Let $k \in \mathbb{N}^{+}$ $S$ be some set, and $f$ be a function in $S^{\{ 0, 1, \ldots, k \}}$ be a function.
I'm wondering whether there's an alternative term for the image of $f$, i.e. the elements $s$ of $S$ which have some $i \leq k$ with $f(i) = s$.
My motivation has to do with me considering the finite sequence of evaluations $f(0), \ldots, f(k)$. I would rather not "remind" my readers of the domain $\{ 0, \ldots, k\}$, and of $f$ as a function - I want to focus on the evaluations and on the minimum set to which they all belong.
Note: A commonly-used term is great, an occasionally-used term is fine, a term you've thought of yourself could be a comment...
An alternative term is the range of $f$. For shorthand, you can let $K = \{0,\dots,k\}$ and denote it $f(K)$; occasionally $f[K]$ is used instead to emphasize that it's the image of a set, not a function evaluation. The latter notation lets you write $f[k+1]$ instead of naming the set $K$, using the set theory convention that each natural number is the set of its predecessors, but you may not want to assume that.