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

📝 Question

Let:

\[ f:\mathbb{C}\to\mathbb{C}, \quad f(x)=x^4 \]

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

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✅ Solution

🔹 Step 1: Meaning of \(f^{-1}(1)\)

Since \(f(x)=x^4\) 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

\[ f(x)=1 \]

\[ x^4=1 \] —

🔹 Step 3: Find Fourth Roots of Unity

Write \(1\) in exponential form:

\[ 1 = e^{2k\pi i} \]

Fourth roots are:

\[ x = e^{\frac{2k\pi i}{4}}, \quad k=0,1,2,3 \]

So, the roots are:

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🎯 Final Answer

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

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🚀 Exam Shortcut

• Solve \(x^4=1\)
• Use roots of unity formula
• Total 4 roots in \(\mathbb{C}\)
• Even powers ⇒ symmetric roots