How to invert a function that contains a sign function?

Question

I have the following function: $$f(w,y-x) = \operatorname{sgn}[y-x]|y-x|^w$$, where $w \geq 0, (y-x) \in[-1,1]$, which results in $f(w,y-x) \in [-1,1]$.

I would like to find its inverse function with respect to $x$. But I got stuck when trying to invert the sign function and the absolute function. Can you show me how to invert the function please ? $$x = ???$$

Answer 1

Note that $\text{sgn}(f) = \text{sgn}(y-x)$ and $|y-x|^w = |f|$, hence $|y-x|=|f|^{1/w}$, hence $y-x = \text{sgn}(f)|f|^{1/w}$, hence \begin{equation} x = y - \text{sgn}(f)|f|^{1/w} \end{equation}