Correct interpretation of the sentence: "$y$ is a function of $x$"

Question

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.

Answer 1

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 $(p, t)$.

Answer 2

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 $X$.