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

📝 Question

Let:

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

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

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✅ Solution

🔹 Step 1: Check invertibility

The function \(f(x)=x^3\) is strictly increasing on \(\mathbb{R}\).

Hence, it is one-one and onto, so inverse exists.

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🔹 Step 2: Find inverse function

Let:

\[ y=x^3 \]

Taking cube root:

:contentReference[oaicite:0]{index=0}

Thus,

\[ f^{-1}(x)=\sqrt[3]{x} \] —

🔹 Step 3: Find \(f^{-1}(1)\)

\[ f^{-1}(1)=\sqrt[3]{1}=1 \] —

🎯 Final Answer

\[ \boxed{1} \]

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🚀 Exam Shortcut

• \(x^3\) is always one-one on \(\mathbb{R}\)
• Inverse = cube root
• \(\sqrt[3]{1}=1\)