if f(x) + f(x) = f(f(x)), What equation could f(x) be besides 0?

Answer 1

f(x) + f(x) = f(f(x))

2f(x) = f(f(x))

I'm going to swap it around so that

f(f(x)) = 2f(x)

This means that putting a function inside of itself is equivalent to doubling its value.

If f(x) = 2x, then

f(x) + f(x) = 2x + 2x = 4x

f(f(x)) = 2(2x) = 4x

f(x) = 2x would work. Unfortunately, I'm not sure that this is the most consistent/empirical way to solve it. Hopefully someone else demonstrates a more reliable process.

Answer 2

David is correct. You can show it by taking derivatives of both sides (for example), apply the chain rule, and integrate them back.

However, it sounds like a Computer Science problem with a variation of Idempotent functions.