Kernel of a substitution map
Since $\mathfrak I = (y-x^2,z-x^3)$ is contained in the kernel, your map corresponds to a map $R/\mathfrak I\to S$. Show that this map is injective.
Hint: If $I = (y-x^2, z-x^3)$, then since $I$ is in the kernel of $f$, we have an induced map $f : R/I \to S$ which is clearly surjective, so we just need to show that it's injective. Now use the fact that every element of the coset $R/I$ contains an element of $R$ which is purely a polynomial in $x$.