This is exercise $4.4$ part (c) of Hartshorne's book. Let $Y$ be the nodal cubic curve $y^{2}z=x^{2}(x+z)$ in $\mathbb{P}^{2}$. Show that the projection $f$ from the point $(0,0,1)$ to the line $z=0$ induces a birational map from $Y$ to $\mathbb{P}^{1}$.
Attempt:
Consider the open subset of $Y$ given by $Y \setminus V(z)$ , that is we set $z=1$.
Define $f: Y \setminus V(z) \rightarrow \mathbb{P}^{1} \setminus \{[1:1],[1:-1]$ by:
$f([x : y : z]) = [x: y]$
Now define $g: \mathbb{P}^{1} \setminus \{[1 : 1],[1 : -1]\} \rightarrow Y \setminus V(z)$ by:
$g([x : y]) = [(y^{2}-x^{2})x : (y^{2}-x^{2})y : x^{3}]$
Question: what if $x=0$? then we get $y=0$ so $[x :y : 1] = [ 0 : 0 : 1]$ but the point $[x^{4} : x^{3}y : x^{3}]$ is not defined at $x=y=0$.