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

🎥 Video Explanation

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📝 Question

Let \( f:\mathbb{R} \to \mathbb{R} \),

\[ f(x)=x^3+3 \]

• (a) \(x^{1/3}-3\)
• (b) \(x^{1/3}+3\)
• (c) \((x-3)^{1/3}\)
• (d) \((x+3)^{1/3}\)

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✅ Solution

🔹 Step 1: Let \(y=f(x)\)

\[ y=x^3+3 \] —

🔹 Step 2: Solve for \(x\)

\[ x^3=y-3 \]

\[ x=(y-3)^{1/3} \] —

🔹 Step 3: Replace \(y\) by \(x\)

\[ f^{-1}(x)=(x-3)^{1/3} \] —

🔹 Final Answer

\[ \boxed{\text{Option (c)}} \]