How is the inverse of a set-valued transformation defined?
For a function $f$, we know that the inverse exists if and only if $f$ is bijective. However, I have found in a book that the inverse of $$S:[0,1]\to[0,1], \qquad S(x)=4x(1-x),$$ is given by $$S^{-1}([0,x])=\left[0,\frac{1}{2}-\frac{1}{2}\sqrt{1-x}\right]\cup\left[\frac{1}{2}+\frac{1}{2}\sqrt{1-x},1\right].$$ It is pretty straightforward to check graphically (by plotting $S(x)$ for $x\in[0,1]$) that the inverse is correct. Nevertheless $S$ is clearly not bijective. So what it the characterization of inverse transformations?
I think that my confusion stems from the fact that I am not used to think of functions as set-valued objects.