Can you get the function equations just based on the graph ( without knowing whether it is linear,quadratic...)

Question

For example, I tell a kid draw a line(it doesn't have to be straight, anything you like) in the coordinate system. Am i able to find the equation for f(x)? Also, if i have a program to which i give an input ( a number) and it gives me an output.Will i be able to find f(x) and how much data(how many input and output pairs) will i need ?

Answer 1

If you have that program which gives you 3 outputs to 3 input points, you can construct a polynomial of second degree matching this program. But there are infinitely many polynomials agreeing with that constructed polynomial on these 3 points whilst not agreeing on other ones.

Since you can do this for every number $n$ you will at least have to have countably many input-output-pairs.

I think there is a theorem telling you that if you have uncountably many input-output-pairs then there is just one polynomial which matches these pairs, but I'm not sure about that