1 should be the answer.

I would use plug any play. Or guess and check. Should be able to do with fractions.

F(1 + 3) = 1 to the power of 2 + 2*1

4=4

I have heard of functional analysis. In I think 10th grade. It’s one of the fundamental ways to do algebra, so I suspect you just didn’t hear/remember that term used. I think some teachers skip that term for it. I still use it in my work as a game developer, and occasionally for other things.

@RedDeerGuy1 , You seem to be approaching the solution. On the left side of your first line you have f(4). On the right side you have values in terms of 1: 1 to the power of 2 and 2 times 1. Similarly, we have f(7) = f(4+3) = 4 to the power of 2 + 2*4.

@Zaku, Here is a link to a definition of functional analysis along with some examples that are not like anything I have seen before.

Here is the solution to the problem. We can replace x with x-3 on both sides of the equation to get f((x-3)3) = (x-3)^2 ] 2(x-3).
f((x-3)+3) is just f(x), so we have:
f(x) = (x-3)^{2 + 2(x-3) = x}2 -4x + 3

@LostInParadise That’s a definition of a functional equation – i.e. an equation in which some of the unknown terms are themselves functions. Your question asks about functional analysis. The example in your question details, was simply:

Find an expression for f(x) such that f(x+3) = x^2 + 2x.

Which reduces to:

f(x) = (x-3)^2 + 2(x-3)

So I thought you were referring to analyzing functions in algebra, i.e. “solving systems of equations” which are often expressed as functions f(x) to solve for.

The term Functional Analysis (in mathematics) seems to me to refer to something a bit different from either of those things, e.g.: https://en.wikipedia.org/wiki/Functional_analysis

i thought functional equations is part of functional analysis. I could be mistaken. This is all new material to me. I have not found any reference that talks about the two fields together, but I asked ChatGPT what branch of mathematics includes functional equations and it said functional analysis.