Semigroup-valued solutions of some composite equations

Let X be a linear space over the field K of real or complex numbers and (S, °) be a semigroup. We determine all solutions of the functional equation $$f(x+g(x)y)=f(x)circ f(y)quad text{for}quad x,yin X$$ in the class of pairs of functions (f,g) such that f : X → S and g : X → K satisfies some regularity assumptions. Several consequences of this result are presented.