so I was attempting a question on Möbius transformations and I've encountered a problem in my workings. The question is "Determine the image of the strip $-1 < \Re(z) < 1$ under $g(z) = (iz+1)/(z+1).$ How should I go about solving this?
A huge help when dealing with Möbius transformations:
The image of every circle-or-line is a circle-or-line.
So if you know the images of three points on one boundary line of the strip (say $1$ and $1\pm i$), then you know the circle-or-line that is the image of that boundary line; similarly, if you know you know the images of three points on the other boundary line (say $-1$ and $-1\pm i$), then you know the circle-or-line that is the image of this other boundary line; and if you know the image of a point inside the strip (say $0$), then you know which side of the image circles-or-lines is the interior of the image region.