That is, is it possible that $\displaystyle f\in R$ but $\displaystyle f\notin R(\alpha)$ for some $\displaystyle \alpha$ other than $\displaystyle \alpha(x)=x$, or $\displaystyle f\in R(\alpha)$ for some $\displaystyle \alpha$ other than $\displaystyle \alpha(x)=x$, but $\displaystyle f\notin R$? If it is, can you give me an example? Thanks.