I'm trying to show that stereographic projection from the north and south pole determine a smooth atlas, and I just showed that the transition map $$ \varphi_N\circ\varphi_S^{-1}(x_1, \ldots, x_n) = \left(\frac{x_i}{x_1^2 + \ldots + x_n^2}\right)_{i=1}^n $$ is smooth.
However, I can't find the inverse map. If the coordinates are written in the above form (ie $x_i$ over the sum of squares) then it's straightforward, but what if I'm just given $(x_1, \ldots, x_n)\in\mathbb{R}^n$.