Finding a function only with arguments and outputs

[casio]

Hello everybody, this isn't something like a homework to be solved, more of a discussion I stumbled upon...

I think I remember from school you could find a function if you had its argument and outputs and derivatives, something like this: y'(0)=1, y(2)=3... and so on. What I'm interested in how would you (if it's possible) calculate a function with just the basic outputs. Like this would be data I have for example y(4)=3, y(1)=7,y(7)=1... Is it possible to get a function just out of that (lets say it's more complicated than the function of the straight line)? If not, would I be able to approach this problem in statistics or probability maybe? If there are some links about this thing I'd be thankful, so far I found too general stuff about functions. And excuse me if I posted in the wrong forum, first time here.

 

[galactus]

It would depend on what sort of function one were looking for, what is given, what is asked for, etc.

But, yes, it is possible. It is used in DE's. Also, you may want to look into the "Lagrange interpolating Polynomial".

You can find the polynomial that models given data. 'Finite differences' is another thing to research.

But, as an example:

Suppose we wanted a quadratic, \(\displaystyle f(x)=ax^{2}+bx+c\), given y(4)=3, y(1)=7,y(7)=1.

Then, we could build three equations with three unknowns and solve.

\(\displaystyle 16a+4b+c=3\)

\(\displaystyle a+b+c=7\)

\(\displaystyle 49a+7b+c=1\)

\(\displaystyle \boxed{9y=x^{2}-17x+79}\)

 

[casio]

Yes offcourse, but it's kind of a done deal when you know what kind of function you are looking for. I'm kind of researching would it be possible to figure out if it could be any kind of operation, exponential, trigonometric, polynomials and everything. But thanks a lot for the Lagrange, it's just the thing I wanted for simple functions.

 

[mmm4444bot]

Sets of function arguments and the associated outputs may not define a unique function type. In other words, the data may fit more than one type of function.

Short of using dedicated software, fitting data to a function is manual labor.