Correct - h(x), the unknown blue curve, is a piecewise function of f(x) from 0 to 75 and g(x) from 75 to 150. The ultimate goal is to fit a single curve defined by a single function.

The reason for using Thiele's Interpolation Formula was that it was a feature in Maple, yes. However, Maple also features polynomial, splines, linear, and so on but Thiele's formula does a perfect job on the regular curves shown in the majority of those pictures I uploaded.

The problem arises when the three primary points of the h(x) - the blue curve - are attempted to be fit for the points - (0,0), (75,25), and (150,50) - fit a perfectly linear line. But if more points from the original two functions f and g are chosen, say, the points of mean value, Thiele's formula does not fit very accurately if it fits at all for it is easy to hit vertical asymptotes as the formula divides by zero.

Polynomial interpolation does a terrible job of fitting the ideal h(x) perfectly.

I have read online about various interpolation methods but most of them are beyond me. One that at least sounded promising is found in this link:

http://en.wikipedia.org/wiki/Whittaker%E2%80%93Shannon_interpolation_formula

However it is quite beyond me. Do you happen to know if that is an idea worth trying? I am willing to learn anything new, I am just out of ideas on my end.

I will be taking differential equations starting next month and I know the course has something to do with finding unknown functions. Since you have probably already taken such a course, do you happen to know if there is anything similar to this in said class?

Thanks a lot for getting back to me as quickly as you did.