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 \(-1\)

Write \(-1\) in exponential form:

\[ -1 = e^{i\pi} \]

Cube roots are:

\[ x = e^{i(\pi+2k\pi)/3}, \quad k=0,1,2 \]

So, the roots 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 exponential form \(e^{i\theta}\)
• Apply formula: \(\theta+2k\pi\)/3
• Total 3 roots in \(\mathbb{C}\)