Find \(f^{-1}(x)\)

📝 Question

Let:

\[ f(x)=4-(x-7)^3 \]

Find \(f^{-1}(x)\).

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✅ Solution

🔹 Step 1: Let

\[ y=4-(x-7)^3 \] —

🔹 Step 2: Solve for \(x\)

\[ (x-7)^3=4-y \]

Take cube root:

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\[ x=7+\sqrt[3]{4-y} \] —

🔹 Step 3: Write inverse

Interchange \(x\) and \(y\):

\[ f^{-1}(x)=7+\sqrt[3]{4-x} \] —

🎯 Final Answer

\[ \boxed{f^{-1}(x)=7+\sqrt[3]{4-x}} \]

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🚀 Exam Shortcut

• Shift and cube structure
• Undo operations in reverse order
• Cube → cube root, subtract → add