I assume from the context that this question concerns a function from two-dimensional real space (RxR) into the real line R. That assumption helps me explain my answer but does not change it.
According to Burkill and Burkill A Second Course in Mathematical Analysis, the word that describes a function, such as f(0, y), which takes values that are identical to another function (in this case f(x, y)) but on a subset of the latter's domain (in this case the line x=0 is a proper subset of RxR) is the restriction of f to the the subset.
I daresay there are other terms used by other authors. In quite a few years of studying mathematics I myself have not encountered in this context either 'simplification' (pace @JeffMorrow) or 'reduction' (pace @Andrew) but it is quite possible that they are used also.
If you want to refer specifically to the values taken by f(0, y), they are its range.