Finding a function given a set of points, without knowing what type of function

Question

If I have a set of points, but am not sure what type of function they could fit, is there a way to find that function? Here are a few of the points...

(0,0)
(3,30)
(4,114)
(5,434)
(6,1655)

I ran it through an exponential regression calculator (though I added 1 to the y values, so it passed through (0,1)) but nothing conclusive came up.

I can tell from the dataset (obviously) it has a positive slope and acceleration, but beyond that, i couldn't find anything to suggest what form the function would take. There are quite a few equations that look a bit like this, including factorials, quadratics, exponentials, etc.

I have a feeling the answer is that there isn't a good way to go about this, but any help is much appreciated!

Answer 1

In betwedn, you can define it arbitrarily. There is no rigidity here.

Answer 2

If you don't have any limitations whatsoever on the type of function you seek, then the possibilities are endless. Even if you require the function to be continuous, there are still infinitely many possibilities that fit your finitely many points.

If we specify the desired values on the points $\{0,3,4,5,6\}$ as in your example, the function could still do all sorts of things on the intervals in between these points. It could shoot all the way up to $10^{100}$ at 2, for example. If you don't want it to do this (or something every crazier) then you need to impose some condition that says it shouldn't.

You may want to fit a particular kind of function to your points, such as a power function, polynomial function, or exponential function. Which one of these is best for you depends on all sorts of things such as whether you want an exact fit, whether you want the function values to be easy to compute, whether you want the function to have a particular form dictated by some physical theory and you are only allowed to tweak some parameter, etc.

Answer 3

If you remove the first point (0,0) the four other perfectly align on a logarithmic scale (Log[y] = a + b * x).