Is the image of a function the codomain of a function?
The image is the subset of the co-domain that the domain actually maps to. Take for instance the following function $$f: \mathbb{N} \rightarrow \mathbb{N}, \quad f(x) = 5.$$

It is simply the constant function that always "returns" 5, no matter what natural number is "passed in." The domain/co-domain defining $f$ is fine, but observe that the only value in the co-domain ever "used" is 5. In this case, the image of $f$ would just be the set $\{5\}.$

"The image of $S$ under $f$" is an explicit reference to the image of $f$ given $S$. That is to say, it is whatever each element in $S$ maps to as defined by $f$.