To elaborate a bit, $A$ is an integral domain, $f$ is a monic irreducible polynomial and $a$ is a root of $f$. $A[a]$ is the smallest subring of $\overline{\mathrm{Quot}(A)}$ (Quot denotes the field of fractions) that contains $A$ and $a$ while $(f)$ is the ideal generated by $f$.
One can immediately define $F:A[y]\to A[a]$ by the rule $F(g(y))\mapsto g(a)$. Everything seems to work until you come to prove that $\ker F=(f(y))$. In other cases, the so-called "modern version" of Gauss's lemma can yield the result but now $A$ is not a UFD! I am not aware of any other post-modern version of that lemma that works in general domains.
The context under which this question arose is the first paragraph of page 99 of D. Lorenzini's book "Invitation to arithmetic geometry" where it is stated that the two rings are isomorphic.