Can We Determine a Function from Known X and Y Values?

[albertrichardf]

Hi all,
Suppose y is related to x by some function, or:

y = ƒ(x).

Now, supposing that we know some values for y and the corresponding values for x, would it be possible to find what the function actually is?
And if so, how would it be done, without any computer programs. (i.e, what process should be followed to do it)

Thanks for any answers

 

[Matterwave]

In general, any finite number of data points is not enough to specify the function. After all, the function, without any restriction, could simply be defined like $f(x_1)=y_1, f(x_2)=y_2,..., f(x_n)=y_n,\quad f(x_i)=0\forall i\notin 1,...,n$

For the n inputs you gave, this function gives you your n expected outputs, but is 0 everywhere else.

However, if you know some additional information about f(x), then some progress can be made. For example, if you know f(x) is linear, then any 2 (unique) points is enough to specify f(x) (after all 2 points define a line). If you know f(x) is quadratic, then any 3 points is enough, etc. For any polynomial equation of degree n, you need, in general, n+1 input and output pairs to obtain the unique polynomial satisfying those n+1 constraints. But the polynomial equations form a countably infinite subset of the uncountably infinite set of all functions. So almost all functions are not polynomial functions.

Thankfully, however, polynomial functions are still pretty interesting and of a lot of use in many areas of math and physics. Also, at anyone point, a large class of functions (the smooth functions) can be expressed as a polynomial expansion (called a Taylor expansion).