Find the image of the set $D_1\cap D_2$, where $$D_1 = \{z : |z| < 1\}$$ and $$D_2 = \{z : |z + 1/2| > 1/2\}$$ under the transformation $$f(z) = \frac{z − i}{z + 1}$$
I have done the picture of $D_1\cap D_2$, but I don't know how to do it. If anybody could help me, please. Thanks!
On
Observe that boundaries of $D_1$ and $D_2$ has the common point $-1\in \Bbb{C}$ and $f(-1)=\infty.$ Therefore image of both boundaries under $f$ must be lines and $f(D_1\cap D_2)$ must be the region bounded by those lines. To find those two lines exactly, you can use another two points on boundaries of $D_1$ and $D_2$ respectively.
We have $f^{-1}(z) = \frac{z + i}{1 - z}$.
Let $w \in f(D_1)$. Then $f^{-1}(w) \in D_1$ and by definition we have $|f^{-1}(w)| = \left|\frac{w+i}{1-w}\right| < 1$, that is
$$|w+i| < |w-1|$$
So we can conclude that $f(D_1)$ contains all points $w$ that are closer to $-i$ than they are to $1$, i.e. the "half plane" $\{x+iy \mid y<-x\}$. Now you can use the same method to determine $f(D_2)$.
Then you only need to think about how $f(D_1)$, $f(D_2)$ and $f(D_1 \cap D_2)$ are related.