if |z|=3 is a circle, how do you know if the region outside the circle is
That's not at all the question you originally asked! The set |z|< 3 is the inside of the circle given by |z|= 3. But the function maps the circle |z|= 3, with center at 0 and radius 3, into the circle with center at and radius . It also clearly maps 0 to 0. 0 is inside |z|< 3 but outside the circle with center and radius . Therefore, the interior of the first circle is mapped to the exterior of the second circle.