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

🎥 Video Explanation

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📝 Question

\[ f(x)=\frac{-x|x|}{1+x^2}, \quad f:\mathbb{R}\to(-1,1) \]

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

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✅ Solution

🔹 Step 1: Case-wise Expression

For \(x \ge 0\):

\[ f(x)=\frac{-x^2}{1+x^2} \]

For \(x < 0\):

\[ f(x)=\frac{x^2}{1+x^2} \] —

🔹 Step 2: Let \(y=f(x)\)

Case 1:

\[ y=\frac{x^2}{1+x^2} \Rightarrow x^2=\frac{y}{1-y} \]

Case 2:

\[ y=\frac{-x^2}{1+x^2} \Rightarrow x^2=\frac{-y}{1+y} \] —

🔹 Step 3: Combine

Using symmetry:

\[ x=\frac{-y}{\sqrt{1-y^2}} \] —

🔹 Final Answer

\[ \boxed{f^{-1}(x)=\frac{-x}{\sqrt{1-x^2}}} \]