Is this transformation problem doable?

[mimsy57]

We've been talking about Mobius transformations and I feel like I get it, but we just got a homework problem that isn't making sense to me.

It says: find the Mobius transformation taking the circle |z|=1 to |z+2|=1 such that T(-1)=-3 and T(i)=-1.

My problem with this is that all Mobius trans. are the composition of translations, rotations, dilations and inversions. But this SEEMS like it is doing something else because, unless I am screwing up the graphing, it is taking a quarter circle to a half circle.

Both the circle and its image have radius 1, with the first centered at the origin and the second at -2. The additional points we are given mappings for are -1 and i, which are the endpoints to an arc of a quarter circle. These are mapping to -3 and -1 which are the bounds for the half circle. This implies the mapping is going around twice for once around the circle being mapped, which would imply it is not a bijection, and Mobius transformations are bijections.

It would be great if someone could point out where I am doing something wrong in my reasoning! I don't think I need any help with the calculation part once I figure this out.

 

[lippyka]

It looks like you're misunderstanding the problem. The Mobius transformation is not taking a quarter circle to a half circle; it's taking the entire circle |z|=1 to the entire circle |z+2|=1. The points -1 and i are just given as specific points to check that the transformation is correct. In other words, if you compute the Mobius transformation, you should be able to check that T(-1)=-3 and T(i)=-1.