$$i(x)\cdot f(y)+1=1(x)+f(y)+xy$$
$\left\{\begin{matrix}f=-\frac{1-x-xy}{y\left(ix-1\right)}\text{, }&x\neq -i\text{ and }y\neq 0\\f\in \mathrm{C}\text{, }&\left(y=0\text{ and }x=1\right)\text{ or }\left(y=-1+i\text{ and }x=-i\right)\end{matrix}\right.$
$\left\{\begin{matrix}x=-\frac{1-fy}{ify-y-1}\text{, }&y=0\text{ or }f\neq -i+\frac{-i}{y}\\x\in \mathrm{C}\text{, }&y=-1+i\text{ and }f=-\frac{1}{2}-\frac{1}{2}i\end{matrix}\right.$
