Necessary criterion for expressing f(a + b)

[sultan]

Homework Statement

You are given a function f(x). Determine the necessary criterion for expressing f(a + b) as a finite combination of a, b, f(a), f(b), and possibly a finite set of known mathematical constants, for any a and b from the domain of definition.

For example, if f(x) = x squared, then f(a + b) = (a + b) squared = a^2+ b^2 + 2ab = f(a) + f(b) + 2ab.
If f(x) = e^x, then f(a + b) = f(a) * f(b). For f(x) = 1/x: f(a+b)=1/ab * f(f(a)+f(b))

Clarification: by "combination" i mean any composition of the form G_f(a, b). For example, sin[(f(a/2)^2 + f(b)^2 + 15].

Relevant Equations

f(a+b)=G(f, a, b)

It's easy to see that any polynomial is function of that class. Also it seems that composition of exponent and polynomial is good as well. G_f(a, b) should be equal G_f(b, a) as the f(a+b) is symmetrical.
Does anyone have information about this or at least related to it?

 

[fresh_42]

Given any function $f$ is way too general to draw any conclusions. In case $f$ is differentiable, we have things like Rolle's theorem or the Mean Value Theorem.

 

[realJohn]

The key is understanding what kind of functions let you write f(a + b) in terms of f(a), f(b) and constants.

 

[WWGD]

I assume your expression of f(a+b) is , unless f is extremely simple, intended to be a good approximation ( Edit: In the Limit), to f(a+b) , rather than equal to it?