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why the real part is positive

My work

I looked into

[tex] g(z) = z^2 [/tex] , natural domain is the complex field

we can see that

[tex]g(z) = g(-z) [/tex] , g is not 1-1

if [tex] z = r e^{i\theta}[/tex]

[tex] -z = e^{i\pi} z = re^{i(\theta + \pi)} [/tex]

so we will restrict the domain to get one-one function so we will have the inverse f

how to restrict it, or how to solve it in another way

Thanks