I'm teaching a College Algebra class in the upcoming semester, and only a small portion of the students will be moving on to further mathematics. The class is built around functions, so I need to start with the definition of one, yet many "official" definitions I have found too convoluted (or poorly written) for general use.

Here's one of the better "light" definitions I've found:

A function is a relationship which assigns to each input (or domain) value, a unique output (or range) value."

This sounds simple enough on the surface, but putting myself "in the head" of a student makes me pause. It's almost too compact with potentially ambiguous words for the student (relationship? assigns? unique?)

Here's my personal best attempt, in 3 parts. Each part of the definition would include a discussion and examples before moving to the next part.

A relation is a set of links between two sets.

Each link of a relation has an input (in the starting set) and an output (in the ending set).

A function is a relation where every input has one and only one possible output.

I'm somewhat happier here: starting with a relation gives some natural examples and makes it easier to impart the special importance of a function (which is "better behaved" than a relation in practical circusmtances).

But I'm also still uneasy ("links"? A set between sets?) and I was wanting to see if anyone had a better solution.

  • Most answers given below will suggest if not outright require the function to be computable. This rules out constructs like the busy beaver function. It might be that for the audience you have in mind, this kind of white lie is quite acceptable. But your original attempt does not have this drawback, so perhaps someone can come up with something that is easier to read / understand / imagine and yet doesn't require or imply computability. – MvG Jul 30 '13 at 23:44

10 Answers 10

For fun, I like to liven-up the "black box"/machine view of a function by putting a monkey into the box. (I got pretty good at chalkboard-sketching a monkey that looked a little bit like Curious George, but with a tail.)

Give the Function Monkey an input and he'll cheerfully give you an output. The Function Monkey is smart enough to read and follow rules, and make computations, but he's not qualified to make decisions: his rules must provide for exactly one output for a given input. (Never let a Monkey choose!)

You can continue the metaphor by discussing the monkey's "domain" as the inputs he understands (what he can control); giving him an input outside his domain just confuses and frightens him ... or, depending upon the nature of the audience, kills him. (What? You gave the Reciprocal Monkey a Zero? You killed the Function Monkey!) Of course, it's probably more appropriate to say that the Function Monkey simply ignores such inputs, but students seem to like the drama. (As warnings go, "Don't kill the Function Monkey!" gets more attention than "Don't bore the Function Monkey!")

The Function Monkey comes in handy later when you start graphing functions: imagine that the x-axis is covered with coconuts (one coconut per "x" value). The Function Monkey strolls along the axis, picks up a "x" coconut, computes the associated "y" value (because that's what he does), and then throws the coconut up (or down) the appropriate height above (or below) the axis, where it magically sticks (or hovers or whatever). So, if you ever want to plot a function, just "Be a Function Monkey and throw some coconuts around". (Warning: Students may insist that that's not a coconut the Monkey is throwing.)

Further on, you can make the case that we're smarter than monkeys (at least, we should strive to be): We don't always have to mindlessly plot points to know what the graph of an equation looks like; we can sometimes anticipate the outcome by studying the equation. This motivates manipulating an equation to tease out clues about the shape of its graph, explaining, for instance, our interest in the slope-intercept form of a line equation (and the almost-never-taught intercept-intercept form, which I personally like a lot), the special forms of conic section equations (which aren't all functions, of course), and all that stuff related to translations and scaling.

Parametric equations can be presented as a way to let the Function Monkey plot elaborate curves ... both in the plane and in space (and beyond).

All in all, I find that the Function Monkey can make the course material more engaging without dumbing it down; he provides a fun way to interpret the definitions and behaviors of functions, not a way to avoid them. Now, is the Function Monkey too cutesy for a College Algebra class? My high school students loved him, even at the Calculus level. One former student told me that he would often invoke the Function Monkey when tutoring his college peers. If it's clear to the students that the instructor isn't trying to patronize them, the Function Monkey may prove quite helpful.

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    this was one of the funnier (and also more useful) things I've read today. Thanks. – Jamie Banks Aug 7 '10 at 1:51
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    "the almost-never-taught intercept-intercept form" - I don't know why either, it's actually quite useful! It is also easily manipulated to a polar-coordinate form. – J. M. is not a mathematician Aug 24 '10 at 2:41
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    +1. This has got to be the most fun way to teach or learn functions especially for non mathematicians!! – Pratyush Sarkar Oct 11 '13 at 3:40
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    "Students love the drama" Yes, we do. – Adam Nov 22 '13 at 17:46
  • @Isaac: I myself am quite content with just imagining the Function Monkey. – Adam Nov 22 '13 at 17:48

The way you've restated the definition is fairly common in contemporary high school books in the U.S. (perhaps changing "links between two sets" to "ordered pairs"). What I've seen a lot of in middle school and earlier algebra settings is the idea of a "function machine." The function machine graphic below is from FCIT (©2009), but a google image search for function machine will show you many different ways the concept can be visualized.

function machine

While this probably pushes the idea that a function has a formula, I'd claim that "rule" could be as general as a specific listing of which inputs map to which outputs, as in your definition. To me, the prevalence of this machine metaphor in middle school contexts suggests that it works well for students who do not necessarily yet have a sense of symbolic algebra. I've seen function machines used as low as 3rd grade.

  • While this is a good visualization I was planning to use, could you turn it into a specific definition? – Jason Dyer Aug 6 '10 at 16:05
  • This is the definition I use in Calc and College algebra. In calc I supplement this definition with a more formal one at the level of sets. In major courses I define morphisms in a general category using contrast to this definition. – BBischof Aug 6 '10 at 16:09
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    @Jason: A function is a device/process/object/thing that produces a unique specific output for each input--that is, if you put the same thing in, you'll always get the same thing out, and putting that one particular thing in can only give that one particular thing out. – Isaac Aug 6 '10 at 16:10
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    I suppose I should qualify all this: I'm assuming that the course you're teaching is generally not for math majors and that a large portion of the students may not even need to take calculus. Under this assumption, my inclination is to use definitions that favor intuition over rigor, but allow building to a rigorous definition later if necessary. I expect that the intuition from a function machine base would lead easily into your first definition and could be brought into alignment with your second definition. – Isaac Aug 6 '10 at 16:15

I just wanted to add a few cents to this post to say that Isaac's statement "I've seen function machines used as low as 3rd grade" is quite true. I teach kindergarten and 1st grade, and I use function machines with my students, mostly when introducing the idea of complements in relation to addition and subtraction. It's a standard part of the "patterns and algebra" portion of the Everyday Mathematics curriculum. Functions are honestly NOT a challenging concept for my students to grasp when presented in this manner, hence, I am absolutely confident that your College Algebra students will be just fine! :)

I would describe it as a verb. People are familiar with nouns and verbs so you get to shovel in a bunch of formal understanding for free that way. Take a noun, a real number or a vector or, even better, a food item, a TV show, or a person – and think of something that operates on it. Then the verb transforms it and it's different.

Really the best thing to do is give them a positive, creative, self-expressive assignment – like an essay. Offer extra credit to people who find examples of functions in the real world ongoing throughout the semester. If they bring in examples of two- or three-place operators then you get to explain why they're right in front of the class and they've just brought up the lecture point for you.

Look at the covers of magazines in the grocery store and you will get the material for examples you should be using. Take weight loss. You could graph the number of calories taken in versus weight gain/loss. Give them a lot of examples of functions and then the definition at the end – then have them look throughout the semester for more examples. This should be in concert with or as part of a weekly one-page (or one-paragraph) essay that they submit to you with an example of something from class that related to something from life.

Here's another trick I've used in teaching about $(x-1)^2$ versus $x^2 - 1$, which will surely come up as well. Decompose it into two functions. $x \mapsto x-1$ relabels the abcissa and you've prepared the board by using a dotted or no ordina. Then $\cdot \mapsto \cdot^2$ allows you to plot the parabola – and it's conveniently shifted to the correct place when you put the original abcissa beneath.

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    Here's another practical function to draw: output is health damage, input is number of cigarettes smoked per day. There is a kink or a bend roughly around 1 pack / day. It's a useful fact that couldn't be expressed any other way than math -- which answers the "what's the point?" question. Also allows you to mention technicalities about what set it's mapping from, transformed scale on the cigarettes (per day as opposed to total), all while being very concrete. – isomorphismes Aug 24 '10 at 22:39
  • If we're going for teaching kids, using cigarettes in your example would not seem to be a good idea. Just sayin'... – J. M. is not a mathematician Aug 24 '10 at 22:44
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    What! Kids deserve to know the truth and make decisions for themselves. (This is quite far afield though, especially since it's college Algebra.) – isomorphismes Aug 25 '10 at 6:26

Isaac's answer is almost exactly the first definition that I give. But what comes after is similar to what you are describing, I discuss each ambiguous term in the definition at great length, replacing the word with synonyms. Hopefully this lets the students take in the slight abstraction of the definition.

After I do the defining and explaining of words I do about 9 examples. Three from each of three classes:

little point-set diagrams, i.e. ovals with points inside them and lines going between them. algebraically with f(x)= graphs

Two from each class of example are functions and I point out all the parts of the definition and what they correspond to. The third example in each class is a non-example, i.e. not a function. I point out where the issue is, thus showing that one point has two arrows coming out of it, one number can be plugged in to get two, and the vertical line test respectively.

I find that this is very successful.

I start with the notion of an expression. An expression is a grammatically meaningful combination of variables and constants. I don't need to tell this audience what I mean by that. An equation relate two expressions. To solve an equation for a variable one manipulates the equation according to rules until one variable is written unambiguously in terms of the others. This may not be always be done.

A function is an equation in two variables in which one variable (y) can be solved as an unambiguous expression in the other variable (x). Thus y can be written as an expression in x. Then I let the students know (a) this definition is not quite good enough for mathematicians, (b) it will work pretty well for all of the applications that we have in mind. Throughout the process examples are given.

A function is something that takes a number , twists it around and spits out another number

Once they grasp that, you can talk about how you can have functions that work with things besides numbers. And how generally, a functions can twist around any object, tangible or mathematical and spit out another object.

Then you can move on to explaining the concept of domain and range. Domain is type of objects that your function can accept, and range is the type of object that your function can conceivably spit out.

A good concrete example. A soda machine is a function that maps (don't use that word though, it scares students at their first encounter with it) the domain of coins to the range of soda's.

Once they have the image in their head and some intuition THEN you go back and discuss the formal definitions.

At least that what I do with my algebra students

Why not use a real "function machine," which each of your students should have -- a scientific calculator? After all, most, if not all the functions in your course will be numerical examples.

For a function of one variable, use any of the trigonometric, squaring, cubing, square roots, or log functions. Use square root, inverse trig and log functions for examples of restricted domains of definition. You don't have to explain what these functions mean at first, just emphasize that you put in one number, and get another number back. Have students make a table of values using the calculator to reinforce this fact. Be sure to include examples that leave values unchanged, e.g. sqrt(0)=0, sqrt(1)=1, sin(0)=0, etc.

For functions of two variables, use the ordinary arithmetic operations: add, subtract, multiply, divide or exponent functions.

All of the above state the usual metaphors with more or less flair; but none convey the idea of a function. Functions have nothing inherently to do with machines or monkeys or rules (whatever they are) or black boxes (whatever they are) or inputs and outputs or the equals sign. There are reasons why we want the invention of a function. There are reasons why we need the invention of a variable. There is a reason why we need the notion of dependence that leads to the idea of an "expression" to provide what we want. The expression is a brilliant device; and it's a shame not to see that explicitly. There are reasons why we consider a constant to be a special case of a function. There are reasons why we commonly limit the word "function" to single valued functions. There are reasons why we expand the notion of a function to a function equation with an explicit dependent variable. And like other objects in mathematics, the same function can have different forms, algebraic and otherwise. Stating conclusions without their sense makes mathematics a mystique instead of a rational interesting science that we can think about and question and explore. One last point: The question should have been how to explain, not "define." No so-called "definition" comes out of nowhere. Presenting a "definition" like Athena springing full grown and armored from the head of Zeus is, to the great misfortune of most of us, standard practice in teaching mathematics.

One way I heard a lecturer describe functions recently was that of the CD player analogy.

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    And? How does this analogy work? – TRiG May 13 '13 at 14:26

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