Let's look at $\phi:\ z\mapsto -i\frac{z-1}{z+1}$ and try to verify in an elementary way that it maps the unit disk onto the upper half-plane:
My problem is the following: I can easily get $\phi^{-1}:\ z\mapsto\frac{i-z}{i+z}$. However, this doesn't mean $\phi$ is bijective, right? Getting an inverse function only works if the domain and the range are the same set?
So I will approach this a little more carefully: It is easy to verify by the definition that $\phi$ is injective. Now, to show that $\phi$ is also surjective on the given sets I need to show that for each $w\in\mathbb H^+$ (upper half-plane) a proper preimage $z\in \mathbb E$ can be found. So let's take some $w\in\mathbb H^+$, then the preimage I get by using the inverse function is $\frac{i-w}{i+w}$. And now we need to obtain $$\left|\frac{i-w}{i+w}\right|<1$$ using only $\text{Im}(w)>0$.
Question 1: How do I do that?
Question 2: Do I need ANYTHING else to obtain the upper statement or does completing question 1 include everything?