What Is the Image of the Upper Half-Disk Under This Mobius Transformation?

[nhrock3]

\(\displaystyle D^+=\left \{z:|z|<1,Im \left \{ z \right \}>0 \right \}\)
so it represents the northen hemisphere of a circle with radius 1.
\(\displaystyle f(z)=\frac{2z-i}{2-iz}\)
i need to find what is the picture of \(\displaystyle f(d^+)=? \)

i tried to solve it like this:
my area is bounded by a line and a curve.
i want to see what each one transforms to
curve points:
f(1)=1
f(-1)=-1
f(i)=i/3
so it will look like this
http://i46.tinypic.com/3129rg4.gif

line points:
f(1)=1
f(-1)=-1
f(0)=-i/2
so it looks like this
http://i49.tinypic.com/dnnh2u.jpg

and when i try and see where the inside goes :
f(i/2)=0
so the answer represents their intersection area

http://i49.tinypic.com/14bi2qr.gif

but my prof thinks otherwise
http://i48.tinypic.com/2jff52r.jpg

who is worng here? why?
so its in their intersections

 

[lippyka]

is the set of f(D^+), since the line and the curve intersect at the point (0, -1/2). The set of f(D^+) is the area between the line and the curve, as shown in your professor's diagram.