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I am struggling to help some of my new precalculus students accept function notation -- something new to them this term. I am looking for strategies to help them adopt this new notation.

Their main complaint: "Why do we need $f$ when we can just talk about $y$?"

For these students, an equation already represents an input/output relationship, so they see no need to introduce a new letter to refer to something they already have a name for. I would like them to feel some ownership over the notation, as though it is useful-enough to them that they actually want to use it.

When you have students with these complaints, how do you help them? What strategies/exercises have worked?

For this level of student, assume they are versed in linear and quadratic equations, and they are learning function notation for the first time, including composition. They are being asked questions related to functions with formulas, word problems, graphs and numerical data.

[Note: While I could list many particular things my students struggle with in this regard, the issue I want to address is helping them feel function notation may be useful and, therefore, something worth really adopting.]

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    $\begingroup$ The kids have a point. And you have a point. But I don't think there is some magic soundbite to change the dynamic. Just tell them they need to know different ways of describing functions and drill/test them. Do a daily 5 min quiz until they get it down. But don't ask them to magically love things or agree with you. Heck...like I said, they actually have a point. And so do you. But don't look for agreement. Look for training. Did Mr. Miyagi drill or explain? How about Gene Hackman in Hoosiers? Heck, even the explanations you do get resonate a different way after doing some work. (Cont. below) $\endgroup$
    – guest
    Aug 7, 2019 at 0:54
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    $\begingroup$ The function $\displaystyle \begin{array}{c}f : E\longrightarrow F\\~~~~~x\mapsto y\end{array}$ is a subset of $E\times F$ whereas $y\in F$. $\endgroup$
    – user5402
    Aug 7, 2019 at 18:43
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    $\begingroup$ If your students are saying this, the reason is probably that the notion of a function is of little utility for the topics you're teaching in your precalc course. Mathematicians did, after all, do a great deal of mathematics, including analysis, for centuries without talking about functions. $\endgroup$
    – user507
    Aug 8, 2019 at 17:43
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    $\begingroup$ How do you feel about this notation: $y=y(x)$? $\endgroup$
    – Peter Saveliev
    Aug 10, 2019 at 0:06
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    $\begingroup$ @PeterSaveliev the notation $y = y(x)$ states that $y$ is a function where, regardless of the input, the output is the function $y$ itself. If you mean anything different, then that equation is a category mistake. $\endgroup$
    – Jordan
    Feb 9, 2020 at 23:55

21 Answers 21

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Start by talking about functions in general, not only about functions that can be expressed by a simple formula in x and y. Examples:

  • The function that maps every non-empty list to its first element.
  • The function that maps every finite set to its size.
  • The function that maps color names to RGB triples.
  • The function that maps days to sunrise times at a particular location.
  • The fubction that maps locations to sunrise times on a particular day.
  • The function that maps the age of a particular person to their height.
  • The function that maps the length of a square to its area.
  • The function that maps my age (in days) to your age (in days).

Some of these can be expressed by a simple mathematical formula, some can be expressed by a complicated formula, or by program, or by a table, and for some ("age -> height") there is no formal presentation at all, even though one can still state certain properties (domain, range, monotonicity, ...) of the function.

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    $\begingroup$ How are these functions? "The function that maps the age of a particular person to their height. The function that maps my age (in days) to your age (in days)." $\endgroup$
    – Peter Saveliev
    Aug 8, 2019 at 16:23
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    $\begingroup$ Am I the only one who thinks this doesn't answer the question asked? The asker wants to know how to convince their students function notation is good to use and how to make it make sense to them. This answer goes primarily into types of functions and their logistics, but little to nothing in how students can accept function notation. $\endgroup$
    – Lex_i
    Aug 8, 2019 at 16:36
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    $\begingroup$ The value of this answer is that it gets students to accept the concept of a function, which is really the important thing. Once they accept that, then it will be obvious that some kind of notation for it will be useful. $\endgroup$
    – Toby Bartels
    Aug 8, 2019 at 18:22
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    $\begingroup$ Most of the examples here can be expressed using just $y$ provided subscripts are used. I think students would just ask the same question once again for that reason. If notation is a concern then we need to justify why particular notation is preferable, otherwise hoping students make a connection they didn't already make in algebra and early precalculus will leave students further confused. $\endgroup$
    – Lex_i
    Aug 8, 2019 at 18:29
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    $\begingroup$ @PeterSaveliev Since every living person has some height y at any time t, there's obviously a function descibing the dependency between t and y (of course, the functions are different for any two people, and there is no closed formula for them). The second example is trivial: y = x + k; where k is the age difference in days. (If I had said "in years", it's no longer a function!) $\endgroup$
    – Uwe
    Aug 8, 2019 at 22:31
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You might remind them that $y$ is just a name for a number. When they draw a plot, they draw a bunch of points: maybe $y=3$ here, $y=5$ there, and $y=-2$ over there. But at some point (no pun intended) we want to talk about the entire shape: we want to say that $f$ is symmetric, that $f$ is concave, that $f$ has an asymptote. We can't do that with $y$; saying "$y$ is invertible" is as nonsensical as saying "4 is invertible".

In short, $f$ lets us talk about a shape instead of a number. Or, if you prefer, $f$ gives us a way to talk about the forest, whereas before we only had $y$ for talking about trees.

Consider $y=x^2+x+7$, an equation about "trees". We might have no idea what number $x$ is, and no idea what number $y$ is, but do we know they're related quadratically. Now we can write equations like $g(x)=f(x-1)+1$, an equation about "forests". Again, we might have no idea what shape $f$ is, and no idea what shape $g$ is, but we do know they're translations of each other. There was no way to express this idea when we could only talk about trees.

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    $\begingroup$ One may still talk about the 'entire shape' by treating an equation like $y=x^2$ as a mathematical object. At the precalculus and calculus level all functions are given explicitly by such equations (or by graphs), so the argument that we need the function concept to talk about the forest cannot be to convincing for a student. $\endgroup$
    – Michael Bächtold
    Aug 10, 2019 at 13:06
  • $\begingroup$ @MichaelBächtold Better? $\endgroup$
    – fvy
    Aug 10, 2019 at 14:16
  • $\begingroup$ I'm not sure I understand the edit, but your last example could be phrased as follows without using functions: Let $y$ depend on $x$ and $u$ on $v$, then $y=u+1$ when $v=x-1$. Here my $y$ is what you would write $g(x)$ and my $u$ is your $f(v)$. Of course this requires that we explain what it means that one quantity depends on another. I didn't use the equivalent phrase "$y$ is a function of $x$" on purpose, since then people will think I'm talking about modern functions. $\endgroup$
    – Michael Bächtold
    Aug 10, 2019 at 15:38
  • $\begingroup$ Saying that "$y$ is a function of $x$" does not mean the same as saying "$y$ is a function". It means that the value of $y$ is determined as soon as the value of $x$ is fixed. For more on this distinction see the discussion here or here. $\endgroup$
    – Michael Bächtold
    Aug 10, 2019 at 15:38
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    $\begingroup$ @MichaelBächtold I agree with all you wrote, but I'm not trying to argue that functional notation is inevitable or the only way to talk about unspecified dependencies between variables. I'm saying that it's useful because it's gives us a way to talk about dependencies without explicating them. I think there's a lot of pedagogical value in students evaluating alternatives, like a "depends on" relationship. But, regardless, they should understand that there's a problem that we're trying to solve, namely that we have this new class of ideas that aren't expressible with just $x$, $y$, et al. $\endgroup$
    – fvy
    Aug 10, 2019 at 18:27
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You should tell them these two main benefits:

(1) Function notation is concise! For example, instead of writing "Find $y$ when $x=5$" one can simply write "Find $f(5)$" This becomes very appreciable when dealing with long or complicated problems asking for a lot of information. We also shorten things like this all the time. For instance, instead of writing $\{x|-\infty<x<\infty \}$ we may prefer to write $x\in \mathbb{R}$. So these nice conventions are things students will just have to get used to (I'm sure that won't be hard once they see there are very good reasons entire math communities agree on specific notation).

(2) It clears a lot of confusion. If we use $y$ for everything then we'd have to use subscripts every time we are dealing with two or more functions. If we were given two functions where one refers to say, the number of blue marbles and the other the number of red marbles, I'd much rather see $B(x), \ R(x) \ \text{than} \ y_1, y_2$. I don't want to think "wait, what which one is which, again?" in my head more than once.

The same issue comes up in just about anything involving functions (function mapping, defining categories, etc.), but let's say we're talking about function composition. $(y_1 \circ y_2)(x)=y_1(y_2(x))$ is far more confusing (especially when there is context from a word problem) than $(f \circ g)(x)=f(g(x))$. It's just not immediately clear nor pleasing to look at.

I'm sure after enough practice, function notation will come naturally to your students.

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    $\begingroup$ “instead of writing $[\sin(x)]^2$ we will write $\sin^2(x)$” – that's a bad example, in particular in this context, because of the confusion $\sin^2 = (x\mapsto (\sin x)^2) = (\cdot^2) \circ \sin$ but $\tan^{-1} = \arctan$. $\endgroup$
    – leftaroundabout
    Aug 7, 2019 at 16:11
  • $\begingroup$ @leftaroundabout I'm aware of that confusion. I meant to use that example purely as an example of more "concise" notation even though it may not be as clear to some students, which is why it wasn't in the second point. But admittedly it isn't the best example, so I made a different one. $\endgroup$
    – Lex_i
    Aug 7, 2019 at 18:32
  • $\begingroup$ You can write $y|_{x=5}$ for "$y$ when $x=5$". One could say that this needs more pen strokes than $f(5)$, but for writing $f(5)$ you first need to introduce an additional letter $f$ and explain the function concept. The second example I find hard to follow. Why would we need to use subscripts? Why can you not write $r,b$ for the number of red and blue marbles, instead of $y_1,y_2$? In your final example about composition your issue seem to be only with subscripts, it has nothing to do with quantities versus functions as far as I can tell. $\endgroup$
    – Michael Bächtold
    Aug 10, 2019 at 13:11
  • $\begingroup$ I imagine they already understand the f part, that they understand a function is a relation between things, considering they've been learning functions at least a year before being introduced to new notation. Second, I used subscripts for $y$ because the original post doesn't say that the students are open to using different letters, it only mentions $y$. Of course, if you use different letters it works much better and bypasses the second benefit I listed, but the first still applies. At least one good reason is good enough for students to be convinced, imo $\endgroup$
    – Lex_i
    Aug 10, 2019 at 14:24
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The crucial thing the students need to realise is that the (e.g.) $x$ that turns up in the function definition is a bound variable. That's what allows it to be freely renamed or indeed omitted without changing the semantics.

Unfortunately, education tends to completely obscure this facet by a) always using the same dumb variable names as if there were a particular meaning to f, x and y (when in fact the power of these abstractions is that the naming is completely arbitrary) and b) by themselves confusing functions with their results.

In particular, I so often hear people talking/writing about “the function $\cos x$”. That's wrong, $\cos x$ is not a function, the function is $\cos$. Why this is important only becomes properly apparent when using higher-order functions, the simplest being indeed function composition. So that's the example I would focus on: give deliberately exercises where the notation would clash without the proper notion of function. Like, just ask them to compose the functions $f : x\mapsto x^2+x$ and $g : x\mapsto x-1$. This clearly doesn't work when just talking about “the $y$s”.

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    $\begingroup$ It's even worse than ‘the function $\cos x$’; even in an introductory course, when they get to applications, they'll start writing ‘$A = A(x)$’, which is literal nonsense. Of course, like ‘the function $\cos x$’, people do say that in practice, and you may want to talk about it at some point; but students have to learn the notation before they can properly abuse it. $\endgroup$
    – Toby Bartels
    Aug 8, 2019 at 18:58
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Because x and y are just variable names

It happens that sometimes y=f(x), but other times z=f(x,y), w=f(x,y,z), or x=f(y) for that matter. All of these variable names are syntactically equivalent, and the mere existence of "x" and "y" in an equation does not necessarily connote that "x" is the independent variable and "y" the dependent. Thinking of the function as a function, making explicit each variable's role, is a new level of understanding that's missing from a mere equation.

The common practice of presenting y as a function of x builds in the minds of many students the notion that y must always be a function of x. Thus, when we introduce f(x) notation, it seems redundant. Textbook authors should deliberately present other scenarios to avoid creating this notion.

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  • $\begingroup$ I fail to get the point of this answer. Yes, $x$ and $y$ are variable names, but so is $f$. The first ones probably have type $\mathbb{R}$ while $f$ might have type $\mathbb{R}^n\to \mathbb{R}^m$. So what? I also don't understand what it means that they are syntactically equal? Certainly when $y=f(x)$ and $u=f(v)$ you are not allowed to conclude that $u=y$. And I don't understand why you shuffle the names of input and output variables while using the same letter $f$ for all functions in your answer. $\endgroup$
    – Michael Bächtold
    Aug 10, 2019 at 10:59
  • $\begingroup$ I interpreted the point of the answer to be that when you study functions the variables used in the equations may be dummy variables, rather than referencing particular quantities. However, I agree that it’s best not to use the same function name “f” for different functions, which some of those must be since they take differing numbers of arguments. $\endgroup$
    – Joe
    Aug 10, 2019 at 16:50
  • $\begingroup$ @Joe I don't know what dummy variable means. Is it the same as bound variables? $\endgroup$
    – Michael Bächtold
    Aug 11, 2019 at 8:26
  • $\begingroup$ @Michael Bächtold, by “dummy variable” I meant, informally, variables that can be replaced by other symbols in the statement with no effect. In that sense, a bound variable is a dummy variable, but since I’ve only ever seen the term “bound variable” in predicate logic, where it refers to a variable bound by a quantifier, I didn’t think it was applicable here. $\endgroup$
    – Joe
    Aug 11, 2019 at 10:56
  • $\begingroup$ @MichaelBächtold By "syntactically equivalent" (not "equal") I meant that _x_,* y, z, and *_w_ are (probably, at this point in students' learning process) real numbers. I used the same letter _f_ but clarified "other times" to indicate that we'll present a scenario to a student using that same letter to mean a different function at a different time. I shuffled the names to show that in each scenario, we are free to use those variable names in many ways. The convention of presenting _y_=f(_x_) (before teaching what "f(x) even means" contributes to an undeserved connotation. $\endgroup$
    – Monty Harder
    Aug 12, 2019 at 17:12
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TL;DR:

  • A function is a verb. It's an action.
  • Variables are nouns, objects.
  • Verbs (functions) connect nouns semantically, i.e. how A (or x) relates to B (or y), how to get from here to there.

Long version

Some context: I learned maths from my father who was a physics / engineering guy at heart, so everything had to be 'tangible' or 'observable' to him.

Thus the lesson would always begin with an "ELI5" overview of the essential concept (and indeed, I was between 6-18...), involving drawings and a lot of gestures, but more importantly was always based on a real-world tangible use-case/example.

Typically, simple physics. That's how I learned most math: as a tool to solve some problem I could 'feel' (often even solve mentally for simple/extreme cases, which is great to build initial understanding). To this day I think¹ of maths mostly in visual terms.

It helped me "internalize" or "own" mathematical objects exactly the way you seek for your students to understand. So that's my recommendation: make them 'feel' it through real stuff.

Below (in ““quotes””) is just one extremely simple example of how I'd do it (it's about going from a tone to the same tone 1 octave higher in music). [my remarks in sq. brackets]

““A musical tone is defined by its frequency in Hertz: for instance, 440 Hz is an A.

When you double the frequency, you obtain the same tone, just 1 octave higher: so 880 Hz is an A too, higher in pitch. We usually number the tones to know which octave on a piano, 440 Hz is A4 and 880 Hz is A5.

So when we go from A4 to A5, we apply a function which doubles the frequency. That's what going 1 octave higher "does".

Let's write this in math:
we define the function Oct of "going one octave higher" as: Oct(x) = 2x with x the frequency of a music tone.
[notice, "going" is a verb, as it should: it expresses a relation between nouns, between 'things' aka variables; also we do away with F and go more literal to open the mind to meaningful abstractions — thereby laying some groundwork for an other important metaconcept, how to select/create/cookie-cut variables to solve problems]

If we input x=A4 we get twice that in result: Oct(A4)=A5

That's the equivalent of moving your finger 12 keys to the right on a piano, or some length down on a cello: that's what this function does in those contexts.
[bonus points if you can show them the movement of the arm, the three physical elements:
key A4, arm movement, key A5; in other common words x, F and y.]

Notice that, we can take any tone, we get the same tone 1 octave higher: this particular function works for all pairs between/from the origin space (say x is bound to octaves 0-7) and/to the destination space (Oct(x), the 'y's (all of them), would thus be bound to octaves 1-8).

Forget about an "equality" between some terms of x and y, only consider the movement from x to y: a relation that takes x and transforms it into y. This relation, this transformation, this change is called a function.””

The important part here, imho, is to de-associate the function itself from variables. Have them see:

  1. a "definition space" (terms like 'origin' or 'from' work well because they prepare concepts like bijection, surjection etc).
  2. a "result space" (again, 'destination' or 'to' work well)
  3. a "connection" or "relation" between the two, a way to get from 1 to 2, from the origin to the destination.

A function is a movement, a process, a change, a variation, a computation, a derivation, an integration, a representation, an abstraction. A fucntion is a verb, an action, something we do. It's not a thing, it's what changes things. It's a very-transparent-box that takes stuff in and chews some other stuff out.

Have them think like:

  • don't ask "how much of x is equal to y?" because that's entirely static, there is no function (verb, change) even if y=2x
  • rather wonder "what do I need to do to x to obtain y?" and that's an action, a verb, you take x and change it into something else and call that thing y.
  • Well, this change is a function.

Later on you can expand on the idea of "for any X" (domain/space of definition).

You could elaborate further on the musical (or whatever) example you used in introduction:

““Now you can also see how there are tons of possible functions in music to describe going from some tone to some other, and these variations more than the tones themselves are what makes music: for proof we can totally change the tonality of a song (play it in "D" instead of playing it in "C") and it's still the same song to everyone listening: because we're in fact listening to the functions, the variations).””

I'd caution against using non-static variables (involving time, progressions), it may be confusing in this particular context (you want 'x' and 'y' as static as can be, like music tones don't ever change by themselves: the only 'moving part' in the picture should be the function).

I don't know where you're going with this (I'm not American so I don't really know in which order you learn math topics), but there are interesting functions to visualize. I personally like:

  • i as a new object defined by a function (sqrt(-1) ). How cool is that.
  • Complex functions, rotations in the complex plane etc.
    Plug with trig or not (I'd say as little as necessary in this context) or compare x,i with traditional x,y linear, show them how functions can appear under different forms yet may remain identical (or close enough for the "feel" like multiplication by i; and how it's also just a rotation of coordinates themselves if you'd rather fix variables in the plane).
    [note: also great electricity math to be done with these, and current works much like water for the most part (e.g. "tension" is a difference of potential between two points, much like gravity/altitude on Earth) so it's very easy to picture the effects of some well-crafted circuits/functions, mentally.]
  • "Divide by two" is more complex but more fun that our 2x example above.
    Like halving 1 infinitely approximates to 0 (or 1/2+1/4+1/8... goes to 1).
  • Likewise the α=1+1/α equation (https://math.stackexchange.com/a/315376). Generally these infinite expressions can be represented as infinite compositions of the same function, infinite recursions, and these kinda 'fractal' movements or evolutions can not only be mentally pictured but via computers as well (or even simple pieces of paper, add one half of the remaining half until you can't and everyone gets the point). Think animation, step by step, whatever the medium.
    Tell the story of "this specific function" in "this specific context", then shift one (function or context), and again, and again, to further isolate the set of variables from the set of functions, in the student's mind.

I'm sorry I took so long but I wanted to arm you with all the terms, views and 'tricks' I could think of, so you can choose. I feel that's easily in the top 5 most important concepts to really grasp in math and any applied field.

¹ Although this is slightly off-topic, I need to plug 3Blue1Brown's video on the Euler identity, it's a visual graal if there ever was one on a math relation.
Link: https://youtu.be/mvmuCPvRoWQ

Edits: clarifications, typos

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    $\begingroup$ "A function is a verb. It's an action." Yes, I used that analogy, that a function commands a series of actions, like a short program. And I used the term 'verb' as well, until a student pointed out that f(x) implies "eff that", and 2 minutes of laughter followed. $\endgroup$
    – JTP - Apologise to Monica
    Aug 8, 2019 at 18:31
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    $\begingroup$ "A function is a verb." No, it is not. Functions are objects in their own right! $\endgroup$
    – YiFan
    Aug 10, 2019 at 8:39
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    $\begingroup$ When I say "The mother of John", then the part "The mother of" is a function, which takes a person as input and returns a person as output. This function is not a verb, nor does it 'do' anything to John, when I apply it. So I don't find this analogy to useful. $\endgroup$
    – Michael Bächtold
    Aug 10, 2019 at 10:42
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(First, I should mention that I've never taught this, so my approach does not come from experience.)

So you have students who think of something like $y = f(x) = x^2 + 3$ as a relationship between two “specific” quantities $x$ and $y$. As intuitions go, that's not so bad: it serves physicists quite well. But it's incomplete, and you're looking for ways to illustrate that. I think a key point is to take $f$ at several points in the same context.

For example, $f$ has an interesting property: for any $x$, $f(-x) = f(x)$. How can this be expressed with $x$ and $y$? It's a bit awkward, isn't it? You have to say that when $x$ is replaced by $-x$, $y$ doesn't change. Now take another example: $z = g(x) = 3 x$. An interesting property of $g$ is that $g(w + x) = g(w) + g(x)$. What could $z$ even be here? “Which” $z$ are you talking about?

You can build up to more complex examples involving multiple functions. For example, take $h(x) = v = x^3$ and $i(x) = u = x^4$. An interesting property of $h$ and $i$ is that $h(i(x)) = i(h(x))$. Which in terms of $u$ and $v$ means that, er, well, if you take $x = v$ then you get $u$ which is the same $v$ as if you take $x = u$? If that makes any sense? Right, it doesn't. But functions let us express this unambiguously.

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  • $\begingroup$ These are nice examples, but I'll play the devil's advocate and rephrase them using only quantities, avoiding functions. The first one: for any value $a$ of $x$ we have $y|_{x=a}=y|_{x=-a}$. The second example: if $u=v^3$ and $v=x^4$ and further $y=z^3$ and $z=x^4$ then $u=y$. The problem with that second example, the way you formulated it, is that if $x,u,v$ were 'specific' quantities in any real applied context, then what meaning should the quantities $h(i(x))$ and $i(h(x))$ have? $\endgroup$
    – Michael Bächtold
    Aug 10, 2019 at 10:36
  • $\begingroup$ @MichaelBächtold First: so you have a way to say “the $y$ for this particular value of $x$”. You write $(x^2+3)|_{x=a}$. I write $(x \mapsto x^2+3)$. You're coming up with functions as a concept and now just have a slightly overly complex notation — corresponding to a redundant baggage to shed that the result of the function should have a specific name, rather than the function itself. Goal well on its way to being accomplished! Second: I don't understand the question, can you give an example of “'specific' quantities”? $\endgroup$
    – Gilles 'SO- stop being evil'
    Aug 10, 2019 at 10:43
  • $\begingroup$ $(x^2+3)|_{x=a}$ cannot be the same as $x\mapsto x^2+3$ since the first one contains a free variable $a$, while the second is a closed expression. Concerning 'specific" quantities, it was a phrase you used, so I assumed you have an idea of what it means. For me, examples of specific quantities are things like the temperature of a lake, the area of a circle, the time to finish my homework etc. $\endgroup$
    – Michael Bächtold
    Aug 10, 2019 at 10:50
  • $\begingroup$ @MichaelBächtold Oh, it's too late to edit my comment, but I meant to write $(x \mapsto x^2 + 3)(a)$. Which is a bit complex and needs to be explained, but if a student comes up with (some variant of) $(x^2 + 3)|_{x=a}$, I think you should make the connection with $(x \mapsto x^2 + 3)(a)$. $\endgroup$
    – Gilles 'SO- stop being evil'
    Aug 10, 2019 at 10:56
  • $\begingroup$ About specific quantities: yes, that's what I meant too. $h(i(x))$ and $i(h(x))$ are calculating the same quantity in different ways. Physical interpretations are limited by dimensionality constraints. Not many “interesting” functions of real numbers commute. It's easier to illustrate with multivariate functions, e.g. $(x,y) \mapsto (y,x)$ and $(x,y) \mapsto (-x,-y)$, or with geometric transformations (but you have to reach the point where a geometric transformation is seen as a function). $\endgroup$
    – Gilles 'SO- stop being evil'
    Aug 10, 2019 at 10:58
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Prior to my final year of high school, I was sent to a maths tutor for a couple of sessions, to give me a headstart on calculus. It helped a lot.

He introduced me to the concept of functions. He described it as a monster, living inside a box, that accepted a thing through one (!) tube, and pushed out a thing through another tube. The monster was consistent - every time it saw a particular thing, it would always push out the same thing.

Monsters that dealt with numbers were just a special case - there were some monsters that would always spit out a number twice as big as the number that went in.

The name of a particular monster was $f$, and $2x$ was a description of its behaviour

It sounds silly, but this description stuck with me and helped a lot (including in my Math major at uni).

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  • $\begingroup$ That doesn’t sound silly to me at all. It sounds like you had an excellent tutor. Although it’s important for mathematical concepts to be defined in precise language, I think it’s important for math teachers to also use analogies to aid in understanding. $\endgroup$
    – Joe
    Aug 10, 2019 at 16:55
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I have never worked with students of that skill-level, so take this with a grain of salt.

I like to thinks of functions as values, just a different kind of value from numbers. This can help demystify stuff like $\circ$ as it just like $+$, except it works on a different type of value. Once you get to vectors you also have a very nice parallel, since they are just yet another type of value.

One thing that you cannot do if you only talk about $y$ is to talk about the operation/computation itself; you can only talk about the result of the computation. When thinking of it this way, when I write $f$, I am talking about some value of function type: it contains an operation that is not yet performed. When I write $f(x)$ or $y$ I am talking about the number that I got from performing that operation on $x$.

Understanding $y=2x$ and $f(x)=2x$

Let's look at the difference between $y=2x$ and $f(x)=2x$. So it all starts with some number $x$. Now, if $y$ is the same as two multiplied with that number, then $y$ must also be a number. Numbers multiplied with numbers give numbers.

On the other hand, when I write $f(x) = 2x$, by the same logic, I get that $f(x)$ is a number. And it is! It is the number you get when you apply the operation. Let's try translating $f(x) = 2x$ into words.

  • $f(x)$ is “the result of the computation/operation/function $f$ performed on the number $x$”
  • $=$ is “is equal to”
  • $2x$ is “two multiplied by $x$”, but let's leave it as $2x$

So what $f(x) = 2x$ means is “the result of the computation/operation/function $f$ performed on the number $x$ is equal to $2x$”.

So what we're saying is that whatever $f$ does to $x$, the result will be $2x$.

I think it is important to see numbers inside the parentheses, e.g. notice how $f(5)$ is the number $10$. This will emphasize why $f(x)$ is a number.

What can you use the function for?

Well just like you can do math on numbers, you can do math on functions. For example, just like you can add numbers, you can combine functions. One example would be:

We have the functions $f(x) = 2x$ and $g(x) = x + 3$.

What is $f \circ g$? Well $f$ and $g$ are functions, so $f \circ g$ is another function. You can then define an $h(x) = 2x + 6$ and explain why $h = f \circ g$.

Note that it can also make sense to use $+$ with two functions, or a function and a number.

What is $f + 3$? Well it is a function that first computes $f$, then adds three to the result.

Now, what might you want to do now that you can talk about a function? Well one thing is that you can talk about the relation between the results when you give it different values.

The function $f(x) = x^2 + 1$ satisfies $f(x) = f(-x)$ for any $x$.

The function $f(x) = 2x$ is one-to-one.

Now you can introduce stuff like inverse functions. Just like the square root, not all functions have an inverse, but some do. One thing I like to be aware of is that all equation solving is just making use of inverse functions.

Let's solve $x + 5 = 10$. Well the left side can be thought of $f(x) = 10$ where $f(x) = x+5$. The inverse function of $f$ is given by $f^{-1}(x) = x - 5$, so we can apply it to both sides.

We now have $f^{-1}(f(x)) = f^{-1}(10)$, and the f's cancel, so we now have $x = 10-5$.

Of course this is much much more verbose than what we normally do, but if you point out how the inverse function is just the function mirrored across the $y=x$ line, then it helps to understand why equation solving corresponds to drawing a horizontal line from the $y$-coordinate and finding where it intersects.

Functions of functions

A function is a value right? So can you put it in a function‽

I don't know if you want to talk about this yet, but when you get to calculus, it can be nice to look back and recognize that the derivative is just a function that takes a function and results in a function.

If you do want to talk about them now, here are some examples of functions of functions.

Finding the inverse can be thought of a function. So you have $\text{Inv}(f) = f^{-1}$.

The composition operator takes pairs of functions. So you have $\circ(f, g) = f \circ g$. Here you can also mention $+(1,2) = 3$, as a weird way of writing plus.

Maximum/minimum value: You can have stuff like $f(x) = x^2$ and $\min(f) = 0$. This is a function that takes a function and returns a number (or negative infinity, which is not a number) (or it doesn't work if it has no minimum; maybe this is just like square root, and doesn't work for all functions)

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Different notation for different things

The key thing that the students seem to be missing is the conceptual distinction between f and y (in this example), so this apparently needs to be explicitly explained to them.

IMHO the way to go at this is to tell them that there are two "things" that we may want to talk about - the transformation process (the function) and it's result with a particular input; and so we need separate names and notation to be clear which we mean.

You need to spend a bit of time to demonstrate both concepts separately, compare and contrast them. Perhaps it's worth to use some very trivial function such as '+2' as an illustrative example; perhaps if you demonstrate the idea of a function as a "machine that does a thing to transform the input", and the need to talk about such "machines" and their properties then that analogy may work better for some students than the abstract math definitions proposed in other answers.

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Many answers already, so I'll keep this one short: it has been realized by researchers in didactic that one difficulty in the concept of function is that it changes status: at first each function is considered as a process (a verb in @ΦDev's answer); they meet several of them, each being akin to a (unitary) operation, not very different from addition or multiplication. Then at some point we start studying functions rather than using them, i.e. we ask student to think of them as a mathematical objects, in the same sense as e.g. numbers or geometric figures: functions will have or not have properties (monotony, continuity, etc.), will be subjected to operations (sums, differentiation, etc.), will be considered in relation to each other (inequalities, asymptotic comparison, primitive-derivative from each other, etc. ), etc. This is the point where one really need a variable name for a function, i.e. to write it as $f$ instead of something specific such as $\sin$, $\cos$, $\exp$, ...

It could help to embrace this shift in the point of view on functions to overcome this difficulty. When I teach math-teachers-to-be, I use the case of relations to have them go through the same process: they already now $\le$, $\subset$, etc. but are not used to consider an unspecified relation and work on it as an object.

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Function notation is a next step in mathematical maturation. In the language of Dubinsky et al., your students are in the process of encapsulating functions as primary objects.

At one point in mathematical development, after learning to count, positive integers are "encapsulated" by children as primary objects. Later, while learning algebra, variables such as $x$ and $y$ become encapsulated as primary objects. Many people never get past this stage. It sounds like your students have.

Then when learning calculus, functions like $f$ and $g$ might become encapsulated (abstracted) as primary objects. But it's a messy process. The very fact that your students are asking this question is evidence that they are in the process of encapsulating functions as primary mathematical objects.

An analogy I like is learning to tie one's shoe laces. At first, it seems very mysterious and difficult. It would just be easier to remove the laces, or go around barefoot. But eventually, the task becomes muscle memory, and it's impossible to recall the learning difficulty. And then it is on to the next struggle.

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From a computer science perspective understanding that functions are first class objects is also pretty difficult. There are cases where functions can be parameters to other functions, the classic example being sort accepting a compare function. This case would be impossible to explain using just the y output. In the case of sort you don't even need to know the compare outputs for all the pairs of inputs, though this is moving away from math a bit.

To be fair I think the students are not at the level of CS to really understand this reason, but maybe you can explain it as a nice teaser?

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Well... $ \def\zz{\mathbb{Z}} $

Let $f : \zz → \zz$ defined by $f(n) = n+1$ for every $n∈\zz$. Then $f(0) = 1$ and $f(f(0)) = 2$ and $f(f(f(0))) = 3$ and so on. It is now obvious why having functions as first-class objects is useful, since we can repeatedly apply them. Similarly, the Mandelbrot fractal is defined in terms of iterating an elegant function.

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To answer the question of why you need f...

... have them consider the region between two graphs. Unless you have a way to distinguish between the different y values, you're going to be hopelessly lost.

Now you don't need to use f... you could use subscripts: $y_{1}, y_{2}$ (and in fact, that's how graphing calculators handle it).

But it's nice to use $R(x)$ to represent revenue and $C(x)$ to represent cost, instead of having to remember that revenue was $y_{1}$ and cost was $y_{2}$.

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  • $\begingroup$ Why not simply use $R$ and $C$ for revenue and cost, without the $(x)$. Why do you need to use the letter $y$ with subscripst? $\endgroup$
    – Michael Bächtold
    Sep 13, 2019 at 13:02
  • $\begingroup$ You can (and do). However, we run into the expression problem: R is revenue (generic), but what if we want to talk about the revenue at two different points in time. $\endgroup$
    – Jeff Suzuki
    Sep 13, 2019 at 14:02
  • $\begingroup$ One could write $R|_{t=a}$ and $R|_{t=b}$, with $t$ being the time and $a$ and $b$ two different values for it. $\endgroup$
    – Michael Bächtold
    Sep 13, 2019 at 14:09
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    $\begingroup$ Sure; in fact, that's how we're supposed to write the definite integral. The problem is...if we write that way, it's not too long before we collapse it down to something like R(a), R(b). Sooner or later, some form of function notation is unavoidable. $\endgroup$
    – Jeff Suzuki
    Sep 13, 2019 at 19:49
  • $\begingroup$ In case someone gets the wrong impression that $R(a)$ means the same as $R|_{t=a}$, you could show them the following example. Let $A$ denote the area, $r$ the radius and $D$ the diameter of a circle. Then $A|_{r=1\mathrm{m}}$ and $A|_{D=1\mathrm{m}}$ are well defined, but $A(1\mathrm{m})$ is not! (The area of a circle is not a function in the modern sense of the word.) $\endgroup$
    – Michael Bächtold
    Sep 14, 2019 at 10:33
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Some rudimentary programming exercises might make it obvious why it's useful to encapsulate functionality. When you write y = f(x) in Python, for example, it's clear that y is just a static result, while f is the thing that does the work. You can't reuse y to change another variable z in the same way - you have to refer to f to do that.

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  • $\begingroup$ While this is true for python, mathematics allows more freedom than most programming languages. For instance, it is perfectly correct to write $3=x$ or $x^2+y^2=1$ in mathematics while almost no popular PL allows that. Also, when you state $y=x^2$ in mathematics, you can very well reuse $y$ for instance by writing $y|_{x=3}$. $\endgroup$
    – Michael Bächtold
    Aug 10, 2019 at 9:08
  • $\begingroup$ In mathematics, $3 = x$ isn't an assignment, it's an equivalence. You can certainly express any mathematics in Python. It's just that the syntax has been developed with other goals than just pure maths. Anyway, that's all besides the point. $\endgroup$
    – l0b0
    Aug 10, 2019 at 9:20
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    $\begingroup$ I absolutely agree that practical exercises in programming in which students learn to create functions that they reuse, rather than typing the same code multiple times, will reinforce the mathematical concept of a function. In my experience, one of the hardest parts of learning math for students that struggle is that it’s too abstract for them. I think that actually writing and executing code will make it more concrete. $\endgroup$
    – Joe
    Aug 10, 2019 at 17:03
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When identifying $y$ with $f(x)$ we implicitly consider a point $(x;y)$ lying on $f$'s graph. However, this doesn't have to be the case. Say, we take $f: x\mapsto x^2$, then the point $(x;y)=(2;f(2))=(2;4)$ lies on $f$'s graph but I can well consider the point $(x;y)=(2;3)$ lying below the graph or the point $(x;y)=(2;5)$ lying above the graph, whose $y$ coordinates are not equal to $f(1)$.

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I hear the problem, but seeing as they are just beginning to learn function notation, they probably are not up to higher level mathematics. I think you can just blame it on the math world being crazy and tell them that as they learn more math it will become necessary for them to use it, so they need to get used to it.

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It is generally easier to learn something when you can see why it's useful. One use mentioned already is that you can effectively describe even and odd functions using function notation.

I most want to help my precalculus students understand function notation so that they can deal with the f(x+h) that they'll see repeatedly when working with derivatives.

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My answer would be: "What if there is no formula?" Just as $x$ might stand for an unspecified number, $f$ might stand for an unspecified function.

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  • $\begingroup$ Can you elaborate? What if there is no formula for what? $\endgroup$
    – Michael Bächtold
    Aug 20, 2019 at 7:13
  • $\begingroup$ @Michael Bächtold When there is a formula, such as $y=x^2$, you might not need a name for the function. When there is no formula, as in "Suppose we have a continuous function.", you often need a name for the function in order to continue to speak about it. $\endgroup$
    – Peter Saveliev
    Aug 20, 2019 at 13:11
  • $\begingroup$ In the example $y=x^2$ we are giving a formula for $y$, which is a function of $x$. If there is no formula for $y$ we might still say, "Suppose $y$ is a continuous function of $x$". I don't see immediately a reason to introduce $f(x)$ because we don't have a formula. $\endgroup$
    – Michael Bächtold
    Aug 21, 2019 at 13:01
  • $\begingroup$ @Michael Bächtold OK, what if there are TWO functions? $\endgroup$
    – Peter Saveliev
    Aug 21, 2019 at 14:47
  • $\begingroup$ Then you would use a different letter than $y$ for that function of $x$, say $u$. In practice (real applications of calculus) it wont happen that the same quantity $y$ is a function of $x$ in two different ways in the same context. $\endgroup$
    – Michael Bächtold
    Aug 21, 2019 at 14:51
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You could start with two tables of values for x (the input variable) and y (the output value) in both. To start, each should represent a permutation on say, the set {1, 2, 3, 4, 5}, but don't use the word "permutation."

Label one "Table A", the other "Table B".

For each line in Table A, introduce the notation A(1)=, A(2)=, etc.

Similarly, for Table B.

Then ask, what is B(A(1)), B(A(2)) etc.?

Then ask what is A(B(1)), A(B(2)), etc.?

Be sure to test the examples before class. You don't want any weird coincidences that you would have to explain away.

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