Fiber of a function ($f^{-1}(y)$)

For a function $f:X\to Y$ and $y\in Y$, its fiber is the set $f^{-1}(y)=\{x\in X:f(x)=y\}$. Fibers partition the domain after empty fibers are omitted. A function is injective exactly when every fiber has at most one element. Quantum hidden subgroup problem algorithms use functions whose fibers are cosets.