Question

Let $f=\{(x,\frac{x^2}{1+x^2}):x\in R\}$ be a function from $R$ into $R$ . Determine the inverse of $f$

Answer 1

Let $y=f\left(x\right)=\frac{x^2}{1+x^2}y(1+x^2)=x^2{x}^2=y+y{x}^2{x}^2-x^{2}y=y{x}^2(1-y)=y{x}^2=\frac{y}{1-y}x=\sqrt{\frac{y}{1-y}}$

$f^{-1}\left(x\right)=\sqrt{\frac{x}{1-x}}$