Show homeomorphism from upper (and lower) hemisphere to n-sphere. Hello, I am stuck on an exercise from my Algebraic Topology book and I am, unfortunately, unable to solve it (at least the (b) part, (a) were quite straight forward for me). Here is the problem:

Consider the map $f:\mathbb{R}^{n+1}\to\mathbb{R}^{n+1}$ defined by
$$f(x_0,..,x_n)=(2x_0x_n,2x_1x_n,..,2x_{n-1}x_n,2x_n^2-1).$$
(a) Verify that $f$ restricts to a map from $S^n$ to $S^n$
(b) Show that the restrictions of $f$ to the upper and lower hemispheres,
$$D_+^n=\{x\in S^n|x_n\geq 0\},\quad D_-^n=\{x\in S^n|x_n\leq 0\},$$
give rise to homeomorphisms
$$D_+^n/\partial D_+^n\to S^n,\quad D_-^n/\partial D_-^n\to S^n$$

Solution:
(a) Let $(x_0,..,x_n)\in\mathbb{R}^{n+1}$ such that $\sum_{i=0}^n x_i^2=1$. then $(2x_0x_n)^2+\cdots + (2x_{n-1}x_n)^2 + (2x_n^2-1)^2=4x_0^2x_n^2+\cdots + 4x_{n-1}^2x_n^2+4x_n^4-4x_n^2+1=4x_n^2\sum_{i=0}^n x_i^2-4x_n^2+1=4x_n^2-4x_n^2+1=1$. $\text{}$
This shows that $f$ restricts to a map from $S^n$ to $S^n$.
(b) I think this should be easy too. But I think my biggest issue is that I don't really know what an element of $D_+^n/\partial D_+^n$ and $D_-^n/\partial D_-^n$ looks like. If someone could help me out with one of the cases perhaps I could work out the second one by myself. Also, I don't really understand why (a) was a part of the same problem. Should I use it to solve (b)?
Any help would be really appreciated. :)
 A: There is a theorem of topology that might be useful, namely if $f : X \to Y$ is a continuous surjective map from a compact space to a Hausdorff space then $f$ is a quotient map. As a corollary, if we take the "point pre-image decomposition" on $X$, namely the decomposition defined by the equivalence relation $\sim$ where $x_1 \sim x_2 \iff f(x_1)=f(x_2)$, then the quotient space $X/\!\!\sim$ is homeomorphic to $Y$. In fact, the map $X/\!\!\sim \, \to Y$ which $f$ induces is a homeomorphism. You can find this material, for example, in Munkres "Topology".
Using part (a), you have a formula from which continuity of $f$ is evident, and furthermore continuity of the restriction of $f$ to any subset of its domain is also evident. You can therefore do part (b) by applying part (a) and the theorem I have quoted. All you have to do is check the hypotheses, namely: $f | D^n_+$ surjects to $S^n$; and the point pre-image decomposition of $f | D^n_+ : D^n_+ \to S^n$ consists of one decomposition element $\partial D^n_+$ plus a single point decomposition element for each individual point of $D^n_+ - \partial D^n_+$.
