Inverse of a function with absolute value function

Question

For $\lambda > 0$ and $y \in \mathbb{R}$:

$$ f(y) = \frac{|y|^\lambda * sign(y) - 1}{\lambda}. $$

What is the inverse function of $f$.

Answer 1

A function $h$ has an inverse iff $h$ is bijective.

The domain of each function can be restricted so that a bijective restriction results.

You should split your functions into bijective restrictions.

The following holds for the partial inverses $f^{-1}$ of $f$:

$$f^{-1}(y)=\cases{?&$ \left( 1+{\it \lambda~}\,y \right) ^{{{\it \lambda}}^{-1}}<0$\cr 0&$ \left( 1+{\it \lambda}\,y \right) ^{{{\it \lambda}}^{-1}}=0$\cr \left( 1+{\it \lambda}\,y \right) ^{{{\it \lambda~}}^{-1}},- \left( 1+{\it \lambda}\,y \right) ^{{{\it \lambda}}^{-1}}&$\left( 1+{\it \lambda}\,y \right) ^{{{\it \lambda~}}^{-1}}>0$\cr}$$

Check the domains of the functions in the bottom line of the curly bracket.