I am trying to come up with a systematic way for solving an implicit "functional" equation. That is, given
$$g(x) = f(x + a \cdot f(x))$$
how would one (numerically) recover the functional form of $f(x)$? $g(x)$ is known and a is a known constant. I've been thinking of defining something along the lines of
$$h \equiv g(x) - f(x+ a \cdot f(x))$$
and using something like Newton-Raphson's method to find the zeros of $h$ as a function of $f$, but that doesn't quite seem to work, because for that I would need $f(x)$ and not $f(x+a \cdot f(x))$ to appear in my $h$.
Any ideas would be appreciated. Thank you!