Given the function $f:\mathbb{C}\setminus\{-i\}\rightarrow \mathbb{C}\setminus \{1\}$, defined by $f(z)=\frac{z-i}{z+i}$.
I'm supposed to find the image for $f(\{z\mid\Im (z) > 0\})$. However I'm fairly uncertain on how to do this.
I know that $z = \frac{-(a+1)i}{a-1}$ for some $a\in \mathbb{C}\setminus \{1\}$, and i've tried writing z on the form $n+im$ for $n,m\in\mathbb{R}$, but that doesn't help me.
Any suggestions on how to solve it?
Hint: With $z=x+iy$ simplify $$u+iv=\dfrac{z-i}{z+i}$$ where $w=u+iv$ is in the range of map. Then find $$1-u^2-v^2=((1-u)^2+v^2)y>0$$ which gives the result.