how to find this?

Question

Let f(z)=(i+z)/(i-z) and c=circle through z1=(1+i), z2=0, z3=i. Find f(c).

note: z belong to complex and i= sqrt(-1).

Answer 1

I got the line y=x-1 (or in complex notation z = x +(x-1)i ). This is what I did:

f(z) = -1 + 2i/(i - z).

So f is the composition of the following functions:

1) f1(z) = -z

2) f2(z) = i + z

3) f3(z) = 1/z

4) f4(z) = 2iz

5) f5(z) = -1 + z

Then just follow the circle along with each fi from f1 to f5. Now f1 and f2 just move the circle around. The important change is f3. It takes circles through the origin (which this is) to lines. Knowing this, just get two points on the line and draw the line. Also multiplying by i rotates the figure by 90 degrees. The rest is easy.

Answer 2

I'm not sure how one finds the function of a geometric figure, which is what this question is asking... find f(circle). This is one of those questions you ask to a robot to make it stutter and have smoke come out its head.