Finding the inverse of a function involving |x|

Question

I need to find the inverse of the following function

$ f:(-1,1) \rightarrow \mathbb{R} $

$ f(x) = \dfrac{x}{1-|{x}|} $

How do I deal with the absolute value here?

Thanks

Answer 1

Because of the absolute value, it's not one-to-one ($f(-x) = f(x)$), so there is no inverse function.

EDIT: The new function is one-to-one. Treat the two cases $x \ge 0$ and $x < 0$ separately. Note that if $y = f(x)$, $y \ge 0$ if and only if $x \ge 0$.