Find \(f^{-1}(-4)\) for \(f(x)=x^2\) on \(\mathbb{C}\)

📝 Question

Let:

\[ f:\mathbb{C}\to\mathbb{C}, \quad f(x)=x^2 \]

Find \(f^{-1}(-4)\).

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✅ Solution

🔹 Step 1: Meaning of \(f^{-1}(-4)\)

Since \(f(x)=x^2\) is not one-one on \(\mathbb{C}\), inverse function does not exist.

Here, \(f^{-1}(-4)\) means the inverse image of \(-4\).

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🔹 Step 2: Solve Equation

\[ f(x)=-4 \]

\[ x^2=-4 \]

\[ x=\pm \sqrt{-4} \] —

🔹 Step 3: Use Complex Numbers

\[ \sqrt{-4}=2i \]

So,

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🎯 Final Answer

\[ \boxed{f^{-1}(-4)=\{-2i,\,2i\}} \]

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🚀 Exam Shortcut

• Solve \(x^2 = negative\ number\)
• Use \(i=\sqrt{-1}\)
• Take both + and − values
• Answer is always a set (not single value)