Map onto the unit circle $|w|<1$ the interior of the circle $|z|<2$ with the circle $|z+1|\leq 1$ thrown out with a cut on the real axis $\left \{ y=0; 0\leq x\leq \frac{1}{2} \right \}$
My attempt: I applied $z \rightarrow z+1$ and moved the circle $|z+1|\leq 1$ to $|z|\leq 1$, while the circle $|z|<2$ will move to $|z+1|<2$. Then apply an inversion which maps the interior of the unit circle to the exterior and vice versa. Let $z \rightarrow \frac{1}{z}$, then circle $|z||\leq 1$ will change to $|z|\geq 1$, and circle $|z+1|<2$ will change to $|z-\frac{1}{2}|<\frac{1}{2}$. And used a stereographic projection which maps the upper half-plane into a unit circle $$z \rightarrow \frac{z-i}{z+i}$$
The half-plane $|z|\geq 1$ transforms to a circle $|w|<1$ and $|z-\frac{1}{2}|<\frac{1}{2}$ becomes $|w|<\frac{1}{2}$. Hence we can write this mapping as a composition of three transformations $$w=\frac{\frac{1}{z+1}-i}{\frac{1}{z+1}+i}$$
Have I found this mapping correctly?