mapping complex functions

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From my text book I have this question:

Let f(z)=$\frac{2 z}{z+2}$. Draw two graphs, one in the z-plane and the other showing the image in the w-plane. (You should only need to calculate the images o 0, -2, 2, infinity and -1-i).

c) the line x=y plus infinity e)the circle with radius 1 centered at 1.

What I have done for c) is to parametrize the line and put it in the function, with the given point I got a rectangle in the w-plane. I dont know if this is correct, there is no solution to the problem (at least I cant find any).

For e) I have written the circle as z=1+$e^{i \theta}$ and used $e^{-i \theta/2}$ to simplify, and the result I got was f(z)= w = $\frac{16 cos(\theta/2)+ 4 sin(\theta/2) i}{12 cos^2(\theta/2)+1}$ after I have rewritten it. From here I dont know what to do. I think I have written f(z) in a bad way.

Thank you for answers.