Given a map $f: \mathbb{P_C^3 \rightarrow P_C^5}$ defined by $(x_0:x_1:x_2:x_3) \mapsto (x_0x_1:x_0x_2:x_0x_3:x_1x_2:x_1x_3:x_2x_3)$ how does one check if it induces a morphism when restricted to a hypersurface $X \subset \mathbb{P_C^3}$ given by $2x^2_0+3x_1^2+x_2^2+x_3^2=0$, or in other words is $f_{|X}:X \rightarrow \mathbb{P^5}$ a morphism? Moreover is $f_{|X}:X \rightarrow f(X)$ an isomorphism?
As far as I know it only needs to be checked if it is defined on the entire $X$ to be a morphism since it is a rational map. But then it is only not defined at points $(1:0:0:0)$, $(0:1:0:0)$, $(0:0:1:0)$ and $(0:0:0:1)$, and they certainly are not on $X$. Am I missing something?
As for the isomorphism part, I don't really know how to show that. Any help? Thank you