Question

suppose \\(h(x) = 3\sqrt3{x} + 2\\).

find two functions \\(f\\) and \\(g\\) such that \\((f \circ g)(x) = h(x)\\).

neither function can be the identity function.
(there may be more than one correct answer.)

\\(f(x) = \square\\)
\\(g(x) = \square\\)

Explanation:

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<reasoning>

Identify the inner function

\[
g(x) = \sqrt[3]{x}
\]

Identify the outer function

\[
f(x) = 3x + 2
\]

Verify the composition

\[
(f \circ g)(x) = f(g(x)) = f(\sqrt[3]{x}) = 3\sqrt[3]{x} + 2 = H(x)
\]
</reasoning>

<answer>
Suppose \(H(x) = 3\sqrt[3]{x} + 2\).

Find two functions \(f\) and \(g\) such that \((f \circ g)(x) = H(x)\).

Neither function can be the identity function. (There may be more than one correct answer.)

\(f(x) =\) <blank>\(3x + 2\)</blank>

\(g(x) =\) <blank>\(\sqrt[3]{x}\)</blank>
</answer>

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"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Algebra",
"Decomposing Functions"
]
}
</post_analysis>

Answer:

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"Decomposing Functions"
],
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</pre_analysis>

<reasoning>

Identify the inner function

\[
g(x) = \sqrt[3]{x}
\]

Identify the outer function

\[
f(x) = 3x + 2
\]

Verify the composition

\[
(f \circ g)(x) = f(g(x)) = f(\sqrt[3]{x}) = 3\sqrt[3]{x} + 2 = H(x)
\]
</reasoning>

<answer>
Suppose \(H(x) = 3\sqrt[3]{x} + 2\).

Find two functions \(f\) and \(g\) such that \((f \circ g)(x) = H(x)\).

Neither function can be the identity function. (There may be more than one correct answer.)

\(f(x) =\) <blank>\(3x + 2\)</blank>

\(g(x) =\) <blank>\(\sqrt[3]{x}\)</blank>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Algebra",
"Decomposing Functions"
]
}
</post_analysis>