On the Wikipedia article for function problem it currently defines a function problem as follows:
A functional problem $P$ is defined as a relation $R$ over strings of an arbitrary alphabet $\Sigma$: $R\subseteq\Sigma^*\times\Sigma^*$. An algorithm that solves $P$ if for every input $x$ such that there exists a $y$ satisfying $(x,y)\in R$, the algorithm produces one such $y$.
My question is this: why one such $y$ - doesn't a function have unique output?
My guess: Does this have to do with the fact that Turing Machines (one possible instance of a model for computation) will sometimes output strings which differ only by trailing zeros or something, but you want to ignore that in these problems?