Prove that the map $y_0 = x_1x_2$, $y_1= x_0x_2$, $y_2 = x_0x_1$ defines a birational map of $\mathbb{P}^2$ to itself. At which points are $f$ and $f^{-1}$ not regular? What are the open sets mapped isomorphically by $f$?
My attempt I tried to invert the equations and express $x_i$ in terms of $y_i$. So I got $y_1/y_0 = x_0/x_1$ and $y_2/y_1 = x_1/x_2$ and it looks like $f$ is always regular, while $f^{-1}$ is not regular for $x_1 =0$ and $x_2 =0$. Is this correct? As for the last question, I don’t really know how to address it: could the open sets be $x_1 \neq 0$ and $x_2 \neq 0$? I am sorry if I made any gross mistakes, but I am an absolute beginner at the subject. Thank you.