Question

Let f : R − {−1} → R − {1} be given by $f\left(x\right)=\frac{x}{x+1}.\mathrm{Write}{f}^{-1}\left(x\right)$.

Answer 1

$$\begin{gathered} \text{Let}{f}^{-1}\left(x\right)=y...\left(1\right) \\ \Rightarrow f\left(y\right)=x \\ \Rightarrow \frac{y}{y+1}=x \\ \Rightarrow y=xy+x \\ \Rightarrow y-xy=x \\ \Rightarrow y\left(1-x\right)=x \\ \Rightarrow y=\frac{x}{1-x} \\ \Rightarrow f^{-1}\left(x\right)=\frac{x}{1-x}[\text{from}\left(1\right)\text{]} \end{gathered}$$