Q = ( (x - 1)^2 + y^2 == 1);
 S = ImplicitRegion[Q, {x, y}];
 a = Region[S, Frame -> True]

f = Evaluate[{x^2 - y^2, 2 x y}] &;
 F = TransformedRegion[F, f];
 b = Region[F, BaseStyle -> Red, Frame -> True]