Find \(f^{-1}(-25)\) for \(f(x)=x^2\)

📝 Question

Let:

\[ f:\mathbb{R}\to\mathbb{R}, \quad f(x)=x^2 \]

Find \(f^{-1}(-25)\).

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✅ Solution

🔹 Step 1: Meaning of \(f^{-1}(-25)\)

Since \(f(x)=x^2\) is not one-one, inverse function does not exist.

Here, \(f^{-1}(-25)\) means the inverse image of \(-25\).

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🔹 Step 2: Solve Equation

\[ f(x)=-25 \]

\[ x^2=-25 \] —

🔹 Step 3: Check in Real Numbers

For all real \(x\),

\[ x^2 \geq 0 \]

So \(x^2\) can never be negative.

Hence, no real solution exists.

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🎯 Final Answer

\[ \boxed{f^{-1}(-25)=\varnothing} \]

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🚀 Exam Shortcut

• Square of real number is always ≥ 0
• Negative value ⇒ no solution
• Answer = empty set