i was wondering if two inverses that do not intersect at all, can they intersect on different points the line y=x.
It is possible for a function and its inverse to intersect at points that do not lie on the lie y = x. It is easy to prove that such intersection points (when they exist) lie on lines of the form y = -x + c where c is a constant.
For example, the intersection points of $\displaystyle f: R \rightarrow (-\infty, 1] ~ \text{where} ~ f(x) = -x^2 + 1$ and its inverse function include (0, 1) and (1, 0). These two points lie on the line y = -x + 1.