I have a question on rational and birational maps: Is the map $\mathbb{P}^1\rightarrow \mathbb{P}^2, (x:y) \mapsto (x:y:1)$ rational? Birational? If birational what is its inverse? Same questions for map $\mathbb{P}^1 \rightarrow \mathbb{P}^2, (x:y) \mapsto (x:y:0).$ My guess is that both aren't birational and that both are rational, but would like to hear another opinion.
Thank you