0

Take a function defined by $y=f(x)$ what does this really mean? Is the variable $y$ entirely defined by $x$ or is it a statement where in the domain of $f$, the relation $y=f(x)$ is true for the values of the variables $x$ and $y$ when $(x,y)∈ graph(f)$?

  • 1
    It means $y$ is the output of $f$ for the input $x$. A function is defined by its inputs, a set containing its outputs and how it relates the inputs to the outputs. And $(x,y)\in dom(f)$ doesn't make sense unless $f$ takes two inputs, you want to say $(x,y)\in f$. – David P Sep 20 '22 at 20:07
  • @DavidP corrected to 'graph', in this case, can $y$ vary outside of being the value of $f(x)$ and we can consider the graph of $f$ as a subset of the possible combinations of $(x,y)$? Can we define this? –  Sep 20 '22 at 20:11
  • 1
    FWIW I don't agree with the downvotes to this question at all; I think this is a very natural question and it's not well-explained anywhere that I know of. – Qiaochu Yuan Sep 20 '22 at 20:25

1 Answers1

0

The honest answer is that the meaning of this notation changes the more math you learn. When you get to the point of learning how to define everything in terms of set theory and so forth the concept of a "dependent" or "independent" variable largely disappears. In set theory a function $f$ is just a collection of ordered pairs $(x, f(x)) \in X \times Y$ (its graph) and given $x \in X$ we can evaluate $f$ on it to get $f(x) \in Y$ and that's all.

The meaningful distinction once you get to this stage is between free variables and bound variables, and that depends on what kind of further statements you want to make about $f$; either $x$ or $y$ could be free or bound in some statement involving $f$.

Qiaochu Yuan
  • 419,620
  • For me it's a little bit strange, in elementary situations it's as if $y$ is defined such that we can only consider it so it has a particular value at a particular $x$. However as we go on $y$ seems like it can vary freely, but the equation $y=f(x)$ is true for $(x,y)∈f$, are the two equivalent. –  Sep 20 '22 at 20:18
  • @user1007028: yes, I agree that it's strange, this sort of thing is not well-explained. You are correct that once we start getting into higher mathematics we allow $y$ to range freely over the codomain $Y$, so that we can, for example, ask questions like "is $f$ surjective?", meaning "for every $y \in Y$, does there exist $x \in X$ such that $y = f(x)$?" Here both $x$ and $y$ have been quantified so they are now bound variables. – Qiaochu Yuan Sep 20 '22 at 20:21
  • I appreciate the answer, I understand better now, I think it's a case that I can make sure to be aware of this. –  Sep 20 '22 at 20:29
  • Really the statement "$y = f(x)$" by itself doesn't mean anything past a certain point. What is $y$? One needs quantifiers! In higher mathematics if we just want to talk about $f(x)$ we just introduce the function $f(x)$ and don't name $y$ at all until we want to say something about points in the codomain. – Qiaochu Yuan Sep 20 '22 at 20:30