Composing functions

Algebra Level 5

Consider all pairs of polynomials \(f(x)\) and \(g(x)\) such that \[\begin{cases} f(g(x)) =f(x)+g(x), \\ f(0) = 5, \\ f(1) = 7, \\ g(0) \neq 0.\\ \end{cases} \] What are the last three digits of the sum of all (distinct) possible values of \(f(20)?\)

Details and assumptions

The empty sum (sum of no elements) is 0.

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