Find \(f\circ f(x)\)

📝 Question

Let:

\[ f(x)=\left(3-x^3\right)^{1/3} \]

Find \(f(f(x))\).

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✅ Solution

🔹 Step 1: Write composition

\[ f(f(x))=\left(3-\left[f(x)\right]^3\right)^{1/3} \] —

🔹 Step 2: Substitute \(f(x)\)

\[ f(f(x))=\left(3-\left((3-x^3)^{1/3}\right)^3\right)^{1/3} \] —

🔹 Step 3: Simplify

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So,

\[ f(f(x))=\left(3-(3-x^3)\right)^{1/3} \]

\[ =\left(x^3\right)^{1/3} \] —

🔹 Step 4: Final simplification

\[ (x^3)^{1/3}=x \] —

🎯 Final Answer

\[ \boxed{f(f(x))=x} \]

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🚀 Exam Shortcut

• Cube root and cube cancel each other
• Recognize inverse-type structure
• Result becomes original input