TITLE: Function "evaluation" just means "composition"?
QUESTION [25 upvotes]: I am self-studying and have a basic or naive question that follows from a simple observation. I have also included tags for type theory, etc because "evaluation" probably has a different meaning in these constructive approaches, which I'm also curious about.
Any set $X$ with elements $x_i$ can be characterized by a set denoted $(X)$ of pairs $(\star, x_i)$, alternatively denoted $(\star \mapsto x_i)$ (for all $x_i \in X$). Each one of these latter pairs encodes a distinct function from common domain $\{ \star \}$ to (each different element of) $X$. If we name such functions after their image, i.e. $(x_i):\star \mapsto x_i$, then the symbol "$f(x_i)$", traditionally interpreted as function evaluation, is reinterpreted as function composition, i.e. $f\circ (x_i)$ and the $\circ$ is often notationally suppressed. And roughly, $f = \cup_xf(x)$, every function decomposes to a unique set of singleton functions (better word?).
Question: Is there anything more to evaluation, or can I exactly equate evaluation with composition and forever de-clutter my mind of the term "evaluation"? Is the above perspective unusual or fruitful beyond reducing terminological clutter?
The notion of evaluation above - perhaps "extensional" is the right term? - means essentially subsetting the function at the chosen argument $x$. Perhaps I will see that a full answer to my question requires considering an intentional (?) or constructive perspective on "evaluation" (as in lambda calculus, etc). Perhaps only there will composition and evaluation legitimately have distinct meanings.
Edit: The notation $(x)$ seemed the most natural way to denote any function with singleton domain because it coheres with the conventional way to write evaluation as $f(x)$. The notation seems unambiguous because there is no context I can think of, other than "evaluation", where a single symbol, say $x$, is enclosed in brackets "()"? It does seem useful to notationally distinguish functions on singleton domain in some way: I would like the notation to be a mnemonic for whether the resulting function composite is a singleton function (i.e. a "value" like $f(x_i) = f \circ (x_i) = (\star, y_j)$) versus a non-singleton function. Perhaps there is a conventional notation for functions from singleton domain, other than $(x)$ and a conventional name for such functions, other than "singleton functions"?

REPLY [5 votes]: Is there anything more to evaluation, or can I exactly equate evaluation with composition and forever de-clutter my mind of the term "evaluation"?

Frankly, there is no way to non-circularly 'get rid' of the notion of "evaluation". A rose by any other name is still a rose. Even Lawvere's axiomatization of ETCS explicitly axiomatizes the evaluation mapping in Axiom 2, just phrased in category theory terms.
I think what you must realize is that certain mathematical notions are core notions and whatever implementation you use for them does not make them disappear at all. For example, it makes no sense to try to argue that ZFC set theory has eliminated the notion of function application because instead of "$f(x)$" we can use "$\bigcup \{ y : ⟨x,y⟩∈f \}$" given by the Union axiom and the Replacement schema (or a slightly more complicated expression if you use Specification and Powerset instead of Replacement).
The whole point is that we design foundational systems to be able to do the mathematics that we want to do. Since we want to have the usual natural numbers with the usual induction, we also want our foundational system to be able to construct a model of PA (for suitable encoding of "model"). And since we want to be able to reason about functions on $ℕ$, functions on $ℝ$ and so on, with the usual notion of "function", including evaluation, composition, function types and equality, we make sure that our foundational system is able to construct representations of the usual functions and prove those usual properties! See also this related post explaining why we care about abstractions rather than how things are concretely implemented in the foundational system.
There is nothing wrong with thinking about weaker systems that cannot do some of these things, but most logicians would not consider these systems viable as foundations for all mathematics, precisely for that reason!