When are variables 'dependent'?

Question

I apologize if this question is a bit too pedantic for here, but this is something that I haven't been able to get a good answer on, relating to the idea of 'dependence' and context, say we look at the graph of a function $f$, we can say that for all $(x,y)$ in $G_f$, the relation $y=f(x)$ should hold. However outside the domain of $f$ this should may not hold.

In this case we can say that '$y$ is dependent on $x$ in $D$'.

My reasoning for this is that in some cases $f$ is not a nice smooth function and may be piecewise or have a strange domain for which not every $x$ can be considered.

If $x$ and $y$ can be thought of as varying freely in the reals, but $f(x)$ has a domain $D=$ {$a,a∈N$} then $y=f(x)$ as a relation has many more constraints, for every $x$ there isn't a $y$. However, we can nicely say that $y$ depends on $x$ in the graph of $f$, and have both $x$ and $y$ varying over the reals independently in our wider context.

If I have a function $f(x,y)$ I could have $x,y$ freely varying over it's domain, and 'independent' (not functions of) each other, however I could introduce a restriction on $f$, $f_r(x,y)$ whose domain is defined as {$(a,a),a∈R$} in this case I could say that $x$ and $y$ are 'dependent' in the domain of $f_r$.

Is the terminology 'function of' and 'dependent' relative to a set of pairs for which the relation is true?

I am struggling with the idea that a variable must be related to another is in the definition of the variable, for example, that $y=f(x)$. Is the idea of one being 'dependent' on another that we set our context to define what is 'dependent' and what isn't?

Answer 1

Is the terminology 'function of' and 'dependent' relative to a set of pairs for which the relation is true?

Yes, basically. There's really no such thing as "dependent variables" per se, only "dependencies between variables" (which we may or may not want to impose at any given time).

Your confusion is natural and related to the following: we use the equals sign to denote (at least) two quite different things, namely identities such as the Pythagorean theorem telling us that two previously defined quantities are equal, and conditions we would like to impose between free variables such as $y = f(x)$. These are conceptually quite different operations and it's really unfortunate that we conflate them like this. The statement $y = f(x)$ by itself has no independent meaning since the variables $x$ and $y$ are not bound; it's not telling you anything you didn't already know, it's just naming a condition you may or may not want to impose relating $x$ and $y$.