Solve the given Functions for Complex numbers.(Complex Functions)

Question

I tried solving for the real solutions and found myself stuck. Here is how i tried to solve it: Re(z)=1

f2(M2)= f2(1+iy) {z=1+iy} i'm not so sure about this.

f2(m2)= (1+iy)^2 {as z^2 is f2}

further solving i get y^2=4x-4 and can't move further.

Answer 1

For $f_2$, the image of $M_2$ is $$\{z^2\mid Re(z)=1\}=\{(x+iy)^2\mid x=1\}\ .$$ If you write $z^2=u+iv$ then doing the algebra gives $$u=1-y^2\ ,\quad v=2y\ .$$ You can recognise this as the parametric description of a ceratin curve, or you can eliminate the parameter to get $$4u=4-v^2\ .$$ This is the equation in the $(u,v)$ plane of a familiar curve.

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You can do $f_1$ in the same way, in fact it will be a bit easier.