Recently, I've developed a habit of trying to separate the idea of a function from the idea of the image of the function. This has mostly just confused me, but I am adamant about sticking to it. I think the two terms, "ring of polynomials" and "ring of polynomial functions," are not equivalent. Here's my reasoning: If we talk about a polynomial, we talk about something of the form ##\sum a_nx^n##. When we talk about a polynomial function, we talk about a function that takes every ##x## in a domain to a member of a codomain of the form ##\sum a_nx^n##. If we were to try to establish a bijection between them, we'd have to forget about the domain and codomain. Is there any reason to think of them differently, or are they really just two terms for the same thing?