Does a bijection $\phi$ which is not a morphism between two projective plane curves $C$ and $D$ induces an isomorphism $\phi^{*}$ on function fields?

Question

I was doing a problem, where I found that $\psi: \mathbb{P}^1 \to Z(x^2z-y^3)$ such that $\psi:(t:s)\to (t^3:t^2s:s^3)$ is a morphism between $\mathbb{P}^1$ and $C:=Z(x^2z-y^3)$ and $im(\psi)=C$. I also proved that $\phi:C\to \mathbb{P}^1$ such that $(t^3:t^2s:s^3)\to (t:s)$ is not a morphism.

I need to understand whether $\phi$ induces an isomorphism $\phi^{*}$ on function fields and I got completely stuck. I would appreciate any help!

Answer 1

$\psi : [t:1]\to [t^3:t^2:1]$ is a rational map $\Bbb{P}^1\to Z(x^2z-y^3)\subset \Bbb{P}^2$. What is its inverse ?