Question
Express the given function \( h \) as a composition of two functions \( f \) and \( g \) so that \( h(x)=(f \circ g)(x) \), where one of the functions is \( 5 x-2 \). \( h(x)=(5 x-2)^{6} \) \( f(x)=\square \)
Ask by Ford Martinez. in the United States
Feb 25,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( f(x) = x^6 \)
Solution
To express h(x) = (5x - 2)^6 as the composition (f ∘ g)(x) with one of the functions being 5x - 2, we can set
g(x) = 5x - 2
Then, since h(x) = (g(x))^6, we choose
f(x) = x^6
This gives us the composition:
(f ∘ g)(x) = f(g(x)) = (5x - 2)^6 = h(x).
So the answer is:
f(x) = x^6.
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Simplify this solution Bonus Knowledge
To express the function \( h(x) = (5x - 2)^6 \) as a composition of two functions, we can identify \( g(x) \) as the inner function and \( f(x) \) as the outer function. Here, we can choose \( g(x) = 5x - 2 \) and then define \( f(x) = x^6 \). So, we have: \( g(x) = 5x - 2 \) \( f(x) = x^6 \) Therefore, the composition is \( h(x) = (f \circ g)(x) = f(g(x)) = f(5x - 2) = (5x - 2)^6 \).
