For that, you can use Reflex4You.com. Normally it is a ray-tracing engine, but you can make it work to find the image of your circle. The identity function looks like this:

A function describing the unit circle with colors on a black background is the following:
if((abs(z)>0.9)*(abs(z)<1.1),z,0)

The function you are invoking has the following representation:

To see what the circle becomes through the function, you can use the following code:
set k = 0; let n = 100; let f = 1/(1-z); set result = 0; let threshold = 0.1; repeat n in set theta = exp(i*2*pi*k/n); set result = if(abs(f(theta)-z) < threshold, theta, result); set k = k + 1; result

Edit it yourself: Click on the image above.
If you know the inverse of the function (if it exist) then you can print it faster. In our case, $f^{(-1)}=1-\frac{1}{z}$, so that you can write the formula:
let w = 1-1/z; let c= if((abs(z)>0.9)*(abs(z)<1.1),z,0); o(c,w)

DISCLAIMER: I am one of the authors of this website which can also serves the purpose of customized object store.