Immersion of $(RP)^2$ in $R^3$

Question

I am working on the following problem.. Let g: $S^2$-->$R^3$ be a map given by the formula g(x,y,z)=(yz,xz,xy), which induces in a natural way a mapping $G$ from $RP^2$ into $R^3$. I need to find six points of the real projective plane $p_i$ i=1,...,6 such that G is an immersion for all the other points in $RP^2$. However my calculations show that we do not need to deduct those points for $G$ to be an immersion. I am propably missing something here, so any help would be much appreciated!