Let $R$ be a ring. If $a \in R$, show that the map $f(x) \longmapsto f(x + a)$ is a ring isomorphism of $R[X]$.
It is easy to show the homomorphism. I see that the bijection is true intuitively, but I don't know how to write formally. Thanks for the help.
We can write down the inverse function $R[x] \to R[x]$, $f(x) \mapsto f(x-a)$. This is a homomorphism because it is the same as $f(x) \mapsto f(x + (-a))$, which said you already showed is a homomorphism.