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

📝 Question

Let:

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

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

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✅ Solution

🔹 Step 1: Meaning of \(f^{-1}(1)\)

The function \(f(x)=x^4\) is not one-one on \(\mathbb{R}\).

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

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🔹 Step 2: Solve Equation

\[ f(x)=1 \]

\[ x^4=1 \] —

🔹 Step 3: Find Solutions

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🎯 Final Answer

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

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🚀 Exam Shortcut

• Solve \(x^4 = 1\)
• Take real roots only (since domain is \(\mathbb{R}\))
• Answer is a set