Find the images of half plane X>0 and y<1 under the linear fractional transformation

Question

My attempt on this question Please help how to proceed further

Answer 1

You might as well check where the boundary lines $x=0$ and $y=1$ go. They will go to either circles or lines...

So, for the half plane $x\ge0$, let's use $0,i$ and $\infty $. So $T (0)=\frac {0+i}{0-i}=-1, T (i)=\frac {i+i}{i-i}=\infty $ and $T (\infty )=1$. So, evidently the line $x=0$ gets sent to the line through $1,-1$ and $\infty $, that is, the x-axis . Use a test point, say $1+i$, to see if the half plane goes above or below . .. $T (1+i)=\frac {1+i+i}{1+i-i} =1+2i$, which is above the x-axis . ..

Do the same sort of thing with $y\le1$... So, let's use $i, 1+i$ and $\infty $... $T (i)=\infty, T (1+i)=1+2i $ and $T (\infty )=1$. Evidently we have a vertical line (the points are colinear)...
Now to check which side use a test point. How about $0$? We already did it... $T (0)=-1$. So to the left of the line...