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There are similar questions here and here asking about a "formal definition of a variable", but the "dependent" makes this unique. If you search the web for "dependent variable" you get hit with things saying,

"In the case $y=f(x)$, $y$ is the dependent variable, $f$ is the function, and $x$ is the independent variable."

Can we formally call $y$ a variable?

I realize this question is digging in the weeds and people often understand what is meant in context. However, formally, is it wrong to call $y$ a variable?

EDIT

I found an old chat session I had. Granted I was asking about functions, the response I got applies here. I think it helps support my curiosity

People who write "y = f(x)" are typically not doing rigorous mathematics. It is unfortunate that historically people started adopting a convention where a relationship between varying quantities in a physical setting is depicted with x being a so-called independent variable (more like a manipulated variable) and y being a dependent variable (more like an observed variable), and if a functional relation is suspected then people would write y = f(x). But that was done long before mathematical functions were made rigorous. Sadly, old habits die hard. The correct way to express such things in modern mathematics is as follows. Suppose you have a function f : R→R. Then you can plot the graph of f in the plane by plotting the set of points { (x,y) : y = f(x) }. The graph of f is not f itself, but merely one way to represent f. We can call this graph the graph of f(x) against x, or the graph of y against x where y = f(x). Either phrasing would be completely unambiguous and rigorous, and correspond to the definition of "graph" as given above.

ryang
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ThatsRightJack
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    A variable is dependent if it is completely defined by other variables. The only way to determine what is and isn't a variable is context. – CyclotomicField Jul 23 '21 at 04:33
  • Along with the above, if we are given $y=f(x)$, we know we can determine $y$ uniquely from $x$. If $f$ happens to be an injection (one to one), then we can also determine $x$ given $y$, taking $y$ from range($f$) – Alan Jul 23 '21 at 05:03
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    "People who write "y = f(x)" are typically not doing rigorous mathematics." That's nonsense. – Jair Taylor Jul 23 '21 at 05:32
  • @JairTaylor It depends on how rigorous is rigorous. People doing formal logic don't write that, as far as I know. – Trebor Jul 23 '21 at 05:52
  • @JairTaylor Also, if you write $y = f(x)$ then you are saying $y$ is a constant (indeed, the value of the function $f$ at the point $x$ of its domain) and then treat $y$ like a "function" is plain, untasteful, nasty, bad maths in my opion. – William M. Jul 23 '21 at 05:55
  • Only terminology... $x$ is the variable and $f$ is the function. Writing $y=f(x)$ we are simply "renaming" function $f$. – Mauro ALLEGRANZA Jul 23 '21 at 05:56
  • @WillM. Just by writing $y=f(x)$ it's not implied that $y$ is a function. It's often written to mean exactly that $y=f(x)$ for a particular $x$.

    Furthermore, when you do write $y=f(x)$ in the sense that $y$ is a function, this is just shorthand for talking about the function $f$ in a notationally convenient way. It's no more or less rigorous than other kinds of common "abuse of notation".

     – Jair Taylor Jul 23 '21 at 06:04
  • The writing $y=f(x)$ is simply due to historical reasons: in the cartesian-coordinates graph we plot $F(x)$ on the $y$-axes. It is simply the graph of function $f$. – Mauro ALLEGRANZA Jul 23 '21 at 06:29
  • @OP You may as well always write "depedent variable" and "independent variable" since these terms cannot be formally defined and are just a mental trick to follow arguments. Of course, always writing the quotation marks will be utterly pedantic. – William M. Jul 23 '21 at 18:06

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A variable is a quantity that varies (over a set of its possible values). Here, as $x$ varies over the domain of the function $f,$ $y$ correspondingly varies over the range of $f.$ Since the value of $y$ is completely determined by the value of $x$ and the mapping operation $f, y$ is variable that depends on $x.$ (To be clear: variable $y$ stands for the value/output of the function $f.$)

Framed this way, $x$ is the independent variable and $y$ the dependent variable.

On the other hand, $y=x^3,$ for example, can be rewritten as $x=y^\frac13$ to discuss instead the behaviour of $x$ as $y$ is varies; in this case, $y$ becomes the independent variable, and $x$ the dependent variable.

ryang
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  • On the other hand, if the dependent variable $y$ had been experimentally determined by varying the independent variable $x,$ then reflecting the plotted curve in the line $y=x$ does not necessarily give the graph of $x$ (now the dependent variable) as $y$ (now the independent variable) varies; this is because $y$ may not be an injective function of $x.$ – ryang Apr 30 '22 at 07:16