0

We like to use terms like '$f$ is a function of $x$' but I do not like this because from my understanding, any function should be applicable to any variable. The definition of a (single variable) function is a rule that takes $x$ to $f(x)$, however although simple this definition does not provide clarity on this. For example this rule could be defined simply as a rule which can only be applied to the value of a variable $x$, this is not clear in the definition, there is no specific meaning of the term rule and to what scope it applies.

Is the language '$f$ of $x$' incorrect as from my understanding we should be able to have $f(x)$, $f(y)$, $f(z)$ without them being fixed constants. With each being independent quantities to $f(x)$.

  • A function is a special kind of relation between two sets, a domain and a codomain. Nowhere does this definition mention $x$, $y$ or $z$, which is just a notational convenience since after all we have to name the elements of the domain somehow to be able to talk about them. – Michal Adamaszek Oct 14 '22 at 10:31
  • 1
    I don't like it either and never use that terminology. If I speak of a function in general then I write "$f$" (not "$f(x)$"). I go for: "$f$ is a function $\mathbb R\to \mathbb R$ prescribed by $x\mapsto x^2$". I hold my shoulders if others do it differently. It is IMV not really an issue as long as it is clear what is meant. – drhab Oct 14 '22 at 10:35
  • A function is a rule that takes some value and produces another value. The rule may depend on other values, but the point is that it maps some value to some other value. If you have $f(x) = x^2$, then $f$ is a function of $x$ that maps $x$ to $x^2$. You can also write $f(y) = y^2$, which means that $f$ is a function of $y$ that maps $y$ to $y^2$. It's the same function, applied to different inputs. In other words, it is a function of any variable you want to give it. Whether you write $f$ of $x$ or $f(x)$, you are saying the same thing. – Amirreza Hashemi Oct 14 '22 at 10:35
  • See answers and links to your previous post. – Mauro ALLEGRANZA Oct 14 '22 at 11:04
  • The terminology "function of x" is (in my experience) exclusively used in the context of multiple possible variables, and the expression is short-hand for naming which variable is currently being applied to (or, in the multivariable context, that the function is constant with respect to the other variables). This is all perfectly comprehensible and consistent within the appropriate context, and that's all that matters. – JBL Oct 14 '22 at 11:41
  • Can you give a single example, including relevant context, in which this expression was used that you believe results in potential unclarity or confusion? – JBL Oct 14 '22 at 11:53

3 Answers3

3

This kind of terminology isn't really "incorrect," because it long pre-dates the modern definition of a function. It comes from saying things like "pressure is a function of temperature and volume," meaning that if you change the temperature or volume of something this will predictably affect the pressure it exerts.

When someone has a concept they need to assign a name to, they normally take a word that's already in use and sort of expresses the same idea and then apply it to that concept. As a consequence, established language that uses the older, everyday sense of the word may not quite make sense when you try to apply it to the new definition. That's just life, at least until we either

  1. start making up new words from scratch every time we need to give something a name, or
  2. impose some kind of central control on the language.
Daniel McLaury
  • 24,656
2

I think what need is acceptance. Functions have a somewhat long and storied history, and the language that we use to describe them reflects that.

Essentially, the notation is torn between:

  • the classic concept of “function of a variable” being hard to define in a universally satisfactory way,
  • having names for your function’s arguments being useful occasionally.

This (and tradition) is why people will use a mix of old (but useful!) and newer notation, and there is no universal agreement on the precise mixture. I think for many (most?) mathematicians, even their personal mixture is not absolutely set in stone. There is no satisfactory universal answers that we could give you. This took me a long time as well, but you have to accept it if you want to progress. You might even notice that you mix old and new notation yourself, at some point.

So what you need to do when you run into unclarity surrounding function notation (and this is important in many other aspects of maths as well!) is to learn to understand what is meant from the context. Someone says “the function $f(x)$”? Learn to read it as “the function $f$” if that is what you prefer. You’re not sure whether $f(a)$ is the value of that function at the point $a$ or $f$ expressed in terms of $a$? Look at the context to figure it out. Usually, this is quite doable and you often even gain additional insight into the material because you’re forced to engage with it (though I would prefer unambiguous notation as well).

Eike Schulte
  • 3,232
1

In general, one can define a "function" $f$ from a set $X$ to another set $Y$ as a subset $R\subseteq X\times Y$ such that for every $x\in X$ there is a unique $y\in Y$ such that $(x,y)\in R$. Then by $f(x)$, one means the unique $y\in Y$ such that $(x,y)\in R$. But when one writes "$f$ of $x$" one means $f$ applied to $x$ where $x\in X$, without specifying which $x\in X$.

kandb
  • 1,349
  • The 'rule' definition is too vague, for example part of the 'rule' could be that $f$ can only be applied to the variable $x$ and its output can only be $f(x)$, but defining it as a set of pairs makes sense. It can appear that using $x$ and $y$ and only writing everything in terms of $x$ and $y$ you could be tricked into thinking the function only is defined for a variable $x$. –  Oct 14 '22 at 10:50