r - How can I determine that Y_n can be represented as a funtion of X_n -

on enter i'm having sequence of pairs (x_n, y_n). consider following 2 graphics of 2 possible sequences.

in first case x_n can modeled f(y_n) while in second case has no sense. question how can determine if trying represent x_n f(y_n) makes sense? probable there criterium or that?

what can done in multivariate situation (i.e. when we're trying represent y f(x_1, x_2, ..., x_k))?

please note trying fit points graphicaly (e.g. on first graph) , seeing if fits data not ok. i'm looking numerical criterium.

please feel free propose variants in either matlab or r. link on page algorithm great too!