Homeomorphism and real projective plane

Let S denote the unit sphere in $\displaystyle {R}^3$ given by $\displaystyle s = \{(x,y,z) \in {R}^3 : x^2 + y^2 + z^2 = 1\}$, and let $\displaystyle f:S \rightarrow {R}^4$ be defined by $\displaystyle f(x,y,z) = (x^2 - y^2, xy, yz, zx)$.

Prove that

a) f determines a continuous map $\displaystyle g: P \rightarrow {R}^4$ where P is the real projective plane.

b) g is a homeomorphism onto a topological subspace of $\displaystyle {R}^4$. (I can show g is injective, which requires a lot of work, so you don't have to do this bit).