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)?
Best Answer
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.