Quasi graph-additive functions with a prescribed branch

Here, we investigate the solutions to equation $$f(f(-x)+x)=f(-f(x))+f(x),\qquad x\in\mathbb{R}$$ that are prescribed on the non-positive half-line. We will refer to this prescribed function as the generator of the corresponding solution. We show that any function taking negative values on the negative half-line and being strictly greater than the identity can be extended to a solution. Nevertheless, the solutions generated by continuous, strictly monotone functions can be well characterized. As our main result, we establish a closed-form expression for these functions.