Can a function be found to fit any set of points? Could a function be found for any set of points, assuming those points didn't contradict the definition of function?
I mean, given a bunch of (x, y) pairs, could a function be found where when you input the x given in each pair, the output is the y?
I've run into many questions like this in highschool mathematics. Questions like "Find a function who's graph goes through the points (-5,10) and (7,-10)". Now I'm wondering if a function could be found no matter which points, and no matter how many points you're given.
Also, is there any method for finding these functions? Any applications of this? Any more information on the idea? I'm just looking for information about this thing I've been wondering about.
 A: Yes, there are many methods to find a function that has particular values at particular points. In general, this is part of a field called 'numerical analysis' and this question is called 'interpolation.' If you have any finite set of points, you can even interpolate the points with a polynomial. The standard way here is to use a Lagrange polynomial interpolation or Newton's Divided Differences (which amounts to the same thing). The link is good, but let me give you the idea - one comes up with a very clever polynomial for each point that is zero at all the points but that point, so that when you add them all together you get a (actually the - it's unique) minimal polynomial that hits them all.
Polynomial interpolation is a big topic, and there are lots of resources on the problem. Much of it is automated. I often link to Numerical Recipes, a freely available numerical analysis text aimed towards the end goal of automating the procedure of interpolation and other numerical problems. I think it's a good place to look - once you find something that you're interested in, google it and find accessible material.
Now, you ask about applications and utility. I would say that interpolation is one of the most useful things out there - it's like the idea of best-fit lines, sort of. It lets you predict and analyze patters of relationships, and it the approximation is smooth enough it even lets you use the techniques of calculus (which is a very powerful tool set). Very useful.
