The question

For the mapping $\displaystyle f(z) = \frac{z - i}{z + i}$, find the image of $\displaystyle \{z \eo C : im(z) > 0\} $

My tutor did this a tricky way, by noticing that:

$\displaystyle |f(z)| = |\frac{z - i}{z + i}| < 1$

Therefore, $\displaystyle |f(z)| < 1$ (which is just the open unit circle).

But how do I attempt this if I don't notice such nifty tricks? I keep making a mess of it.

Thanks.