Find a conformal mapping from the quarter-disk $Q=\{ |z|<1 : rez>0,im z>0 \}$ onto the upper half plane set $U=\{im z>0\}$

Find a conformal mapping from the quarter-disk $Q=\{ |z|<1 : rez>0,im z>0 \}$ onto the upper half plane set $U=\{im z>0\}$

I'm guided through this problem:

First I need to find the image of the quarter disk $Q=\{ |z|<1 : rez>0,im z>0 \}$ by $f(z)=z^2$?

My answer to this is a semi-circle (edit:disk), with unit radius.

Now I need to find the map of the halfdisk $D=\{ |z-1|<1 :im z>0 \}$ onto the the first quadrant $\{im z>0, rez>0\}$.

I'm stuck on this bit:

I tried: rewriting the halfdisk as $1+e^{it}$ for $0\leq t \leq 2\pi$ , but I didn't get far.

Lastly, I need to somehow put the first two together, to get a map from the quarter disk onto the upper halfplane

Consider the map $f_1=z^2$, then the map $f_2=-{1\over 2}(z+z^{-1})$.