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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 ?

Milan
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  • it's possible for the linear function but not for other types – Vasili Mar 30 '18 at 19:04
  • but why?( how do you know) – Milan Mar 30 '18 at 19:04
  • this question is way too broadly phrased.

    No generally not, in special cases (polynomials,...) yes

     –  Mar 30 '18 at 19:05
  • because there is exactly one line that goes thru two data points – Vasili Mar 30 '18 at 19:05
  • i know how to find a linear equation, i am asking why cant you do it if you dont know the power of x – Milan Mar 30 '18 at 19:06
  • There is infinite number of ways to connect two points so it's impossible to guess a function without additional info. Interpolation can help you but it's going to be an approximation. – Vasili Mar 30 '18 at 19:08
  • @MilanStojanovic Show us the line and graph about which we are talking.. – Narasimham Mar 30 '18 at 19:43
  • There are a lot of simple functions which have familiar graphs which you can guess but in general given any number of points you can make infinitely many functions. – kingW3 Apr 01 '18 at 12:19

1 Answers1

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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

  • how can i find that theorem – Milan Mar 30 '18 at 19:13
  • as already said I'm not sure! –  Mar 30 '18 at 19:15
  • if someone know it would be very helpful – Milan Mar 30 '18 at 19:19
  • if you know it's a polynomial of degree $n$, you need $n+1$ points do define a unique polynomial that goes thru those points. google method of successive differences or refer to this question: https://math.stackexchange.com/questions/685472/find-n-degree-polynomial-from-n1-points – Vasili Mar 30 '18 at 19:21
  • This isn't the best way to approximate your function. Since the polynomial still may not converge to your function even if it agrees with it on each of those points. Bernstein polynomials are more appropriate in this case, as they uniformly converge to $f$. – Kitegi Mar 30 '18 at 20:48