Find \(f^{-1}(1)\) for \(f(x)=x^3\) on \(\mathbb{C}\)

📝 Question

Let:

\[ f:\mathbb{C}\to\mathbb{C}, \quad f(x)=x^3 \]

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

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✅ Solution

🔹 Step 1: Meaning of \(f^{-1}(1)\)

Since \(f(x)=x^3\) is not one-one on \(\mathbb{C}\), inverse function does not exist.

Here, \(f^{-1}(1)\) means the inverse image of 1.

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🔹 Step 2: Solve Equation

\[ f(x)=1 \]

\[ x^3=1 \] —

🔹 Step 3: Find Cube Roots of Unity

The solutions of \(x^3=1\) are:

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🎯 Final Answer

\[ \boxed{f^{-1}(1)=\left\{1,\;\frac{-1+i\sqrt{3}}{2},\;\frac{-1-i\sqrt{3}}{2}\right\}} \]

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🚀 Exam Shortcut

• Solve \(x^3=1\)
• Use cube roots of unity
• Total 3 solutions in \(\mathbb{C}\)
• Answer is a set of all roots