Reasons for (not) distinguishing $f$ from $f(x)$

Question

Formally, if $f$ is a function, $f(x)$ is a value. So for instance, $f$ can be continuous, but not $f(x)$. In teaching at school and university, notation is quite often mixed up, e.g. the function is called $f(x)$. This may be problematic but can also have benefits. What good reasons for the use of the different notations do you know?

This question raised from the discussion of this discussion on notation of derivatives.

Answer 1

To distinguish not too strictly between $f$ and $f(x)$ allows to operate more easily with functions built up from other functions.

For example, one might want to say things like:

Let us consider the function $\sin (3 x^2)$ on the intervall $[0,1]$.

This can be considered as sloppy in a formal sense, but I think it is still clear and in some contexts preferable over more correct alternatives, in my opinion.

Answer 2

It was mentioned in other answers that having a more sloppy notation is better to not complicate the communication. This is okay for people who had really understood the concepts of mathematics.

I have the feeling that understanding what a function is one of the those things students don't really understand. Common mistakes (and in my opinion all of them are led by the fact of the sloppy notation) are:

• "A function is something you can write down". For example, most students have a lot of problems if you define a function in a more indirect way: For every argument, the function takes as a value the minimizer of some functional (dependent on the argument). In abstract algebra or differential geometry there are also very sophisticated examples of a very abstract usage of functions.
• By all the time taking $x$ as argument, some students are not able to think about another variable as argument of a function which is the case in many applications.
• The lack of not distinguishing between function and its evaluation leads often to a lack of the ability to use theorems, as it was also pointed in this question about the chain rule: Revisiting topics from previous courses.
• If students fail to understand the deep meaning what a function is, there is no chance that they can understand basic concepts in e.g. functional analysis, where "values" are functions.

Answer 3

Consider the following observations:

• People shorten things with increasing frequency of usage. For example, the most frequent words are short. This is a way of making communication more effective, e.g. see Huffman coding. For example this list has its first 5-letter word at 39th place (first three are for comparison): $$\begin{array}{c|c|c|c} \textbf{place} & \textbf{word} & \textbf{letters} & \textbf{count} \\\hline 1&\text{you}&3&1222421\\ 2&\text{I}&1&1052546\\ 3&\text{to}&2&823661\\ 39 & \text{about}&5 & 142750 \\ 70 & \text{really} & 6 & 68311 \\ 93 & \text{something} & 9 & 54736 \\ 95 & \text{because} & 7 & 53566\end{array}$$
• If there is only one free variable in expression $\mathcal{E}$, saying "function $\mathcal{E}$" is unambiguous. The key word here is "function" which let us know that what follows should be treated as a function, and not just as an expression (with appropriate context it might be not necessary). Such formula might be unambiguous even with multiple free variables, if the convention designates one as such (e.g. $x$ versus $a$), compare \begin{align} &\text{function } &x^2&+\sin x &\text{ok}\\ &\text{function } &ax^2&+\sin bx&\text{ok}\\ &\text{function } &c^2&+\sin bd&\text{ambiguous} \\ &\text{function } &d\mapsto c^2&+\sin bd&\text{ok} \end{align}
• Human thinking is fuzzy. Our brains do not process mathematical concepts formally (even if we are capable of it). There is a whole thread about notation abuse at math.se, in particular it includes a great quote by Gila Hanna: \begin{align}\text{The student of mathematics has to develop a tolerance}\\\text{for ambiguity. Pedantry can be the enemy of insight. }\end{align}

I suspect that there are also multiple other reasons, but the above seem to be the most important. That being said, I think it is alright to tolerate such abuse in speech (which is ambiguous and ill-suited for math anyway) or if time constraint is severe. However, I would not accept such a style in homework assignments. Moreover, I urge the teachers to write correctly on the board during their classes, it's not a significant burden and it might help, e.g. to avoid confusion.

I hope this helps $\ddot\smile$

Answer 4

It is interesting to see how computer algebra systems deal with this kind of thing. In Maple, for example, you can do the following:

• Define f := x^3
• Enter diff(f,x) to get the derivative
• Enter subs(x=3,f) to evaluate.

Alternatively:

• Define g := (x) -> x^3
• Enter D(g) to get the derivative
• Enter g(3) to evaluate

You can convert between the two idioms by f := g(x) and g := unapply(f,x).

When using Maple for research I find myself mixing the two idioms according to what is convenient for the immediate purpose. This feels faintly unsatisfactory, but it works. I used to teach a Maple-based undergraduate course, which also used both idioms. The difference between them did cause some confusion, but it was not especially prominent compared to other things that the students found difficult.

Answer 5

Though I also like to use the notation "the function $x^3$" when appropriate, sometimes this causes problems. For example when I teach inverse functions, it is difficult to grasp that the $x$ in $x^3$ is not the same $x$ as in $\sqrt[3]{x}$.

I always have problems with first year students who are used to the notation that $x$ is in the domain and $y$ is in the range of the function.

Answer 6

Perhaps it is worth mentioning here there is another notation which at first glance seems non-rigorous. For example, $f = x^2+\sin(y)+z$. Here $f: \mathbb{R}^3 \rightarrow \mathbb{R}$ is a function built from the coordinate functions $x,y,z$ each of which is itself a function defined in the obvious manner: $$ x(a,b,c)=a, \qquad y(a,b,c)=b \qquad z(a,b,c) = c. $$ At first I resisted this notation, but as the semester wears on I grow to enjoy its simplicity. I encountered in in Barrett O'neill's Elementary Differential Geometry. There is always a tension for me in teaching manifold theoretic topics, it's hard to strike the right balance of notation; too much kills intuition, too little makes calculations seem impossible.

Answer 7

The reasons for not distinguishing $f$ from $f(x)$ are historical.

1. For roughly 250 years, from the first use of the word "function" in mathematics (by Leibniz ~1680) until ~1930, the official (and only) definition of the word was: A quantity $y$ is called a function of $x$, if its value is uniquely determined as soon as $x$ assumes a value. Look into any book from that period. You might find small variations, but we can all agree that according to those definitions $x^2$, $\cos x$ and $f(x)$ are functions of $x$.
2. It was normal to drop the "of $x$" part when it was clear from the context. So people would simply call $x^2,\cos x$ and $f(x)$ functions. This is familiar from everyday language, where we also drop the "of" 's. We might for instance speak of "the owner of the car" and later just say "the owner".
3. The notation $f(x)$ for an arbitrary function of $x$ was introduced by Johann Bernoulli some 40 years after Leibniz introduced the word "function". Bernoulli decided to call $f$ the characteristic of the function $f(x)$, as did Euler and Lagrange after him. But this terminology didn't catch on, since people didn't feel the need to treat $f$ as an object by itself before ~1890. So they never talked about $f$ in isolation. They were satisfied with treating $f(x)$ as an object, and for that they had the name "function of $x$". (Actually Bernoulli, Euler and Lagrange wrote $f\,x$, without parenthesis. Parenthesis became part of the standard notation only around 1790. In a sense there was nothing revolutionary about Bernoulli writing $f\,x$ for an arbitrary function of $x$, since people were already writing $r\,x$, $l\,x$ and $s\,x$ for the root, the logarithm and the sine of $x$ long before.)
4. Around 1840, Jacobi introduced a confusing notation in relation to partial derivatives, with the good intention of causing less confusion. You can read more about it here and here. Probably inspired by this notation he writes $y(x)$ for what should be $y$ in the same article (using different letters). As far as I know, this is the first implicit but conscious use of the $y=y(x)$ abuse. Soon after that (1850) one finds Cayley write $\Omega = \Omega(r,v,y)$ explicitly as an abbreviation for "$\Omega$ is a function of $r,v,y$" and many others adopted this convention. Nevertheless, I haven't seen Euler and several other important mathematicians (before and after Jacobi) confuse $f$ for $f(x)$. For them that would have been like confusing $\cos$ for $\cos x$. I'm also certain Jacobi understood the difference. He probably just didn't foresee the confusion his notation might generate.
5. Around 1890 Dedekind, Cantor, Peano and Frege all isolated the $f$ in $f(x)$ as an independent mathematical object and gave definitions. But each of them decided to give it a different name:

• Dedekind called $f$ a mapping and kept the name function for $f(x)$.
• Cantor called $f$ an allocation and kept the name function for $f(x)$.
• Peano called $f$ the prefix sign for a function (in line with the original name characteristic of a function) and kept the name function for $f(x)$.
• Only Frege decided to call $f$ a function and claimed that variable quantities like $f(x)$ have no place in mathematics. (See his 1904 article "What is a function?".) Frege was well aware of the distinction between $f$ and $f(x)$, but he was also aware of the already existing abuse $y=y(x)$. Since he convinced himself that variable quantities like $y$ denote nothing, he concluded that mathematicians must be referring to $f$ when they speak of functions. Be that as it may, the fact that among these four mathematicians only Frege decided to call $f$ a function suggest it was a bad choice.

6. Probably due to Freges influence among logicians, the name "function" for $f$ was adopted by mainstream mathematics during the rise of set theory and logicism (1900-1940). At the same time the notion of something being a function of something else disappeared from the official mathematical language. Today many people still speak of functions "of something", but are probably unaware that they're not talking about modern functions $f$.

Summary: We've been calling $f(x)$ a function for much longer that $f$ and it's officially wrong only since ~1940, by some unfortunate choice of names and notations. It's hard to get rid of this heritage.

Answer 8

In most contexts, it makes sense to say that

"$x + \sin(x)$ is a continuous function".

Consider the alternatives:

"the function taking $x$ to $x+\sin(x)$ is a continuous function"

"$\text{id} + \sin$ is a continuous function"

Take your pick.

Answer 9

Several arguable points where made in previous answers, mostly that the abusive notation $x+\sin x$ for $x\mapsto x+\sin x$ is in many circumstances lighter, while the more formal notation avoids some confusions that are common among students.

In my opinion, the most important point is to adapt the notations and the rigor to the kind of students we deal with. In front of first and second year students, I try to always distinguish functions from values (or from expressions with free variables), because the very notion of function is shaky in their minds. I ask them to write things rigorously too, until they master them. With higher level math majors, I start to accept more sloppiness, as seems convenient, and sometimes reminding students the abuse made to notations. In a graduate course in differential geometry, abuses would probably be everywhere (but I did not teach such a course yet).

Answer 10

To my mind, this abuse of notation is more similar to how I think when I'm doing physics. Imagine a bottle of gas. It has a temperature $T$, a volume $V$, a pressure $P$, a number of moles $N$ and an entropy $S$. They are related by various relations such as the ideal gas law. Which one is a function of which? Well, any one can be a function of any other, under varying circumstances. Insisting that I am going to write everything as a function of temperature, or everything as a function of volume, or whatever, only introduces confusion.

I also tend to think in terms of many related quantities, no one a function of any other, when doing algebraic geometry (my research area) and differential geometry, but those aren't relevant to students in an ordinary calculus course.