I'm trying to construct a quotient map $f$ from the $n$-sphere $S^n$ to the $n$-disk $D^n$, in which $f(x_1,x_2,\ldots,x_n,x_{n+1})=f(x_1,x_2,\ldots,x_n,-x_{n+1})$. The goal is to show that $S^n/\sim_f$ is homeomorphic to $D^n$. (Is this the right way to go about it?)
I've been looking at the stereographic projection for inspiration, but haven't come up with anything. I only just started learning about topology. Any hints?