Question

find the inverse of f(x)=x/x+1

Answer 1

Given:

$f(x)=\frac{x}{x+1}$

For inverse function the value of x,y is interchange then solve for y :

$\begin{gathered} f(x)=y \\ y=\frac{x}{x+1} \\ x\rightarrow y \\ y\rightarrow x \end{gathered}$

$\begin{gathered} y=\frac{x}{x+1} \\ so\colon \\ x=\frac{y}{y+1} \\ x(y+1)=y \\ y=xy+x \\ y-xy=x \\ y(1-x)=x \\ y=\frac{x}{1-x} \end{gathered}$

So inverse function is:

$\begin{gathered} f(x)=\frac{x}{x+1} \\ f^{-1}(x)=\frac{x}{1-x} \end{gathered}$