Case 1a: The points on |z-C|=R can be written as z=C+R⁢ei⁢θ. They are mapped to the points w=a⁢C+b+a⁢R⁢ei⁢θ which all lie on the circle |w-(a⁢C+b)|=|a|⁢R.
Case 1b: The line Re⁢(ei⁢θ⁢z)=k are mapped to the line Re⁢(ei⁢θ⁢wa)=k+Re⁢(ba).
Case 2a: Consider a circle passing through the origin. This can be written as |z-C|=|C|. This circle is mapped to the line Re⁢(C⁢w)=12 which does not pass through the origin. To show this, write z=C+|C|⁢ei⁢θ. w=1z=1C+|C|⁢ei⁢θ.
Case 2b: Consider the line which does not pass through the origin. This can be written as Re⁢(a⁢z)=1 for a≠0. Then a⁢z+a⁢z¯=2 which is mapped to aw+a¯w¯=2. This is simplified as a⁢w¯+a¯⁢w=2⁢w⁢w¯ which becomes (w-a/2)⁢(w¯-a¯/2)=a⁢a¯/4 or |w-a2|=|a|2 which is a circle passing through the origin.
Case 2d: Consider a line passing through the origin. This can be written as Re⁢(ei⁢θ⁢z)=0. This is mapped to the set Re⁢(ei⁢θw)=0, which can be rewritten as Re⁢(ei⁢θ⁢w¯)=0 or Re⁢(w⁢e-i⁢θ)=0 which is another line passing through the origin.
Then f=f3∘f2∘f1. By Case 1, f1 and f3 map circles to circles and by Case 2, f2 maps circles to circles.