Show that $\displaystyle \phi : K \rightarrow K^3$ given by $\displaystyle \phi(t) = (t,f(t),g(t))$ with $\displaystyle f,g \in K[x]$ is an algebraic set of $\displaystyle K^3$
let $\displaystyle p(x,y,z)=y -f(x), \ q(x,y,z)=z-g(x).$ clearly both $\displaystyle p(x,y,z)$ and $\displaystyle q(x,y,z)$ are in $\displaystyle K[x,y,z]$ and the image of $\displaystyle \phi$ is the solutions of the system: $\displaystyle p(x,y,z)=0, \ q(x,y,z)=0. \ \ \Box$