Mapping question I don't understand

[Genericcoder]

Hi I uploaded the question and its answer first of all I am not good at all in mapping I don't understand it that much like here we want the output of the function under that specific domain that's what we want to see.
I understand until PI < alpha < 5/2 * PI but I don't understand how did he get
f(D) = {|z| < 2 , -PI < Arg(z) < PI/2} I don't get that could someone please explain because I need to understand this.

 

[LCKurtz]

If you take the range $\pi < \alpha <\frac 5 2 \pi$, you can subtract $2\pi$ from the $\pi$ and $\frac 5 2 \pi$ and it is the same region in the plane. So $-\pi < \alpha < \frac \pi 2$ describes the same region.

 

[Genericcoder]

Oh I see that makes a lot of sense thank you !

 

[Genericcoder]

could you please also like explain maybe more about this question about how did he arrive to this solution coz I am not sure that my understanding is correct or intuition is right..

 

[Mark44]

Which part don't you understand?

 

[Genericcoder]

So I was wondering how do we systematically solve stuff like this ? are we looking for how some function output under certain set of conditions ? like for example f(x) = x^2 and we restrict the domain to be something like
[0,2] and see what is the range of output we are trying to get ?? I don't really understand what's going on here I just know how to solve in some scenrios but I don't get are we actually doing ...

 

[Genericcoder]

Is there systematic way to do these kind of problems?