Do you assume the ideal is homogeneous? No power of S belong to (X,S-1). 

If you assume that the ideal is homogeneous, then S^m is in the ideal (f,g) for m=deg f+deg g-1. Indeed all homogeneous polynomials of that degree are in the ideal (f,g). This a simple Hilbert series computation, just use the fact that the Koszul complex resolves K[X,S]/(f,g) and use it to compute the Hilbert series of K[X,S]/(f,g).