free_to_fly,

Plug in x for z and multiply (numerator and denominator) by the conjugate of the denominator, which is x + i). Then separate into real and imaginary parts, and take the modulus of the expression. It turns out the modulus of w is 1, (i.e. ||w|| = 1). Thus the image (in the w plane) of the real axis under the mapping defined by w = (z+i)/(z-i) is the unit circle.

Best,

m_s_d