Pexider equation

Remarks

Example: $f(x) = 5x + 3$, $g(x) = 5x + 1$, $h(x) = 5x + 2$ is a solution, since both sides equal $5x + 5y + 3$. Non-example: $f(x) = x^2$, $g(x) = x^2$, $h(x) = x^2$ is not a solution, since $f(1 + 1) = 4$ while $g(1) + h(1) = 2$. Setting $y = 0$ and then $x = 0$ gives $g(x) = f(x) - h(0)$ and $h(y) = f(y) - g(0)$. Then $A(x) = f(x) - g(0) - h(0)$ is an additive function, which gives the general solution $f = A + a + b$, $g = A + a$, $h = A + b$ with $a = g(0)$ and $b = h(0)$. The same method solves the equation for maps from a group to an abelian group.