To decompose the function [tex]\( h(x) = (9x - 10)^3 \)[/tex] into the form [tex]\( h(x) = f(g(x)) \)[/tex], we need to identify two component functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] such that when [tex]\( g(x) \)[/tex] is applied first and then [tex]\( f(x) \)[/tex] is applied to the result of [tex]\( g(x) \)[/tex], we obtain [tex]\( h(x) \)[/tex].
Let's go through this step-by-step:
1. Identify the inner function [tex]\( g(x) \)[/tex]:
Observe the expression inside the parentheses of [tex]\( h(x) \)[/tex]. It is [tex]\( 9x - 10 \)[/tex]. Thus, we can choose this to be our inner function [tex]\( g(x) \)[/tex].
So, let
[tex]\[
g(x) = 9x - 10
\][/tex]
2. Identify the outer function [tex]\( f(x) \)[/tex]:
After applying [tex]\( g(x) \)[/tex], we see that the entire expression [tex]\( (9x - 10) \)[/tex] is then raised to the power of 3. Therefore, the outer function [tex]\( f \)[/tex] can be described as raising any input to the power of 3.
So, let
[tex]\[
f(u) = u^3
\][/tex]
Here, we use [tex]\( u \)[/tex] as a placeholder to emphasize that [tex]\( f \)[/tex] is an independent function that acts on the result of [tex]\( g \)[/tex].
3. Combining both functions:
Now, we confirm that applying [tex]\( g(x) \)[/tex] first and then [tex]\( f(u) \)[/tex] matches [tex]\( h(x) \)[/tex]:
[tex]\[
h(x) = f(g(x)) = f(9x - 10)
\][/tex]
Substituting [tex]\( g(x) \)[/tex] into [tex]\( f(u) \)[/tex]:
[tex]\[
f(g(x)) = f(9x - 10) = (9x - 10)^3
\][/tex]
This precisely matches our original function [tex]\( h(x) \)[/tex].
Therefore, the correct decomposition of [tex]\( h(x) = (9x - 10)^3 \)[/tex] into component functions is:
[tex]\[
h(x) = f(g(x))
\][/tex]
where
[tex]\[
g(x) = 9x - 10 \quad \text{and} \quad f(u) = u^3
\][/tex]
Thus, the correct answer choice is:
[tex]\[
f(u) = u^3 \quad \text{and} \quad g(x) = 9x - 10
\][/tex]
