# Math Help - Image of a function

1. ## Image of a function

The question:

Find the image of the following function:

f(z)= z/(1-z)^2

I solved this but I am not 100% sure.

My solution

if w=z/(1-z)^2 then for z!=1 then wz^2-(2w+1)z + w=0 and therefore

z=(2w+1 + sqrt(4w+1))/2w or z=(2w+1 - sqrt(4w+1))/2w for every w!=0
and for w=0 we can pick z=0.

So for every w in C there exists a z in C so w=f(z) and therfore the image of f(z) is the entire complex plane C.

I can't seem to contradict this, but still it dosent seem right to me...

The question you posted isn't even very precisely formulated anyways. What is the domain of the function? The Riemann sphere? The complex plane? Do you include the singularity $z=1$ in the domain? (If so, you have to include the point $w=\infty$ in the image.)