Rigorous Definition of "Function of"

The modern approach is, as you say, to view a function as a relation. Thus $f\subseteq A\times B$ is a function if it satisfies that if $(a,b)\in f$ and $(a,b')\in f$ then $b=b'$. It is then common to write $f(a)=b$ instead of $(a,b)\in f$.

This is a way to formalize the notion of $f$ defining its output as a function of its input. If you like then, this is the actual definition of 'function of'.

It is helpful to keep in mind the long history of the development of the notion of function. During the early days of the calculus a function $f:\mathbb R \to \mathbb R$ was vaguely defined to mean something like: f is a process that transforms the input $x$ to some output $f(x)$ and moreover $f$ does so in a very smooth way (almost always differentiable).

This historical approach to function, while not rigorous, is more in-line with $y$ being a function of $x$. The modern approach of a function as a relation, while very rigorous, is more static. This may be viewed as a shortcoming of this rigorous definition. However, the formalization of function is simple enough and easily allows abuse of concepts to actually think of a function as some process while it is formally not.

This situation is somewhat similar to the definition of a random variable. A random variable is nothing but a function with a particular domain and codomain. Thus, according to the relational definition, it is a very static thing. Nonetheless, we think of a random variable as a highly variable thing, even as if it's value is not yet known or is uncertain. However, this formalization of random variable within the rigorous confines of measure theory is highly useful, allowing one to correctly argue about uncertain events. This goes to show just how powerful the modern axiomatization is - there is enough flexibility in the interpretation of the notion of function to accomodate many situations.

"$y$ is a function of $x$" means the value of $y$ is determined by that of $x$. For example, to say that the area of a circle is a function of the radius implies that all circles with the same radius have the same area.

There certainly is a discrepancy between the formal set-theoretic definition ("giving" a function by giving its graph), and the informal use. Another important aspect of the informal use of "function" in practice is to ascertain when one thing $y$ is not "a function of" another thing $x$, which ordinarily means that "when $x$ changes", but everything else is "kept constant", $y$ does not change. A synonymous phrase is "$y$ does not depend on $x$".

How to ascertain whether $y$ "depends on/is a function of" $x$? There is no universal algorithm, and unless the relationship or lack thereof is described adequately, even specific examples are not resolvable. This is especially true of physical measurements, where correlation and causality are not always easy to distinguish.

In purely mathematical situations, often there is some difficulty in "finding" a thing $y$, and one is interested in being able to use "the same $y$" while other things in the environment/context vary. Giving upper bounds or lower bounds or counting something... with an outcome independent of, that is, not a function of, some other thing $x$... is a simpler story. It is not always obvious whether or not this is possible, so it is reasonable to ask the question.

In introductory physical science and engineering discussions, it is typically mathematically useful insofar as it simplifies things to assume (tentatively? heuristically? as a good approximation?) that one thing is independent of another, that is, "is not a function of". The archetype for this is a situation in which one will differentiate implicitly, but, if everything depends on all parameters, a uselessly complicated expression comes out. Using some experimental/physical sense about the physical realities often allows a practically useful approximation by declaring that this doesn't depend on that.