Given is the projection $$u: \mathbb{C} \rightarrow \mathbb{R}: z \rightarrow Re(|iz|) - |iRe(z)|$$ which is $$u: \mathbb{C} \rightarrow \mathbb{R}: z \rightarrow |z| - |Re(z)|$$ when simplified.
We are suppossed to check the projection for injectivity (pretty easy to disprove) and surjectivity. Now, I'm stuck when it comes to proving whether or not the function is surjective. My approach of trying to prove that the statement $$f(z) = a: z \in \mathbb{C}, a \in \mathbb{R}$$ leads nowhere as $z$ can't be isolated, so I figure that this isn't going to result in anything usefull.
What would be the right way to approach this problem?
Thank you for any help!
It is not surjective to $\mathbb R,$ for if $a,b\in \mathbb R$ then $f(a+ib)=\sqrt {a^2+b^2}\;-\ |a| \geq 0.$ It is surjective to $[0,\infty),$ as the image of the imaginary axis is $\{\sqrt {b^2}: b\in \mathbb R\}=[0,\infty).$