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When I was a university student, I learnt that a function is the data of three pieces of information:

  1. the rule that tells how to associate an object $x$ to its image $f(x)$,
  2. A domain $E$ where live the values of $x$ that are transformed by $f$,
  3. and a codomain $F$, a space where all images can live.

So, an usual notation for an function is $ f:\begin{array}{rcl} E & \to & F \\ x & \mapsto & f(x) \\ \end{array}$.

But, in high school textbooks (at least the ones I know: Stewart's Precalculus, Larson's Precalculus and Axler's Algebra and Geometry), as well as undergraduate textbooks, a function is defined just by a formula, $y=x^2$ or $y=\sin(x)$. The domain is seldom part of the definition, but something the students have to compute, hence seen as an inherent property of the formula.

The codomain is, as far as I know, almost never mentioned. Most textbooks focus on the range. Since the range is the smallest possible codomain, I guess it may be for "economical" reasons, but it seems to me that it leads to misconceptions about functions. It also leads to shocking sentences like "the function is one-to-one, so it has an inverse" :(.

My question : what the pros and cons of not teaching the notion of codomain of a function?

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    $\begingroup$ A great question. Perhaps at least part of the answer is that $\mathbb{R}\to\mathbb{R}$ is so common it is overlooked in favour of instilling algebraic manipulation. In my opinion it is important to investigate functions that have radically different domains and codomains (perhaps assigning a primary colour to a number or something like. ) But I agree it definitely should be taught earlier. $\endgroup$ Commented Apr 3, 2015 at 14:43
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    $\begingroup$ I think the bigger problem is the one you point out "the domain is ... seen as an inherent property of the formula" which compounds students' confusion between functions and formulas (a confusion which is roughly ok till precalculus and then sets one up for failure in calculus and beyond). In fact, even through calculus, the codomain can be taken as $\mathbb{R}$ without serious interesting real-world-oriented counterexamples. $\endgroup$ Commented May 1, 2015 at 18:57
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    $\begingroup$ You don't talk about codomains in high school?! That's news to me. What country are you in? Here the concept of function is introduced in middle school, with domain and codomain and the two-line notation you used yourself in your post (although 95% of the functions encountered in middle school happen to be real linear functions). $\endgroup$ Commented Oct 1 at 9:27
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    $\begingroup$ @Stef: I am French and I was also told the full notation in high school in the late 90s (middle school was only dealing about linear functions if I remember well). I am now teaching outside of Europe using US textbooks (Stewart's Precalculus now, Larson's Precalculus and Axler's Algebra and Geometry before). I do teach a bit about codomains and onto functions as extra-material in my precalculus course. $\endgroup$ Commented Oct 2 at 4:00
  • $\begingroup$ The thing is, you can always limit the domain to whatever you want. The range comes from your domain. The codomain is kinda unnecessary unless you require conditions for bijections. $\endgroup$ Commented Oct 13 at 19:12

3 Answers 3

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Pros: People will potentially better understand ideas required for fairly abstract mathematics.

Cons: It pretty much doesn't matter to any field outside of pure mathematics. It would add to an already fairly gigantic list of things that we need to teach at the high school level. It is a pretty abstract concept to teach.

In general, range gets the concepts across well enough that if a student goes on to study mathematics in college they should be able to modify their definitions to include co-domain and properly apply it. Most of the country will never need that in ever. There's no reason to go beyond a basic overview of it at the high school level.

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    $\begingroup$ Definitely matters outside of pure mathematics. Keeping careful track of domains and codomains is important in programming. Especially in functional programming. $\endgroup$ Commented Apr 4, 2015 at 5:27
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    $\begingroup$ Forgive me for kind of venting my frustration on a random post, because this really has to do with a whole lot more. Anyways, what I want to say is that we should all benefit greatly from dispelling the "usefulness" criteria in choosing a mathematical curriculum. First of all, the actual set of topics for any school subject, is rarely what is "useful" later in life. Rather, general notions and skills (still regarding any subject) are what we gain. In mathematics, said notions or skills are simply the act of thinking. Just thinking, defining, laying out a scheme or what have you. $\endgroup$ Commented Apr 6, 2015 at 18:08
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    $\begingroup$ ... This goal I think is best achieved by presenting concepts in as loyal a form, to that as which they appear in "real mathematics", as possible. $\endgroup$ Commented Apr 6, 2015 at 18:10
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    $\begingroup$ @GPerez If we dispel the “usefulness” criteria, then we probably wouldn’t need a curriculum at all. Why talk about mappings at all and not simply spend math class solving puzzles or playing logic games like Sudoku or working on a Rubik’s Cube. I’m not opposed to that, but perhaps there’s a way to combine what you call “thinking” with content that’s also “useful.” $\endgroup$ Commented Sep 30 at 7:24
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    $\begingroup$ I mean, a curriculum has other benefits, like not leaving teachers to have to do all of the planning themselves. I don't remember exactly what I had in mind 10 years ago, but yes I agree a combination is probably the best $\endgroup$ Commented Oct 2 at 12:46
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Codomain may be a relatively recent precision of the language.

You're interpreting "range" as only meaning the image, but "range" has also been used to mean the codomain. according to a couple of cites here:

Wikipedia for Range of a Function.

I agree it can be troublesome when the books are not carefully distinguishing these ideas. However, many students will be tasked with domain and range concepts from pre-algebra courses when their knowledge of functions is usually limited to discrete cases or linear real-valued functions. In later courses, students will see and write things like $\frac{x^2}{x-1}$ is a function from $\mathbb{R}\to\mathbb{R}$. This kind of serves as describing the function as "real inputted" with the origin set, and it provides the codomain explicitly. (Students are usually tasked to find the domain (meaning pre-image) and range (meaning image)).

A source of these confusions may be that the codomain is so often $\mathbb{R}$, that it is deemed automatic.

One way to tease out the difference with a class is to discuss the function of $student \to birthmonth$. Here you can say that the domain is students in the class (noting that you, the teacher, is not an element of the domain). The codomain is the set of twelve months. And the image is the actual set of months acquired, which may not be the entire codomain. (best if your students don't cover all 12 ;)

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  • $\begingroup$ A great idea. You could then change the codomain to the season you were born in. $\endgroup$ Commented Apr 5, 2015 at 16:58
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Why codomain might not be taught

A couple of practical reasons why some thing is not pointed out:

  • The thing is the same in all cases encountered.
  • The thing makes no difference in actual practice.

These principles apply beyond teaching mathematics. One of issues for learners is that one becomes habituated to the seeming uselessness of an unused idea.

I think codomain falls into that category in courses on the high-school algebra thru single-variable calculus track. I don't recall having seen it until later. (This would have been in U.S. schools and international schools with a "U.S." curriculum in the late 1970s.) My current students usually have not seen the word "codomain" or admit to not knowing what it is. The mathematics-interested students in Linear Algebra right now seem interested in the concept and wonder how in the world the idea of codomain could be useful enough for them to have to learn it. Luckily, it is a useful idea in L.A., and I am glad they have such curiosity. I am satisfied that now is the time they are first learning it, and it doesn't seem to be a problem that they have not seen it before.

Historical observation

The OP remarks, in a deprecatory way, about someone who might say, "the function is one-to-one, so it has an inverse." Well, the University of Illinois Committee on School Mathematics (1959) defines a function to be "a set of ordered pairs no two of which have the same first component." The domain and range are derived from the set of ordered pairs. "Codomain" is not mentioned. A function has an inverse if the set of order pairs derived from the function by reversing the order of each pair is itself a function. Thus, in this presentation, if a function is one-to-one, it has an inverse. One thing to keep in mind is that people who were taught this approach are probably still teaching mathematics.

Precalculus motivation

I think the description of the codomain often found in internet searches is confusing. The one I find says that the codomain is the set of all possible values of a function and the range is the set of all actual values of the function. To my mind, the actual values are the only possible values, and I don't see any distinction between codomain and range being made. I have a different way to describe the difference.

Functions in precalculus often arise in the form $y = f(x)$, that is, they model equations. As equations, the value of $y$ is given and $x$ is to be determined. So for each value of $y$, we obtain a problem $y=f(x)$ to be solved. Then the codomain represents all the possible problems that may be posed, the range represents all the problems that have solutions, and the domain represents the set of solutions to all the problems.

For instance, $-1=x^2$ does not have a solution, but $4=x^2$ does. And saying, "there are no real solutions," is a common response to an equation.

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  • $\begingroup$ (1/n) Thank you for sharing your experience. I have some issues with the second part: I have seen the definition of function you mentioned in a set theory context (IIRC, in a textbook by Delahaye), or in an affine geometry course where it is called partial function, but never in a Calculus textbook. Also, why choose a 65-year-old book? That sounds like cherry-picking, the "people who were taught" with this book are probably retired now... (I am not saying that we should discard a point of view simply because of its age). $\endgroup$ Commented Oct 12 at 5:00
  • $\begingroup$ (2/n) More importantly, if "a function is a set of ordered pairs", in which Cartesian product are these ordered pairs living? It does not change that a function is a triple $(A,B,f)$ where $A$ and $B$ are sets, and $f$ a subset of ordered pairs of $A\times B$ (also known as a relation) with some properties. Changing one may change the function and its properties. $\endgroup$ Commented Oct 12 at 5:01
  • $\begingroup$ (3/n) As defined in the question, a function is a relation with two properties; and it is invertible if its inverse relation is itself a function. It is no surprise that if you remove one property (changing the definition of a function to a "partial function"), then the criteria for being invertible is weaker. $\endgroup$ Commented Oct 12 at 5:03
  • $\begingroup$ (4/n) My second problem with the answer is the last part. Mathematicians in the late 19th-early 20th centuries (I think; I am bad at History of Mathematics) made efforts to distinguish equations and functions, and I think it is a good thing. So the discussion on equations seem beside the point to me. (We can use functions to understand equations, but that just reinforce the need to have precise terminology to speak about functions). $\endgroup$ Commented Oct 12 at 5:10
  • $\begingroup$ (5/n) Finally, I agree speaking about the codomain as the set of all possible values and the range as the set of actual values is confusing, and I avoid using it. And I should have formulated the definition of functions better in my question (starting by introducing $E$ and $F$). $\endgroup$ Commented Oct 12 at 5:15

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