Show that the image of the open disk |z + 1 + i | < 1 under the transformation w = (3 - 4i)z + 6 + 2i is the open disk |w + 1 - 3i| < 5.
The method done by the book is as follows, through an inverse transformation:
z = (w - 6 - 2i)/(3 - 4i)
This is substituted into the first given inequality.
Simplifying down to |w - 6 - 2i + (1 + i)(3 - 4i)| < 5, which now goes to |w + 1 - 3i| < 5.
My method was to use the transformation w = .... directly and substitute that into |w + 1 - 3i| < 5
Doing it this way and simplifying it, we're left with the initial inequality |z + 1 + i| < 1, which means it's correct, but is the method right?
I do not fully understand this inverse transformation idea. If we have something (a point/a plane/whatever) in the z-plane, and want to show the corresponding image of it in the w-plane, why are we using an inverse transformation z = ... ?
I cannot get my head around this. Shouldn't we just the actual mapping function w = ... to prove that the image is the correct one?
Unfortunately not, seeing that you did not mention the magic word.