How Does the Function f(z) = (z + 1) / (z - 1) Map Complex Regions?

[jaejoon89]

For z not equal to 1
f(z) = (z + 1) / (z - 1)

How do you show the function maps {z ϵ C : Re(z) < 0} into {w ϵ C : |w| < 1}
and
{w ϵ C : |w| < 1} into {z ϵ C : Re(z) < 0}?

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I don't even know how to start this one besides that "into" means 1-1.

How do you show the mappings?

 

[HallsofIvy]

Well, the one thing you say you know is wrong! "Onto" means 1-1. "Into" does not have to be. If the real part of z is negative, we can write z as -x+ iy where x> 0. z+1= 1- x+ iy= (1-x)+ iy and z-1= -x+ iy- 1= (x-1)+ iy.

Now,
$$\frac{z+ 1}{z- 1}= \frac{(1-x)+iy}{-(x+1)+ iy}$$
Multiply both numerator and denominator by -(x-1)- iy. What does that give you?