$Iff:C\to Csuchthatf\left(z\right)=z-\overline{z}\forall z\in C,thenfis$

$\begin{array}{l}Iff:C\to Candf\left(z\right)=z-\overline{z}\\ Let{z}_1=x_1+iy\Rightarrow \overline{z_1}=x_1-iy\\ f\left(z_1\right)=x_1+iy-x_1+iy=2iy\\ and{z}_2=x_2+iy\Rightarrow \overline{z_2}=x_2-iy\\ f\left(z_2\right)=x_2+iy-x_2+iy=2iy\\ \therefore Differente\lim entshavesameimage\\ \therefore fisnotone-one\\ f\left(z\right)=z-\overline{z}isimaginary\forall xandy\\ \therefore Range\ne Codomain\end{array}$