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Often in introductory mathematics textbooks or papers when authors are talking about functions they write the sentence "$y$ is a function of $x$". Is it correct to interpret this sentence as "there exists a function $f$ such that $y=f(x)$"? I know this is a bit of an elementary question but there are so many different interpretations of this sentence I just wanted to get the communities feedback on if my understanding is correct.

J. W. Tanner
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Taylor Rendon
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  • You say that there are "so many different interpretations". It would help me to understand the scope of the question if you could give some examples. – Xander Henderson Aug 17 '23 at 21:39
  • @XanderHenderson - Of course: I’ve seen “$y$ is a function of $x$” written down as “$y$=$y(x)$ (which is bad notationally, but it’s out there often), and another example would be taking $y$ to be the function (for example, $y : \mathbb{R} \to \mathbb{R} $) such that $y(x)$; or in other words, $y$ is a function with argument $x$. – Taylor Rendon Aug 17 '23 at 21:44
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    These are comments which should be edited into your post. That said, I don't have a problem with $y = y(x)$ (it really says the same thing as "there is a function $f$ such that $y = f(x)$", just in a shorthanded kind of way. And saying $y : \mathbb{R} \to \mathbb{R} : x \mapsto y(x)$ is just a third way of saying that same thing. These are all just different ways of expressing the same basic idea. – Xander Henderson Aug 17 '23 at 21:52
  • @XanderHenderson - I’d be happy to edit these comments in. And just to be clear, you think interpreting “$y$ is a function of $x$” as “there is a function $f$ such that $y=f(x)$” is correct? Also, thank you for your time. – Taylor Rendon Aug 17 '23 at 22:06
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    I do not think that it is incorrect to interpret the phrase "$y$ is a function of $x$" to mean "there exists a function $f$ such that $y = f(x)$". I don't think that this is quite correct either ($y$ is a variable, $f(x)$ is the value of the function $f$ at a point $x$, so there is a bit of a type error here), but it is an entirely understandable interpretation. – Xander Henderson Aug 17 '23 at 22:09
  • @XanderHenderson - That’s great feedback, to which makes me curious how you would interpret more application based sentences such as considering “the position $x$ of a ball is a function of time $t$” - would you write $x=f(t)$ for a function $f$? – Taylor Rendon Aug 17 '23 at 22:17
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    Usually, if someone said "the position $x$ of a ball is a function of time $t$", I would write $x = x(t)$, and get on with my life. Why complicate the notation by adding extra letters which don't help me to understand what the heck is going on? I understand the sentence to mean that position is a function (call it $x$), and that this function takes a single input (call it $t$). So, really, there is a function $x : T \to Y : t \mapsto x(t)$. But this is a all really pedantic, and no one, I think, would ever bother to formalize it in this way. – Xander Henderson Aug 17 '23 at 22:33
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    It seems you’ve already read my answer here, but I’ll refer you to it again. In math, we define function as a precise object in set theory. The phrase ‘function of’ and its various uses then simply become a language issue. At this stage I offer you the following suggestion: don’t follow the masses and be your own person/be robotic and use completely formal language and notation throughout until you get to the stage where you’re willing to be more cavalier with formality for the sake of brevity/conformity. – peek-a-boo Aug 17 '23 at 23:39
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    I could talk about the reasons for why one uses language in the manner they do, be it historical, pedagogical etc, but really these are all subjective matters. Ultimately math is about communicating (see what I say in my answer there). One place where the standard notation “$y=f(x)$” could get confusing is when you’re dealing with complicated objects, like maps between function spaces. – peek-a-boo Aug 17 '23 at 23:42

2 Answers2

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You've asked a more complex question than it might at first appear. The real issue is not with the meaning of the word "function" in the sentence "$y$ is a function of $x$" but with the meaning of the symbols $y$ and $x$. What are these referring to? They're supposed to be "variables," but what is a "variable"?

Let's restrict our attention to a more specific case where things will be clearer: suppose we're discussing the trajectory of some particle traveling along a line (to keep things simple) over time. So there are two "variables" here (whatever those are), position and time. We'd like to say "position is a function of time." What does this mean? It means that there is a trajectory (or a set of possible trajectories), which is a subset $\Gamma \subset P \times T$, where $P$ is the set of possible positions and $T$ is the set of possible times, and to say that position is a function of time means that there exists a function $f : T \to P$ such that the trajectory is the graph of $f$, or symbolically

$$\Gamma = \{ (p, t) : p = f(t) \}.$$

What it means for $p$ and $t$ to be "variables" here is that they vary over the possible points $(p, t)$ in the trajectory. The complication here is that the trajectory is being left implicit and not being named explicitly even though it is the actual object of interest. We are not talking about a single point $p \in P$ and a single point $t \in T$, nor are we even talking about a set of points in $P$ and a set of points in $T$, we are talking about a set of pairs $(p, t)$.

Qiaochu Yuan
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I suppose tipically this sentence is not said in a vacuum. Presumably, if $x$ is in some set $X$, and $y$ in another set $Y$, then, $y$ is a function of $x$ means precisely what you said. There is a function $$f:X \rightarrow Y$$ Of course calling $y$ the function is a slight abuse of language. For example, when you say "position is a function of time", you surely mean there's a set of times $T$ and a set of positions $X$ such that tehre's a function $f: T \rightarrow X$ and you write $x=x(t)$ to name the function the same thing you name the elements of $X$.

Lourenco Entrudo
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