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

🎥 Video Explanation

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📝 Question

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

\[ f(x)=x^2-3 \]

• (a) \(\sqrt{x+3}\)
• (b) \(-\sqrt{x+3}\)
• (c) \(x+\sqrt{3}\)
• (d) none of these

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✅ Solution

🔹 Step 1: Check Injectivity

\[ f(x)=x^2-3 \Rightarrow f(2)=1,\; f(-2)=1 \]

Different inputs → same output ⇒ ❌ Not one-one

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🔹 Step 2: Conclusion

Inverse exists only if function is one-one.

Since function is not injective on \(\mathbb{R}\),

❌ Inverse does not exist

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🔹 Final Answer

\[ \boxed{\text{Option (d): none of these}} \]