We can distinguish the two functions by looking at the sections y = k for k ∈ R a constant.
(a) For the function f (x, y) = sin x - sin y these sections are z = sin x - sin k. In other words the graph z = sin x is shifted up and down as y varies.
(b) For the function f (x, y) = sin(x - y) these sections are z = sin(x- k). In other words the graph z = sin x is shifted to the right and to the left as y varies.
So we conclude that the right correspondences are (1) – (b) and (2) – (a).
