# Proving that a map is birational

Prove that, given an algebraic plane curve $$X$$, the map $$f: \mathbb{A}^1 \to X$$ defined as $$f(t) = (t^2, t^3)$$ and the map $$f: \mathbb{A}^1 \to X$$ defined as $$g(t) = (t^2 - 1, t(t^2 -1))$$ are birational.

My attempt For the first one I wrote $$x= t^2$$ and $$y=t^3$$ and then $$\frac{y}{x}= t$$. I think I can’t go on from here, I don’t even know if expressing $$t$$ in terms of only $$x$$ and $$y$$ is helpful.

• Please define the concept of "birationality". – Arman Malekzadeh Aug 6 at 11:11
• A rational map is birational if it has an inverse rational map. – cip Aug 6 at 11:15
• "I can't express $t$ in terms of only $x$ and $y$." But didn't you already do that when you wrote $t=\frac{y}{x}$? – Lazzaro Campeotti Aug 6 at 13:28
• Yes, sorry, that’s definitely what I did. What I meant instead is that I don’t know if expressing $t$ like that is going to be helpful to find the inverse. – cip Aug 6 at 13:38
• Hint: you already found the inverse. – Lazzaro Campeotti Aug 6 at 14:15