Find the Inverse of f(x) = x/(1−x)

\(f^{-1}(x) = \dfrac{x}{1 + x},\; x \ne -1\)

Explanation

Step 1: \(y = \frac{x}{1-x} \Rightarrow y(1-x) = x \Rightarrow y - yx = x \Rightarrow y = x + yx = x(1+y) \Rightarrow x = \frac{y}{1+y}\).

Step 2: \(f^{-1}(y) = \frac{y}{1+y}\,(y \ne -1)\).

Step 3: \(f^{-1}(x) = \frac{x}{1+x}\,(x \ne -1)\).