Solving for the functional shifts and its inverse

Solution 1:

$\renewcommand{\phi}{\varphi}$Write

• $\phi(z) = z + f(z)$, so that $f(z) = \phi(z) - z$, and
• $\gamma(z) = z - g(z)$, so that $g(z) = z - \gamma(z)$.

Then $f(x) = g(x+f(x))$ becomes $$ \phi(x) - x = g(\phi(x)) = \phi(x) - \gamma(\phi(x)), \quad \text{that is,} \quad \gamma(\phi(x)) = x. $$ Similarly $g(x) = f(x - g(x))$ becomes $$ x - \gamma(x) = f(\gamma(x)) = \phi(\gamma(x)) - \gamma(x), \quad \text{that is,} \quad \phi(\gamma(x)) = x. $$ So it seems $\phi$ and $\gamma$ have to be functional inverses of each other. And conversely, once you have such a pair $\phi, \gamma$ of functional inverses, $f(x) = \phi(x) - x$ and $g(x) = x - \gamma(x)$ will satisfy your functional equations.