TITLE: What does it mean for a variable to be a function of another?
QUESTION [5 upvotes]: When I hear someone say "$y$ is a function of $x$," I think of the notation $y(x) = 2x + 4$. But I've seen some people also say that $y = 2x + 4$ is a function $y$ of $x$. That's confusing to me because surely that's an equation and not a function. You can change it to be $x = \frac{1}{2}y - 2$, can you now call it $x$ being a function of $y$ even though nothing changed except for where the variables are or is that just outright incorrect and an equation can't be considered a function of another variable like that? I've seen these two being used interchangeably most often when plotting graphs of polynomials, sometimes the $y$-axis is even labelled $y(x)$ even though I didn't know you could have a function as an axis.
A little more broadly, how can I know when something is a function and when it is an equation, and are there any notable differences or problems when you misuse them (e.g. when a function was used when an equation should have been)?

REPLY [4 votes]: The confusion arises because there are two inconsistent, albeit related, uses of the word function. The older use is what may be called a dependent variable, in (for example) the  phrase “$y$ is a function of $x$” or, more specifically,  “the function $y=2x+4$”. This usage is still common among non-mathematicians who employ mathematics. This language tends to be avoided by present-day mathematicians, because it implies, in this case for example, that a function is a kind of real number (which depends on another, freely specifiable, real number). The function here is not $y$ but (in simple terms) the rule that specifies how $y$ is obtained from $x$. In the modern sense, a function can be precisely defined as a kind of mathematical object, which is quite distinct from the values (e.g. $y$) associated with the function.