Let $C$ be a plane curve parametrized by $x=f(t),y=g(t)$ where $f(t),g(t)\in k[t]$. We can easily see that the coordinate ring of $C$ is isomorphic to $k[f(t),g(t)]\subset k[t]$. So $C$ is isomorphic to the affine line $\mathbb{A}^1$ if $k[f(t),g(t)]=k[t]$. For example, the curve given by $x=t+t^4,y=t^2$ is isomorphic to the affine line.
My question is that whether all curves that are given by $x=f(t),y=g(t)$ with $k[f(t),g(t)]=k[t]$ are of such kind, i.e., satisfy a relation of the form $x-ay^n=bt+c$ or $y-ax^n=bt+c$ for some $n\in \mathbb{Z}$ and constants $a,b,c$?
In algebraic terms, I am asking the following question:
If $k[f(t),g(t)]=k[t]$, does it necessary that the relation of $f(t),g(t)$ and $t$ is of the form $f-ag^n=bt+c$ or $g-af^n=bt+c$?
Note: The Abhyankar–Moh theorem states that if $k[f(t),g(t)]=k[t]$, then $\deg f$ divides $\deg g$ or $\deg g$ divides $\deg f$. We may take $k=\mathbb{C}$ if necessary.
No, you can take for example $f=t+t^2$, $g=t^2+f^2$.