$R[x]$ is simply the free left $R$-module on the set of symbols $\{x^k:k\geq0\}$, on which a certain multiplication operation is defined. There is nothing analytic about that!
In particular, the elements of $R[x]$ are not functions. When $R$ is commutative, there is a somewhat canonical ring homomorphism $\phi:R[x]\to \mathcal F$, where $\mathcal F$ is the ring of all functions $R\to R$ with pointwise operations. But in general $\phi$ is not injective.
$R[x]$ is simply the free left $R$-module on the set of symbols $\{x^k:k\geq0\}$, on which a certain multiplication operation is defined. There is nothing analytic about that!