqrt[3]{x - 9})^3 + 9 \][/tex]
[tex]\[ g(\sqrt[3]{x - 9}) = x - 9 + 9 \][/tex]
[tex]\[ g(\sqrt[3]{x - 9}) = x \][/tex]
Therefore, [tex]\( g \circ f(x) = x \)[/tex]. For example, for [tex]\( x = 2 \)[/tex]:
[tex]\[ g \circ f(2) = 2.000000000000001 + 2.220446049250313e-15i \][/tex]
### (c) [tex]\( g \circ g \)[/tex]
The composition [tex]\( g \circ g \)[/tex] means [tex]\( g(g(x)) \)[/tex].
1. First, find [tex]\( g(x) \)[/tex]:
[tex]\[ g(x) = x^3 + 9 \][/tex]
2. Next, substitute [tex]\( g(x) \)[/tex] into [tex]\( g(x) \)[/tex] again:
[tex]\[ g(g(x)) = g(x^3 + 9) \][/tex]
3. Use the definition of [tex]\( g(x) \)[/tex]:
[tex]\[ g(x^3 + 9) = (x^3 + 9)^3 + 9 \][/tex]
For [tex]\( x = 2 \)[/tex]:
[tex]\[ g(g(2)) = 4922 \][/tex]
So, summarizing the results:
- (a) [tex]\( f \circ g(x) \)[/tex] yields 2.0 for [tex]\( x = 2 \)[/tex].
- (b) [tex]\( g \circ f(x) \)[/tex] yields approximately 2.0 with a small imaginary component for [tex]\( x = 2 \)[/tex].
- (c) [tex]\( g \circ g(x) \)[/tex] yields 4922 for [tex]\( x = 2 \)[/tex].
