I'm trying to find a way to compute the inverse of $|y|^p-|y|=x$, I can try my best to compute this numerically, but I was wondering if there was any analytical way to determine this particular inverse. The problem does have any motivation beyond just regular curiosity.
EDIT: Since the function is not one-to-one, how would I go about finding the inverse on each segment of the function that is one-to-one? So for the intervals: $$ \left(-\infty, -\left(\frac{1}{p}\right)^{1/(p-1)}\right)\\ \left(-\left(\frac{1}{p}\right)^{1/(p-1)}, 0\right)\\ \left(0, \left(\frac{1}{p}\right)^{1/(p-1)}\right)\\ \left(\left(\frac{1}{p}\right)^{1/(p-1)}, \infty\right) $$