Question

Let $f:R-\left\{-\frac{3}{5}\right\}\to R$ be a function defined as $f\left(x\right)=\frac{2x}{5x+3}$.

Write $f^{-1}:\mathrm{Range}\mathrm{of}f\to R-\left\{-\frac{3}{5}\right\}$.

Answer 1

$$\begin{gathered} \text{Let}{f}^{-1}\left(x\right)=y...\left(1\right) \\ \Rightarrow f\left(y\right)=x \\ \Rightarrow \frac{2y}{5y+3}=x \\ \Rightarrow 2y=5xy+3x \\ \Rightarrow 2y-5xy=3x \\ \Rightarrow y\left(2-5x\right)=3x \\ \Rightarrow y=\frac{3x}{2-5x} \\ \Rightarrow f^{-1}\left(x\right)=\frac{3x}{2-5x}[\text{from}\left(1\right)\text{]} \end{gathered}$$