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

📝 Question

Let:

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

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

✅ Solution

🔹 Step 1: Understand Meaning of \(f^{-1}(25)\)

Since \(f(x)=x^2\) is not one-one on \(\mathbb{R}\), inverse function does not exist.

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

🔹 Step 2: Solve Equation

\[ f(x)=25 \]

\[ x^2=25 \]

\[ x=\pm 5 \] —

🎯 Final Answer

\[ \boxed{f^{-1}(25)=\{-5,\,5\}} \]

🚀 Exam Shortcut

• If function is not one-one ⇒ inverse image can have multiple values
• Solve \(x^2 = given\ number\)
• Always take both + and − roots