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 \\(x^5 - 2\\).

\\h(x) = \sqrt4{x^5 - 2}\\

\\(f(x) = \\) (simplify your answer.)
\\(g(x) = \\) (simplify your answer.)

Explanation:

Identify the given functions and composition structure

We are given the composite function:

$$ h(x) = \sqrt[4]{x^5 - 2} $$

We need to express \(h(x)\) as a composition of two functions \(f\) and \(g\) such that:

$$ h(x) = (f \circ g)(x) = f(g(x)) $$

We are given that one of the functions is \(x^5 - 2\).

Determine the inner and outer functions

Let the inner function be \(g(x) = x^5 - 2\).
Substituting \(g(x)\) into \(f(g(x))\) to get \(h(x)\):

$$ f(g(x)) = \sqrt[4]{g(x)} $$

Therefore, the outer function \(f(x)\) must be:

$$ f(x) = \sqrt[4]{x} $$

Verify the composition

We check if \(f(g(x)) = h(x)\):

$$ f(g(x)) = f(x^5 - 2) = \sqrt[4]{x^5 - 2} = h(x) $$

This matches the given function \(h(x)\).
Thus, the functions are:

$$ f(x) = \sqrt[4]{x} $$

$$ g(x) = x^5 - 2 $$

Answer:

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 \(x^5 - 2\).
\(h(x) = \sqrt[4]{x^5 - 2}\)

\(f(x) =\) <blank>\(\sqrt[4]{x}\)</blank> (Simplify your answer.)
\(g(x) =\) <blank>\(x^5 - 2\)</blank> (Simplify your answer.)