Talk:Inverse function

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Mathematics rating:	B Class	High Priority	Field: Foundations, logic, and set theory
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Please update this rating as the article progresses, or if the rating is inaccurate. Click to show/hide comments. Please add to or update the comments to suggest improvements to the article. Upgraded to B class. Needs better references. Jim 09:14, 16 September 2007 (UTC)

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/Archive 1

Contents 1 Fixed subtle logic flaw in the definition 2 Major expansion and revision by Jim Belk 2.1 After a new important change by Jim 2.2 Suggesting a method to save the best of previous and new contents 3 Section by section review 4 Function and Inverse function 4.1 Comparison between original and new definition 4.2 Summary of previous debate about introduction and definition 4.3 Jim's point of view 4.4 Secondary discussion about definition of injection 4.5 First step 4.6 What's the best recipe for the main definition? 5 A new beginning 6 The introduction 7 Unnecessary reference

[edit] Fixed subtle logic flaw in the definition

I believe there's a subtle logic flaw in this definition:

Formally, if f is a function with domain X and range Y, , then f ^{- 1} is its inverse function if and only if for every we have:

and for every we have:

Notice that Y is explicitly defined as the range of f, and at the same time used as codomain of f in . Thus, actually the definition contains three conditions, in this order:

1. the codomain of f must coincide with its range
2. f must be "reversible" (it must be possible to undo it)
3. f ^{- 1} must be also "reversible" (with )

The problem is that, as far as I understand, the first condition is not necessary in the definition, because the other two conditions are sufficient to define a (fully) invertible function. Condition 1 is a consequence of 2 and 3.

Another problem is that the definition does not explain clearly the reason why an injection is not invertible (unless it is also a surjection and hence a bijection). I mean that a non-surjective injection is immediately rejected because it does not meet condition 1. On a didactical standpoint it is advisable, in my opinion, to skip condition 1 and realize that a non-surjective injection meets condition 1 but does not meet condition 2.

Also, as explained here, if you arbitrarily decide to replace the codomain of a function by its range, any injection becomes a bijection. The definition seems to suggest not to worry about injections, because you can use a trick to turn them into bijections...

I edited the article and moved condition 1 in the section "Properties", expressing it as just one of the many consequences of 2 and 3. Please correct me if I am wrong.

Paolo.dL 18:18, 1 August 2007 (UTC)

All the above is correct. But the addition of the fact that an invertible function's range must equal its codomain is equivalent to saying it's onto or surjective, but this property is already covered in the "Existence" section which says a function is invertible iff it is a bijection (= surjection + injection). Paul August ☎ 18:52, 1 August 2007 (UTC)

This partly coincides with what I wrote: the property R = Y (i.e. condition 1) is actually a consequence of the true definition (conditions 2 and 3), and the sentence "a function is invertible iff it is a bijection" is just an "encoded" way to enunciate the definition.

But you are right: the property R = Y is immediately implied by the fact that an invertible function is onto, and this is clearly stated in the "Existence" section. However, actually that property is not explicitly stated in the article, and I believe it should be. Some readers (like me) may fail to see immediately the meaning of the word "onto" and may need some help in the decodification. I moved my sentence in the "Existance" section, and condensed it. Let me know if you agree. Thanks for your feedback. Regards, Paolo.dL 19:02, 1 August 2007 (UTC)

This is better. By the way I didn't explicitly state it before but your change of "range" to "codomain" in the definition, was a good catch and an important correction. Paul August ☎ 21:54, 1 August 2007 (UTC)

Thanks both for your encouragement and for your precious contribution. It's a pleasure to be of service, receive useful advices, and find some friendly editor. With kind regards, Paolo.dL 23:57, 1 August 2007 (UTC)

Final refinements. I refined and rearranged sections "Definition" and "Simplifying rule", and renamed the latter to "Equivalent definition". (11:56, 2 August 2007). I also moved again the above mentioned sentence about R = Y into the "Properties" section, and inserted there a second sentence about X and Y having same cardinality (it is another consequence of being bijective). Paolo.dL 08:51, 4 August 2007 (UTC)

[edit] Major expansion and revision by Jim Belk

Note to Readers: All of the ensuing discussion concerns a revision that I made on September 16, 2007. For reference, here is the revision:

• 04:12, 16 September 2007

and here is the version from before the revision:

• 17:31, 14 September 2007

Jim 17:20, 22 September 2007 (UTC)

I think that the introduction and the main definition were much clearer before. The introduction became ambiguous again, as it was some weeks ago. If a function is "reversible", this does not mean that it is invertible. At least this is what you learn when you read the article. But the first sentence in the lead states the contrary (i.e "reversible = invertible"). Either you say that two different definitions exist (see talk:function (mathematics)), or you use in the lead a definition equivalent to the one you use in the article.

There was a major expansion and several figures were added. I have not read all the new article, but I think that the lead and the definition in the previous revision were a much better starting point. The previous version was also better wikified and the language was more accessible.

Also, it is not clear how the rule given in the definition implies symmetry and why it is equivalent to the rule given in "Characterization". The latter, in my opinion, is much more readily understandable by non-mathematicians, because it makes the symmetry immediately evident. This "characterization" was the main definition given in the previous revision. Paolo.dL 21:00, 16 September 2007 (UTC)

I'm sorry that you dislike the new wording of the introduction—it's hard to strike exactly the right balance between precision and clarity in an introductory paragraph. My feeling is that the first few sentences are too early in the article to be making the subtle distinction that you're concerned about.

There are two reasons that I switched the definition to current one (i.e. that ƒ^–1(y) = x if and only if ƒ(x) = y). First, I think the new definition is a bit more concrete. After all, a function is primarily a mapping of elements, and the new definition states explicitly which elements map to which under ƒ^–1. Second, the definition in terms of compositions requires a good understanding of the composition of functions. The new definition does not, which makes the article somewhat more accessible. It is true that the old definition makes the symmetry between ƒ and ƒ^–1 more apparent, but I'm not sure that this should be the most important consideration.

I don't understand exactly what you mean when you say that the previous version was better "wikified". I've not encountered this criticism before, and I'd love some suggestions on how to improve the syntax of my editing. (I've only been editing for a few months now, so I'm still not completely familiar with the wiki markup.)

As for the language, I'd be happy to work together on improving it. Because of the major expansion, much of the language in this version is new, and could probably use revision and clarification by several different editors. My main goal here was to expand the article, and to suggest some new possibilities for the introduction and definition. Jim 00:17, 17 September 2007 (UTC)

I like the informal introduction. I believe that in the intro it is better to keep things informal rather than precise. Jim, thank you for the many very nice pictures also. Oleg Alexandrov (talk) 01:13, 17 September 2007 (UTC)

The article is nice now. Thank you Jim. Sam Staton 14:22, 17 September 2007 (UTC)

What about an informal, simple, and also non-ambiguous introduction? The previous one was like that. Basically, it was: "The inverse of a function f can reverse and at the same time can be reversed by f". This is the concept that now is missing. Another useful sentence was: "Note that a "reversible" function is not necessarily invertible."

It is difficult to understand how the new rule "answers" to the following question: is a square root (or any non-single-valued function) invertible? The answer is not immediately evident. Many readers are likely to read only the intro and the definition of the rule (skipping the rest of the article) and they might not see that a square root is not invertible.

Possibly, you don't think that this "symmetry" or "bi-directionality" of the inverse is important because you are mathematicians, and you already know it. But I have seen a lot of non-mathematicians who confused the concept of (monodirectional) "reverse" with the concept of (bidirectional) inverse. Paolo.dL 07:41, 17 September 2007 (UTC)

I've added two more examples to the "Definition" section to try to address your concerns, one of which is the function ƒ(x) = x². Jim 17:04, 17 September 2007 (UTC)

"The inverse of a function f can reverse and at the same time can be reversed by f". This is the concept that now is missing. Another useful sentence was: "Note that a reversible function is not necessarily invertible."

Well, I think this is kind of confusing. You mean to say of course that reverse is left or right inverse, and to be reversed is the other one. But I think these fine points will be lost on a reader, and instead of thinking that function inversion is same as reversion the reader will be forced to consider what is the difference between reversing and being reversed. Oleg Alexandrov (talk) 17:11, 17 September 2007 (UTC)

I went through the source and changed &fnof to ƒ, &ndash to –, and &mdash to —. Does this help the wikification? Jim 17:16, 17 September 2007 (UTC)

Oleg, thanks for sharing your doubts. My point is that you are actually thinking the opposite way, with respect to the average reader. When you read, you already know that "inverse" is a bidirectional concept. A non-mathematician knows very well the concept of "doing-undoing", or "forward-reverse" or "forward-rewind". And in english, the concept of inverse is monodirectional. It is a synonym of reverse. In the mind of the average reader, "forward-reverse" and "forward-inverse" are the same thing. In your mind they are not, but you need to force the reader to accept a new definition of a word that is commonly used in current english with a different meaning. The "difference between reversing and being reversed" is quite simple. And I think that this is right the point that the reader must be forced to consider. Personally, I was very surprised to learn that inverse was a bidirectional concept in mathematics (by the way, Rick Norwood wrote that, unfortunately, an alternative definition exists; see Talk:Function (mathematics)). Please let me know your opinion.

As for the new definition, an example may be useful. Please compare:

where squareroot(x) = (multi-valued). It is difficult for a non-mathematician to see the reason why the second is wrong. Whatever is the sign I arbitrarily choose for the square root, the square can correctly reverse it. If you use the old definition (or at least you accept to change the intro as I suggested), the message is explicit and this mistake becomes impossible.

Wikipedia is for those who don't know. It is not as a book, where the author assumes that the reader knows what is written in the previous pages. Paolo.dL 17:52, 17 September 2007 (UTC)

Jim, I used the word "wikify" to mean "adding internal links". But this is not the important point. You did a very good job. The readers will fix the details. I mainly would like to discuss your choice for the intro and main definition. Clear examples are useful, but the point of symmetry was already explained in your first version of the definition. The problem is that, although it is quite an easy concept, it is not explained in the intro.

Also, the examples should be easily deducible from the rule. I believe the best definition is the one which makes the examples obvious and not indispensable.

Moreover, your example about non-invertible functions is useful but not sufficient. Nobody will ever maintain that a square is invertible, because the square is clearly not even (unidirectionally) reversible. The problem is to answer my question about multi-valued functions. Paolo.dL 18:30, 17 September 2007 (UTC)

Paolo, the thing is that a multivalued function isn't actually a function (see the article on multivalued functions). The reason that squareroot(x) =  doesn't fit into the definition is that both ƒ and ƒ^–1 are required to be functions. I have changed the wording of the "note on functions" to try to make this clearer. There is an article on inverse relations which might be a better place for the distinction you're suggesting. Jim 22:44, 17 September 2007 (UTC)

I still think the current intro identifying inverse with reverse is best. Oleg Alexandrov (talk) 14:12, 18 September 2007 (UTC)

Assumptions and deductions. That's better, but the main problem remains not addressed, and I still need to deduce something that is not explicitly stated by the definition. In the introduction, for instance, you still assume that the readers know that you are using the word "function" with its proper strict meaning. But they don't. And they typically don't know the strict definition of function. The readers of an encyclopedia typically have not read the "previous chapters of the book". By the way, as you know, multivalued (non-single-valued) functions and partial (non-total) functions do not comply with the strict definition of function. The word function is not always used "properly" by mathematicians. Why should I assume you are using it properly, this time? Why should I assume that the adjective "inverse" is not "weakening" the strict definition of the word function? (see comment by Trovatore dated 15:01, 11 September 2007 on Talk:Partial function).

Similarly, you wrote that your rule implies that an invertible function must be bijective. This seems to be a conclusion that you can draw only if you assume that partial functions are not considered to be functions. If you read the previous version, you will see that these deductions were not required.

Readability. Also, your definition is quite difficult to read:

• f^-1(y) is equal to x iff the f(x) is equal to y

Consider what I think when I read it. First, I am puzzled by the iff. Then I wonder: "wouldn't it be clearer if it were stated as follows"?

• the inverse of f is a function f^-1 such that, if f(x) is equal to y, then f^-1(y) is equal to x.

The definition given in the previous version of the article was absolutely clear. Nothing puzzled me. I did not need to spend time trying to guess why you used iff rather than if, and I did not need to turn the sentence upside down to understand it. Honestly, decoding your definition required ten times the time needed to decode the previous one (and after decoding it, I am still puzzled by the iff...; there must be something hidden behind it! probably something that I won't be able to remember because I can't understand the rationale...; probably a trick :-).

Conclusion. I am still convinced that you did a wonderful job, but the authors of the previous version did a better job when they wrote the introduction and the definition section. Their introduction and definition were not perfect, but less ambiguos, more easily readable, and did not require assumptions and previous knowledge. It was a lot of perfectible but good work. Our mission should be to make things simpler for the readers. Paolo.dL 10:15, 18 September 2007 (UTC)

OK, I think I might see what you're saying, so I've changed the language in the definition to be a bit more straightforward. What do you think? Jim 15:24, 18 September 2007 (UTC)

[edit] After a new important change by Jim

Jim, I appreciate your efforts. Now your rule is much better and I can see why you like it. And I like it too. But not as much as the wonderfully simple definition given in the previous article version.

There's still a problem in your definition. The rule appears to be generic and you need to complete it by immediately referring to the concepts of injection and surjection, which are extremely complex for non-mathematicians, and which most people don't know. I remember that I studied them three times, in different periods of my life, before fixing them in my mind. I could permanently learn them only when I started editing this article (see discussion with Paul August above). In the previous version of the article, the connection with the concepts of surjection and injection was given in a separate section, with useful explanations which helped me not only to better understand the definition of inverse function, but also the definiton of injection and surjection.

I noticed that the condition invertible = bijective disappeared. Your definition now seems to accept non-surjective functions. But this is not coherent with the "characterization" given in the ensuing section (which coincides with the definition given in the previous article version):

How can you fail to see how much simpler and more explicit this definition is, compared to yours? It was so easy to see in this definition the requirement of symmetry! And now I can't even understand whether it is a requirement or not.

You are still ignoring the fact that there are two different definitions in the literature ("invertible = injective", and "invertible = bijective"). And you are still ignoring my comments about the introduction. You are free to do whatever you like, of course, but please consider how difficult it was for you to understand the complexity of your rule, which was absolutely evident for me. You and Oleg are trained mathematicians. You simply don't care when you see a definition turned upside down or incomplete or based on implicit assumtions. I do the same when I read words with typos.

Forgive me if I share a negative thought, but sincerely I am afraid that you are not going to understand the point of view of the generic readers. It is a very hard job. Only the best teachers can do it. And until you fail doing so, you won't be able to make the article as useful as it was before. Generic readers will be puzzled and will fail to extract the basic points from your collection of assumptions and unneeded references to other concepts that they don't know. On the other hand, mathematicians will keep using their favourite textbook... but I hope I am wrong. Luckily, nobody else seems to share my negative opinion.

Please consider that I am only referring to the changes in the introduction and definition. I still believe that your expansions and new pictures were a good job. Paolo.dL 22:01, 18 September 2007 (UTC)

Arbitrary deletion 1. By the way, you deleted the section on "Properties", which was very useful to me when I read the paper for the first time. Paolo.dL 22:24, 18 September 2007 (UTC)

I agree with some of your criticisms, so I have reworded the definition yet again. The discussion now reviews the notions of one-to-one and onto, it mentions the connection between invertible and bijection, and it explicitly states that non-onto functions are still sometimes considered invertible. I'd like to thank you for your many comments—I think this version of the definition is much better than the one I originally posted.

I still agree with Oleg that the current introduction is probably for the best. The introduction is a place for communicating the main idea of inverses as succinctly and clearly as possible, which is what the current introduction accomplishes. The distinction between "reverse" and "inverse" is important, but is also subtle, and comes from a deep understanding of the two concepts. I don't think it helps to mention it at the very beginning.

I didn't really delete the properties section—the properties are now incorporated into text of the article. Many of the properties on the list that were previously covered in one sentence now have their own subsection. It's true that others have been omitted, so let me know if there are any that are missing that you'd like to see discussed in the article. Jim 04:39, 19 September 2007 (UTC)

The important sentence for the introduction is this:

• "The inverse of a function f can reverse and at the same time can be reversed by f".

So, you think that this sentence is too difficult? Well, then let's delete the entire article, because people cannot understand the meaning of the word reverse, nor the difference between an active and a passive tense!

Arbitrary deletion 2. You also removed this sentence:

"The superscript "−1" is not an exponent. Similarly, except when dealing with trigonometry or calculus, f ²(x) means "do f twice," that is f(f(x)), not the square of f(x). For example, if f : x → 3x + 2, then f ²(x) = 3 ((3x + 2)) + 2, or 9x + 8. However, in trigonometry, for historical reasons, sin²(x) usually does mean the square of sin(x). The prefix arc is sometimes used to denote inverse trigonometric functions, e.g., arcsin x for the inverse of sin(x). In calculus, f ⁽ⁿ⁾(x) is the nth derivative of f."

Today, Norwood added a new sentence (a WARNING) explaining this concept. But this sentence explained it much better. I liked it a lot when I read it some months ago. Possibly, many participated in its compilation. But you deleted it without even bothering to explain why! You summarily and arbitrarily removed a lot of interesting stuff. And we don't know what you removed because your edits where so dramatically extensive that the history comparison became useless. Paolo.dL 13:46, 19 September 2007 (UTC)

[edit] Suggesting a method to save the best of previous and new contents

I would love you to restore the previous version, and just add your new sections, without deleting the previous ones. Your definition can be added to the "Alternative definitions" section. After that, section by section, you will decide, by discussing here (or just reporting), section by section, separately, whether to delete previous contents or new contents in case of redundancy. —Preceding unsigned comment added by Paolo.dL (talk • contribs) 14:05, 19 September 2007 (UTC)

Look, Paolo, I'm very sorry if I stepped on your toes here by rewriting this article. You and the other editors obviously put quite a bit of thought into the old version, and I tried to incorporate as much of that thought as possible in the rewrite. Though it may not look it, I arrived at the version here by methodically going through the old version and asking questions like "Should this part be expanded?", "Should this be explained more?", and "What can I do to make this part better?".

I put a lot of effort into this, because I really want to make this article better. The current version is nearly twice as long as the version from a week ago. It has considerably more content, several new pictures, and generally clearer explanations. The terminology is now more standard, and the emphasis on different ideas more closely aligns with what you would find in a mathematics textbook. For example, the definition of inverse function given here is the same one used in Stewart's Calculus, Munkres' Topology, and Strichartz's The Way of Analysis, which are the first three books that I checked.

My philosophy in rewriting this article was that I should be bold in making changes (see Wikipedia:Be bold). I'm a professional mathematician known for good expository work, and I would be qualified to teach this material in an undergraduate course or to write a mathematics textbook that covers this material. For this reason, I consider myself qualified to make major changes to Start-class mathematics articles. My editing here was not meant to show any disrespect to you or disdain for the previous version—I simply wanted to improve the article to be a professional-quality exposition of inverse functions.

I would strongly object to your proposal to revert the article to its original form. Frankly, I think you are being overly possessive about the orignal contents of this article (see Wikipedia:Ownership of articles). In my opinion, most outside observers would agree that the current version of the article is a major improvement over the previous version. Given this, I think we should work from this version if we want to improve the article further.

If you are upset about things that were removed from the old article, I invite you to either bring them up on this talk page or insert the original content directly into the current version of the article. I've already changed this version a lot in response to your comments, and I am happy to continue doing so. However, I really don't think that we should "start over" from the original version. Please don't let this pleasant and constructive discusssion devolve into a revert war. Together we can make this article better. Jim 18:34, 19 September 2007 (UTC)

I did not propose to revert the article to its "original" form, nor I suggested to "start over". Contrary to what you did to the previous version, (which forces me to "start over"), my proposal was absolutely conservative with respect to your expansions and most of your changes. But don't worry, I won't fight. I gave you my suggestion, you did not want to accept it, and that's it.

No, I was not upset, but I was very disappointed about the fact that, as I showed, there are at least two parts that were extremely useful in my opinion and now are missing, and I don't know how many other (possibly interesting) parts you just deleted, because you unilaterally thought that they were not necessary, and indeed they were not such for you, for other mathematicians, and for your students, but they possibly were extremely useful for the readers of Wikipedia. And I am not the only one who misses the explanations that you deleted (see latest edit by Norwood). I guess that most of our readers are neither mathematicians, nor undergraduates in mathematics who thake classes in the required order according to a program. This does not allow me to trust your judgement, and that's why I proposed a method to discuss your changes (and your deletions) section by section, separately.

Yes, you did a good job, possibly now the article is globally better than it was before, but in some parts the old version was better or, as I showed, was simply more complete. Of course I see that your intentions are good, but your method was destructive, and I don't like it. I don't like at all working this way. Why should I bother improving this version, as I did for the previous one, if some other mathematician will be allowed in the future to unilaterally decide such a distructive change? I don't want to be forced to put together the broken pieces of my previous work again and again. Of course I am aware that what I do can be undone, but I would like to be able to understand the reasons or discuss the changes section by section, if needed. You made this impossible. You even changed the names of the sections. Paolo.dL 20:41, 19 September 2007 (UTC)

I'm sorry—I got a bit upset when you suggested reverting the article. I'd be happy to discuss the article section by section with you. I think it's great that you have an outside perspective, since the goal is to make the article as accessible as possible to a general audience. If we work together, we ought to be able to make this article meet the Good Article criteria. Jim 22:39, 19 September 2007 (UTC)

[edit] Section by section review

For reference, here is what the article looks like right now:

• 16:25, 19 September 2007

and here is the version from before the revision:

• 17:31, 14 September 2007

For now, I think we should put the discussion of the introduction and definition behind us. We've argued it to death, and it's not clear that we're ever going to agree. Instead, let's start by focusing on the new material by Rick Norwood, and the related material that you have re-inserted from the old version. (Possibly next we can discuss the "Properties" section from the old article, and which of those properties appear here.)

I think that this material is important for the article, and I regret leaving it out of my rewritten version. I have the following general comments and questions regarding its current form:

• I agree that "notation" should be a subsection instead of just a "note". It's even possible that it should be moved out of "Inverses and Composition" and into its own section. I'm actually not that happy with the "inverses and composition" section of the article: it's possible that the "characterization" should be its own section, and the "inverse of a composition" should be part of some sort of "properties" section. What do you think?
• Second, what would you think of moving the paragraph on multiplicative inverses forward to the beginning of the notation subsection?
• I've noticed that the article on function composition doesn't discuss the analogy between composition and multiplication very coherently. If we put a longer discussion of the analogy in that article, we could link to it from here.

I also have some issues with the current wording, but it seems like we should figure out the best organization first. Jim 22:39, 19 September 2007 (UTC)

You ignored some of my previous suggestions. I mean, you may disagree, but at least I would like to know why. Moreover, the introduction and definition sections are still much worse than the previous version, in my opinion. You maintained that your one-sided definition was adopted in the first three books you happened to browse. Can you deny that the two-sided definition is used by other authors? I thought we were discussing about what's the best definition for an article on an Encyclopedia, rather than for a university textbook. I wanted to discuss about the rationale for a decision in this context, not about statistics on the decisions taken by others in a different context. I don't like working this way. Others can help you, if they like. [By the way, the subsection "characterization" was already a separate section before your revision.] Paolo.dL 10:01, 20 September 2007 (UTC)

[edit] Function and Inverse function

Over at Talk:Function we have been discussing inverse functions as well, and rather than have two separate discussions I thought it best to come here.

My first remark is that I very much appreciate the figures user Jim.belk (talk · contribs) has added. They substantially enhance the article, and I know from personal experience the kind of work that can go into illustrations. I would offer three (minor) suggestions. (1) Place the figures in Wikimedia Commons so that they can easily be shared. (2) Either use the SVG vector format instead of the PNG raster format, or use a higher image resolution (and let the MediaWiki software automatically reduce to desired size). (3) In the first image, Image:Inverse Function.png, flip the bottom half so we see the arrows reverse (admittedly a matter of taste).

My second remark is that I also appreciate the effort Jim has made to rewrite the article. Some qualities have been gained and some lost, but on the whole it is an improvement. I see evidence of a struggle to lie comfortably in the Procrustean bed of wiki mathematics markup, and also a struggle to accomodate the awkward expository demands of a Wikipedia article; these struggles are always with us. I'll try to address both below.

My final observation is that, among mathematicians, there is little confusion about what we mean by "inverse function", but we can easily overlook variations, and we can easily confuse non-experts (like Paolo). This relates to our expository struggles.

Let's take the easy stuff first: markup.

• I was uncomfortable seeing "ƒ" used inline, even through it reads more clearly than "f", because of concerns that for many readers the visual result would be a missing character glyph. However, a quick check suggests that most fonts actually do include the variant character, so it may be safe. When using special characters with HTML entity names (like ƒ), feel free to use either direct UTF-8 or the name, as you prefer. (See here for a table, with available names marked.)
• Our common practice is to use TeX markup for displayed equations. It's bigger, prettier, and suffers no baseline alignment problems, unlike TeX inline. At the moment, I see only a handful of violations, but they stick out.
• Conversely, our common practice is to avoid TeX markup inline. This is true even if the TeX does not generate an image. Thus we prefer −√x to , for example; and we also prefer 2π to 2π. Again, the few violations stick out.
• A few exponents have special glyphs, but we choose not to use them because of readability and consistency. Thus we prefer ƒ² to ƒ². Again, …
• We have a pretty way to write inline fractions so long as they are simple enough. We write the numerator as a superscript, the denominator as a subscript, and use the ⁄ (fraction slash) character between them, like so: ¹⁄_x. I don't think this is documented anywhere, but it is nice to know.
• My practice is to indent a table rather than center it. Since we do not center equations, this seems more consistent; but tastes vary.
• I see use of the alignat rather than the align environment; the latter is simpler and handles the line spacing automatically.
• TeX's \quad (or \qquad) is easier and prettier than using a table.
• Although it is not wrong, it seems unnecessary to place the trailing punctuation (comma or full stop) in \text.
• It is slightly preferable to use "\,\!" instead of "\," alone to force PNGs.

Now for the challenges of writing. As always, the guiding principal is to know what we have to say and to whom we are saying it. In writing for Wikipedia, this can be extraordinarily difficult. Among mathematical articles, a frequent exemplar for this problem is the beginning, especially the first sentence.

• In deciding what to say, we confront a problem uncommon to books and journals: Mathematics is broad and inconsistent in its definitions and notations. Much as we like Humpty Dumpty's approach ("When I use a word, it means just what I choose it to mean — neither more nor less"), here we are documenting common (and sometimes uncommon) practice. Thus we must describe and explain the common variations. What a pain.
  1. We may be unaware of variations we ourselves do not use.
  2. We may forget variations we use rarely.
  3. We may need to establish how commonly a variation occurs.
  4. We may confuse the hapless reader by trying to document all the variations.
• Our audience is hopelessly broad. We may have a curious youngster, a university undergraduate, a teacher, a professional mathematician, a physicist or chemist or engineer or other science professional, a scholar of ancient texts, a taxi driver, or a brilliant housewife. Furthermore, some do not read English easily. Egads; how can we possible write for all of them at once?!
• Wikipedia is an encyclopedia, not a textbook; but what are the implications of that? If we state bare facts without explanation or examples or gradual development, we abandon many of our readers. Yet it is impractical to fill in all the necessary background, and we wish to document the complex as well as the simple.
• A first sentence is not an article; rarely can it carry all burdens. Do we want a meticulous formal definition? (What if we have more than one?) Do we want a gentle introduction? (What should we assume known?) Do we speak informally and intuitively? (What if the informal language suggests invalid inferences?) The fact is, we need a full article, not just one sentence or one paragraph, to do justice to our subject. And the mathematics community has decided, as documented in our manual of style, that it is best to start gently and informally.

I have improved some of the markup, and am willing to participate in some of the content discussions. The first question I would ask is, setting detailed wording aside, is the content and organization adequate? For example, an earlier version had a "Properties" section; have we lost anything important? As for details, the tricky topics to cover include bijections, one-sided inverses, and partial functions: what do we say when and how? This is stuff a bright ten-year-old should be able to understand; with respect, perhaps Paolo can act as a surrogate until one comes along. Anyway, it's not uncommon to see wording tweaked by an endless series of editors, with no version clearly superior to the others. At some point we should declare "good enough" and move on. --KSmrq^T 21:54, 20 September 2007 (UTC)

[edit] Comparison between original and new definition

KSmrq, Thanks for coming along. I am mainly interested in the introduction and definition. These must be extremely clear. Everything else stems from a clear and explicit definition. An "alternative definitions" section should then give non-equivalent (if they exist) and equivalent different definitions. But the main definition should be clear enough to be understood by a bright ten-year-old, as you wrote.

I beg you to read this with attention:

• Enigmatic sandwich The definition in the old version was very clear. And it was not based on the concepts of injection and surjection or on the strict definition of function. After reading it, I immediately knew that a square was not an inverse of a square root. And I didn't need to deduce it based on the concept of true function, and on the assumption that an inverse function must be a true function (as opposed to a partial function or multivalued function). Now, I read Jim's definition and I am lost, and one of the reasons is that it is in the middle of a sandwich between three other definitions (see below for other reasons). By the way, Jim's definition of injection is wrong! [I am sorry, I wrote this stupid sentence at about 3 o'clock in the night (UTC+2); Jim's definition of injection is perfectly correct! - Paolo.dL 13:31, 21 Sept 07]
• Arbitrary deletion 3. Before Jim's revision, a separate section explained brilliantly why only an injection can be reversed and only a surjection can reverse. This explanation is now lost (yet another desaparecido). The obvious conclusion was that an inverse function must be bijective. Another section defined left-inverses and right-inverses, and that definition was wonderfully easy to understand because it was very closely related to the concepts explained previously with other words. The mosaic tesserae were adjacent and ordered from the lighter to the darker. At the end, you were able to see in the dark. The difference with respect to the new version is evident to me. Rather than a mosaic, it resembles a puzzle to be solved by the reader.
• One-sided vs two-sided. You know what's the problem in the new definition: Jim was not able to explain the connection between the monodirectional (one-sided) definition which he decided to adopt and the bidirectional (two-sided) one which was previously used. Jim wrote that they are equivalent, and this is just unbelievable according to what I know. So, either there's something I don't know, or Jim is wrong. In both cases, he didn't do a good job when he wrote the definition. The difference with respect to the old version of the definition should now be evident.

Until you write a clearer definition, I believe that this article will be useless to non-mathematicians. Paolo.dL 01:14, 21 September 2007 (UTC)

[edit] Summary of previous debate about introduction and definition

Thank you KSmrq for your many helpful comments, and for fixing some of the markup in the current version of the article. On the subject of the pictures: I've recently been working on options to convert pictures that I make to SVG. I changed the pictures I made for the linking number article earlier today (and uploaded the SVGs to the Commons), and I should getting around to fixing these sometime in the near future. I also like your suggestion for the initial picture—I'll implement it when I change to SVG.

On some of your typographical comments: I was a bit worried about the ƒ also. If you think that it is likely to present a problem, we could easily go through and change back to f's. For the inline TeX markup, the unicode π on my Firefox display is very ugly, with the π looking much much better, but it might be different on other browsers or other computers. Thanks for all of the other suggestions, especially the trick for writing fractions.

We'd love to have your help with the content discussions. Paolo and I have been butting heads over the first part of the article for several days now, with no end in sight. Let me try to summarize this debate, although Paolo should feel free to add comments here if he thinks I'm mischaracterizing it:

1. I firmly believe the current version of the introduction is quite good, and Paolo firmly believes that this version of the introduction is quite good. This is not surprising: we each like the version of the introduction that we wrote. [NOTE by Paolo: I did not write the previous intro; I just made it more similar to the definition; the words "does the reverse" were not mine, and can be found also in the section about inverses in Function (mathematics); in other parts of the article the verb "undo" was also used]
2. Paolo also dislikes the corresponding definition section [NOTE by Paolo: the reasons are briefly summarized in my previous comment; I did not write the previous definition; I just added a few very short notes which were coherent with the definition that others decided to adopt]. I myself am not completely happy with the definition section, so I keep trying to improve it. (See the original version, the first rewrite, the addition of the "Note on Functions", the second rewrite, and the third rewrite and the current version.) It is unlikely that Paolo will be satisfied with any new definition, since his most basic objection is to the introduction [NOTE by Paolo: not true; see below].
3. There is also some debate about whether any other important material was lost in the rewrite. This appears not to be as important as the main debate about the beginning of the article.

We would very much appreciate any help you could offer in resolving these disputes. Jim 06:52, 21 September 2007 (UTC)

It is not true that my "most basic objection is to the introduction". Please read what I wrote in my summary above. The introduction is secondary. Since the introduction should summarize the definition, we should first fix the definition.

My introduction may not be compatible with your definition. As I explained in detail in my summary, your new definition and characterization are so unclear and appearently contradictory that I don't even know if my introduction would be compatible with it. You wrote that your definition is equivalent to the previous one (which became your "characterization"), but I strongly doubt it. Paolo.dL 14:27, 21 September 2007 (UTC)

[edit] Jim's point of view

Paolo, we need to settle our differences about the introduction, so I'm going to explain my point of view in great detail. For me, there are two things that are fundamental to the definition of an inverse.

1. If ƒ sends x to y, then the inverse sends y back to x.
2. The inverse does nothing else. That is, ƒ^–1(y) = x only if ƒ(x) = y.

You often refer to the first statement by saying "ƒ^–1 reverses ƒ". This is a good way of putting it (if slightly nonstandard), which is why I incorporated the word "reverses" into the first sentence of the rewritten introduction.

You refer to the second statement by saying "and ƒ^–1 is reversed by ƒ", which is the part that I don't like as much. This seems to me to be a complicated way of putting it, more of a conclusion from the definition than part of the definition itself. It should be possible to understand inverses by only thinking about one reversal.

Look at the picture in the upper right of the article. The function shown is a bunch of arrows from elements of the domain to elements of the codomain. The inverse is obtained precisely by reversing the directions of the arrows. There's just one reversal. It's true that if you reverse the arrows twice then they end up where they started, but you shouldn't need to include that in the definition—it's just how reversing arrows works.

Look at the function ƒ(x) = x + 8. This inverse of this function is ƒ^–1(x) = x – 8. The reason is that subtracting 8 is precisely the reverse of adding 8. The original function ƒ has an arrow from 3 to 11, so the inverse has an arrow from 11 back to 3.

The only reason that things get complicated is the requirement that the inverse be a function. You see, there's this rule that defines a function: every element of the domain is supposed to have exactly one arrow coming out of it. If you take a function ƒ and reverse all of its arrows, the thing that you get might not be a function. In fact it usually won't: ƒ needs to satisfy some very special conditions. (These conditions I'm about to talk about aren't really part of the definition of the inverse, they're just the conditions under which it works. I've stated the definition of the inverse already.)

If you want "reversing the arrows" to produce a function, what you need is for every element of the codomain to have exactly one arrow going into it. Let's give these things names: ƒ is a function with domain X and codomain Y (i.e. ƒ: X → Y). So, what you need is for every element of Y to have exactly one arrow going into it. That way, the inverse ƒ: Y → X will have exactly one arrow coming out of every element of Y, which is what you need for ƒ: Y → X to be a function. (Note: Some further superfluous explanation has been removed at this point.)

These are the ideas that I'd like to express in the article, but I'm not allowed to be nearly as loquacious there. Has this helped to explain my point of view?

By the way, Paolo. I really apologize if I sometimes come off as rude. You must realize that I'm even newer to Wikipedia than you are, and I really don't have much experience in discussing things on talk pages. In fact, this page represents by far the majority of the talking that I've done. I keep trying different strategies to make this conversation work, because I haven't yet developed a consistent set of habits for Wikipedia interaction. In retrospect, I think I should have shown you more deference as the existing editor of this article. Probably I should have written a draft on my user page and then posted a link to it from this talk page indicating the suggested rewrite, so that we could argue this out before I replaced your work. If you really actually want to go back to the old version for now, I won't revert it. Jim 08:07, 21 September 2007 (UTC)

The old version was not my work! Others wrote the article and selected the definition. I just improved it slightly, but it was already a very good article when I read it for the first time. As for your suggestion to go back to the old version, I never wanted to, as I already repeated twice! I would like to revert only the introduction and definition, and restore some deleted parts, but Oleg Alexandrov and Sam Staton prefer your version. Although I can't understand their rationale, I will not ignore their opinion. And we still need to know the valuable opinion of KSmrq, and the authors of the old version. I agree that the most relevant changes (expecially the change in the definition section) should have been discussed in this page.

Your first three sentences are very intersting. But your explanations about the definition of function were not necessary. I am not a mathematician, but I perfectly know the standard strict definition. I even edited extensively the article about Function (mathematics) and contributed to dramatically improve its readability, by removing redundances and simplifying its structure. Before my edit, the definition section did not contain figures and did not explain that a true function is a single-valued and total relation. Also, I participated in an interesting discussion in Talk:Partial function about the different meanings given to the word function. Paolo.dL 12:36, 21 September 2007 (UTC)

[edit] Secondary discussion about definition of injection

A function is called one-to-one (or injective) if every element of Y has at most one arrow going into it (and possibly no arrows at all). For example, maybe half the elements of Y have single arrows going into them, and the other half are completely left out. If you try to reverse the arrows for a function like this, what you get in general is a partial function. Some elements of Y will have an arrow coming out of them, but some won't. (I can assure you that this is the correct definition of injective. I do this for a living. If you've heard another definition, let me know and I'll try to explain why it's the same thing.) Jim 07:52, 21 September 2007 (UTC)

You do that for a living, but the definition of injection that you wrote in the definition section is wrong. When you will understand it, you will be a short step forward. Other steps may follow, in which, I hope, you will eventually understand what I already wrote. This is your definition: "A function is one-to-one if each value of y corresponds to at most one value of x."

Paolo.dL 12:36, 21 September 2007 (UTC)

Maybe this is a wording problem? Let me state this sentence precisely: a function ƒ: X → Y is one-to-one if, for every element y ∈ Y, there exists at most one element x ∈ X such that ƒ(x) = y. That really is the definition of one-to-one. How would you define it? Jim 15:25, 21 September 2007 (UTC)

You are right. your sentence is correct. Please forgive me for writing the contrary. I apologize also for all the time I forced you to waste to discuss this point. I appreciate your patience. Actually it was a decoding problem. I wrote that stupid "by the way" sentence at about 3 o'clock in the night (I live in a UTC+2 time zone), and I was stubbornly convinced you switched y and x by mistake! Probably, I was biased and that's too bad. Actually, now I can see that my idea does not make sense at all, and that your short summary in the article about injection and surjection is very well written. I like the quasi-symmetry between the two sentences. This is probably the best way to compare the two concepts. However, don't worry, we perfectly agree about the concepts of injection, surjection and bijection. As far as I know, they were "recently" introduced by Bourbaki's group, and luckily they have only one standard definition... So, now it's your turn. You won this "by the way" fight, and you gained the right to decide the first step! :-) I promise that I will read your text more carefully in the future. Paolo.dL 16:23, 21 September 2007 (UTC)

It's really sort of your move. I tried to summarize my point of view on inverse functions as much as I could in the explanation above, with the hope that we can reach some sort of similar outlook on the introduction and definition. You could either critique my explanation, try to communicate your own point of view in more detail than you have so far, or suggest something else to talk about. If you want to put the discussion about the introduction on hold for now, we could talk about the old "properties" section, and the possibilities for a new one. Jim 21:14, 21 September 2007 (UTC)

[edit] First step

Again, this is a short step. I would try to show that your definition of inverse function in the article appears inconsistent with the ensuing definition of invertible function, and with the alternative definition given in the characterization section. Please see my comment on "one-sided versus two-sided" in my summary. This is not my main concern (as you can see by reading my summary), but we need to untie this knot before going on. For the other points in my summary, we need the help of others (I hope to have the opinion of KSmrq, Paul August, Oleg, Sam, Trovatore and Carl)

My point is that you eventually gave a clear and consistent definition, but you wrote it here, not in the article. Here's what you wrote in this talk page:

1. If ƒ sends x to y, then the inverse sends y back to x.
2. The inverse does nothing else. That is, ƒ^–1(y) = x only if ƒ(x) = y.

In the article, however, you wrote explicitly only the first part of this definition. For a non-mathematician, your rule just defines a left-inverse. It appears absolutely identical to the rule given at the end of the same article for the left-inverse.

There's also a huge problem in the subtle difference between a "function inverse" and an "inverse function" (see related discussion on Talk:Function (mathematics), and the section about inverse functions in Function (mathematics), that I recently edited to make this point clearer). It's impossible for a non-mathematician to guess that the definition of left-inverses and right-inverses given in the article is not actually a definition of left-inverse and right-inverse functions! Paolo.dL 22:17, 21 September 2007 (UTC)

By the way, you did not write explicitly that "invertible" means "that has an inverse". This is another arbitrary deletion of yours. Paolo.dL 22:25, 21 September 2007 (UTC)

Okay, so first of all "invertible" is mentioned in the introduction. Is the problem that it's no longer in the definition section? In any case, it's hard to imagine the reader being confused about the fact that "invertible" means "able to be inverted". After we finish talking about the introduction and definition we should go back and have a unified conversation about these "arbitrary deletions". Jim 23:14, 21 September 2007 (UTC)

Ok, I apologize. It was not a deletion. I did not notice that it was just a displacement. It's ok if you have it in the introduction. Paolo.dL 23:24, 21 September 2007 (UTC)

About the introduction, here is my thought process:

1. The second part of the definition is a bit subtle. For this reason, I think it's better to brush over it in the introduction and not address the point until the definition section. The introduction's job is to convey the most essential information: the inverse is the function that sends y back to x. It takes a lot of thought to even realize that there's an issue here, so I think talking about the second part in the introduction is likely to confuse a reader who is completely unfamiliar with the concept.
2. The definition section should be clear about the second part, but should not focus too much on notions like "left inverse" and "right inverse". Addressing part #2 in some fashion is important for a rigorous definition, but the definition section is not necessarily the place to discuss all of the possible subtleties that may arise.

Now let me try to indicate how I have tried to address part #2 in each of my versions of the introduction:

1. In the original version, the second part is handled by the words "and only if", i.e. ƒ^–1(y) = x if and only if ƒ(x) = y.
2. At some point, you objected to the complexity of this definition, which I interpreted as meaning that you didn't like the use of the mathematician's phrase "if and only if". This led to the current phrasing.
3. What the current phrasing says is "The inverse of ƒ is the function defined by the following rule: If y = ƒ(x), then ƒ^–1(y) = x." The fact that this rule "defines" the inverse is the content of the second part of the definition. For example, suppose I said:

In this statement, the words "defined by" imply that ƒ does nothing other than take reciprocals. For example, you know from this definition that 0 isn't in the domain of ƒ. You can't argue that ƒ(0) might be 3, because the function is "defined by" the rule ƒ(x) = 1/x, which means that ƒ(0) is undefined. For the same reason, a function "defined by" the rule currently given in the definition must be precisely the inverse. It cannot be just a left inverse, since a left inverse would have additional values of y for which the function is defined.

I'm happy with either of these two phrasings. The "if and only if" phrasing is actually what's in the textbooks that I looked at. I'd also be happy with making the second part more explicit in the definition section, as long as we don't change the introduction. Jim 23:14, 21 September 2007 (UTC)

This is exactly what I wanted to show: all of these subtleties were not needed in the original version of the definition. Neither knowing exactly the strict definition of a function, nor the "defined by" stuff, nor the difference between inverse function and function inverse, nor the definition of injection, surjection, bijection. They were gradually unveiled later (see "Arbitrary deletion 3" in my summary).

My recipe You just write two easy to understand conditions such as:

1. , for every x in X (f-1 undoes f)
2. , for every y in Y (f undoes f-1)

Then you say that the inverse function must meet both of these rules, and everything is immediately clear. Readers don't need to be careful about terminological tricks, they can easily decode your definition even if they are not students or colleagues of yours. Good night. Paolo.dL 00:12, 22 September 2007 (UTC)

Okay, I worked on this for a while longer. After thinking about it, I decided that part of the problem is that there's a simpler definition of inverse for functions in calculus, which is then modified after codomains are introduced in higher mathematics. I therefore rewrote the definitions section to present the calculus definition first, and then the set theory definition involving codomains. (In the original calculus definition, the subtle part is dealt with by specifying that the domain of the inverse is the range of ƒ. I think this gets to the heart of the matter, and is surprisingly accessible for calculus students.) I'm very happy with how the first part turned out, but I'm not so thrilled with my writing in the later "inverses and codomains" subsection. Can you think of any way to make the later part clearer?

I also did some general editing throughout the article. Here's a semi-comprehensive list:

1. The "note on functions" has been removed, since the same material is now covered in the "inverses and codomains" subsection.
2. The examples in the definition section have been reworded and shortened, and I re-added the squaring function.
3. The "inverses and compositions" material has been moved into the definitions section, and I added a sentence on category theory after the compositional identities.
4. The explanation of multiplicative inverses has been largely replaced with a link to the multiplicative inverse article. Some of the wording explaining the notation and the notational issue with trig functions has been changed.
5. I created a "Properties" section to hold the "inverse of a composition".
6. I added a subsection on "symmetry" and a subsection on "self-inverses" to the section on "Properties".
7. I removed some of the superfluous entries from the table of inverses in calculus. I think the table now communicates just the right information.

Feel free to undo any of these changes, and let me know what you think of the reorganized definition section. Jim 06:23, 22 September 2007 (UTC)

Arbitrary deletion 4. Jim, you deleted again (and again without warning and without explanation), a part that I restored explaining the reasons in this talk page. This behaviour is inacceptable.

I'm sorry—which part are you referring to? Jim 16:11, 22 September 2007 (UTC)

(Looks at Paolo's edit to the article.) Oh, I see. Paolo, I moved the text about the meaning of ƒ³ and ƒ⁽ⁿ⁾ to the end of the previous section, and I reworded it some. This wasn't a deletion. Now that you've reinserted the old text, these notations are discussed twice over the course of two paragraphs.

I can explain my motivations for the rewording if that would be helpful. Jim 16:58, 22 September 2007 (UTC)

Excessive boldness. Also, you introduced a non-equivalent definition, which obviously increases the complexity of the article, and you did that without asking for the opinion of others in this talk page. I know you mean well, but there must be a limit to boldness. Paolo.dL 11:12, 22 September 2007 (UTC)

Paolo, this is Wikipedia. The way we make articles better is by changing them until a consensus is reached. If you dislike some of my changes, go ahead and undo them. See, for example, the guidelines at Wikipedia:Consensus and Wikipedia:BOLD, revert, discuss cycle. Jim 16:11, 22 September 2007 (UTC)

I think that major changes should be discussed in the talk page. And minor changes should be done separately, section by section, and explained in single edit summaries. Otherwise, it becomes too difficult to understand what you did. Paolo.dL 22:43, 22 September 2007 (UTC)

I'm tired of this discussion. Paolo, make the article however you want. I'm removing it from my watchlist. Jim 22:27, 22 September 2007 (UTC)

[edit] What's the best recipe for the main definition?

Sincerely, I know you mean well, but this is still a mess. You seem to use the article as a notepad. The notes of a mathematician are not understandable to non-mathematicians. You introduced a non-equivalent definition, without warning the reader that you use two different definitions. And the first (and threfore more important for the reader) definition you give is exactly the same as the definition of left inverse. In a section called "inverse and composition" you actually give equivalent definitions that are not equivalent to the main definition. Moreover, in the same section, you use Y = range. This is a useless restriction which was already discussed above. The question is:

• Alternative (equivalent or not) definitions can be given in a separate section, but what's the best recipe for the main definition?

I believe there are four possibilities:

A) "invertible=bijective" - original recipe,

B) "invertible=bijective" - Jim's recipe,

C) "invertible=injective" (the current main definition)

D) "invertible=surjective" (see comment by Norwood below)

My 22 September recipe (see above) coincides with A. I would like to know the opinion of others. Paolo.dL 10:16, 22 September 2007 (UTC)

Let me point out that an encyclopedia should say how the phrase "inverse" is actually used in major textbooks and in research articles, and should give references. Since Munkres uses the word in one way and Halmos in another, and since both are major authors, the article should reflect this. The lede should just say "The idea of the inverse of a function is the idea of reversing the action of a function, but different books have different definitions of exactly what "reversing the action of a function" means in the case of functions that fail to be injective or that fail to be surjective. If a function is bijective, then all of the definitions of "inverse" agree." Rick Norwood 12:18, 22 September 2007 (UTC)

What does that imply for the definition section? Do you suggest to give three (or more) different definitions? Or to give the strictest definition (A or B) in the main section, and less strict definitions elsewere? (e.g. the section about left and right inverses might say that some authors use those one-sided definitions for the generic inverse) Paolo.dL 14:04, 22 September 2007 (UTC)

Okay, we should maybe talk about the math of this a little bit more. The current definition given in the article is not at all the same as the definition of left inverse.

After thinking about it, what I realized is that not everyone agrees that the codomain is an intrinsic part of a function. It's certainly not standard terminology in calculus or pre-calculus. See for example Function (mathematics)#Is a function more than its graph?. For a function without a codomain, it doesn't even make sense to talk about whether it's onto or bijective—the only question you can ask is whether it's one-to-one.

I looked through some precalculus books, and it appears to be standard in precalculus classes to state that a function is invertible if and only if it is one to one. The additional requirement that the function be onto isn't an issue until functions begin to have codomains, which is mostly an issue for subjects like algebra, topology, and category theory. I think a large number of analysts and applied mathematicians would disagree that two functions with the same graph can be different. Because most precalculus and calculus students are unfamiliar with the idea of a codomain, it seems like it will make the article more acceptable if we first give the simpler definition of the inverse, and then explain what happens for codomains a few paragraphs later.

The current definition is not the same as the left inverse, because it specifies explicitly that the domain of the inverse is equal to the range of ƒ. In fact, if you look at the first definition of inverse given, the function is required to be a bijection from its domain set to its range set. (A function always maps onto its range.) There really aren't multiple definitions of an inverse: there's just the naive definition currently given first, and the more sophisticated defintion that I have now moved a little bit later. They aren't inequivalent—they just apply to slightly different contexts. Jim 16:33, 22 September 2007 (UTC)

[edit] A new beginning

Usually the opening is best deferred until the rest of the article is in good shape, and aside from details of the initial definition and perhaps a few more references (to reflect the diversity of usage) that seems to be accomplished.

If I must choose between "does the reverse" and "reverses", then I would say the latter, for three reasons: brevity, clarity, and impact. But I see little benefit in mentioning reversal, since we must then immediately explain what that means for our purposes. If I must choose between an opening with equations and one without, then I would prefer to not depend on equations, or to only use them as a supplement (much like figures). Can we do all this? I have made an attempt.

Previous discussions about "reversal" obscured an essential point. Given ƒ: A→B it is not enough to choose just any function from B to A; we must choose ƒ⁻¹ to cancel ƒ. Therefore, I have tried to shift the emphasis. Jim has cleverly worded a definition that uses the range as the domain of the inverse; I have tried to give an informal description that covers the essence of all inverses. For generality, what I don't say is as important as what I do say.

In writing for novices, examples often convey more than definitions and equations, so I have included two examples, both easily understood (I hope) by most readers. The first example, besides being familiar, foreshadows inverting a composition. The second example naturally demonstrates some of the fine points discussed on the talk page, but with no technical language. Without mentioning domain, codomain, range, injection, or surjection, it shows their relevance. And, where the first example was a continuous real function, the second is discrete.

Please feel free to draw upon the examples later in article. --KSmrq^T 00:50, 23 September 2007 (UTC)

I'm not going to participate much here, but I'd like to thank Paolo for the fine discussion, and KSmrq for jumping back in. I think I've contributed all the ideas I can to this article, but I'll check back from time to time to see if there's anything I can help with. Keep up the good work! Jim 01:07, 23 September 2007 (UTC)

Great. The reference to composition using the intuitive "round trip" concept is very smart, useful, and needed in the lead. The text is so well written that I was not able to change a comma (I wanted to add some words about non-single valued and non-total relations, but then I realized they would disturb the elegance and simplicity of the example). I will just add two links within parentheses. If you don't like them, feel free to undo.

KSmrq, you wrote that the definition by Jim is "cleverly worded". Does this mean that you want this to be the main definition in the article? (please see previous section). Paolo.dL 16:01, 23 September 2007 (UTC)

I also propose to start with "In mathematics, an inverse function is a function which reverses [or "undoes", if you prefer] another one. If ...". I do not agree with Ksmrq that this gives little benefit. But I will not do that unless somebody else agrees with me. KSmrq's work deserves respect. I would also like to add somewhere in the lead a sentence such as "Another property of f^-1 is that it can always be reversed [or "undone"] by f. Formally, the inverse f^-1 of an invertible function f is always invertible and its inverse is f." A reference to the symmetry of the round trip which has a great intuitive advantage as I tried to show previously. But there's a problem: is this true for all the existing definitions of inverse function? It seems that a positive answer can be deduced from Jim's latest comment, but I am not sure Norwood and KSmrq agree. Paolo.dL 16:24, 23 September 2007 (UTC)

[edit] The introduction

I guess I came late to the party, however, the current intro

In mathematics, if ƒ is a function from A to B, then an inverse function for ƒ is a function in the opposite direction, from B to A, with the property that a round trip (a composition) returns each element to itself. Not every function has an inverse; those that do are called invertible.

sounds plain bad to me. One should not jump to notation and symbols right from the first sentence. For the sake of non-mathematicians, there's got to be at least a plain English paragraph explaining (if not very accurately mathematically), what an inverse function is about. Oleg Alexandrov (talk) 00:58, 29 October 2007 (UTC)

I completely agree that clear introductions are our goal. But I don't think that sentence is an example of unacceptably cryptic notation. For example, we could trivially replace the symbols, leaving

In mathematics, given a function from one set to a second set, an inverse function for the original function is a function in the opposite direction, from the second set to the first, with the property that a round trip (a composition) returns each element to itself. Not every function has an inverse; those that do are called invertible.

But I think this is less clear, not more; the judicious use of a few variables makes the sentence easier to follow. I don't see an easy way to make that sentence objectively better by rewording it; do you have one? — Carl (CBM · talk) 01:20, 29 October 2007 (UTC)

I liked what was there before (the italicized first sentence in the next paragraph):

In mathematics, the inverse of a function ƒ is a function which "reverses" ƒ. More precisely, if ƒ is a function from A to B, then the inverse function of ƒ, denoted by ƒ⁻¹, is a function in the opposite direction, from B to A, with the property that a round trip (a composition) returns each element to itself. Not every function has an inverse; those that do are called invertible.

Starting with a simple sentence, while not very accurate, is much better in my way than saying "f from A to B". Oleg Alexandrov (talk) 04:26, 29 October 2007 (UTC)

I agree with Oleg, even though KSmrq has a different opinion (see the 00:50, 23 September 2007 comment). I would also specify that A and B are sets, so we get:

In mathematics, the inverse of a function ƒ is a function which "reverses" ƒ. More precisely, if ƒ is a function from the set A to the set B, then the inverse function of ƒ, denoted by ƒ⁻¹, is a function in the opposite direction, from B to A, with the property that a round trip (a composition) returns each element to itself. Not every function has an inverse; those that do are called invertible.

The underlined words were added by me. -- Jitse Niesen (talk) 11:48, 29 October 2007 (UTC)

I appreciate the intention. The implementation fails. Here's why.

• "Reverses" is a placeholder, a distraction, a waste of time. Nobody knows what the word means until we explain it, and then we don't need it. Kill it, and the text is immediately stronger. Furthermore, it mistakenly implies that inverses are about reversal, when the core idea is really identity under composition. In fact, it's wrong for a right inverse. Bad mathematics and bad writing combined — this we do not need!
• Naming A and B is necessary so we can say what "opposite direction" means. Adding the qualifier "set" erects a barrier for readers who are not yet comfortable with sets, and is understood as part of the meaning of "function" for more experienced readers. Thus this extra clutter helps no one.
• Although I originally introduced the ƒ⁻¹ notation in the first sentence, I have since come to feel uncomfortable with it on technical grounds. The problem is that the sentence deliberately allows for one-sided inverses, but this notation is usually restricted to two-sided inverses. I'd like to introduce the notation, but not at the expense of misleading, nor at the expense of an excursion into sides — something I went to great lengths to avoid in the opening! One reason I want the notation is because I use it in the examples that immediately follow. Awkward, yes?

Finally, there is another factor. As often happens in Wikipedia, to understand the state of the text one must review the process from which it was born. The relevant threads are on this page and on Talk:Function (mathematics) (and archives). Both Jim Belk and I had our fill of one problem editor, and having found something that seemed to put an end to the problem, I have no wish to revisit the horror. Once or twice a year I encounter an editor who is such a bloody nuisance I'm tempted to depart Wikipedia; this was such an editor. Where were you guys when it was happening? Missing in action. Why the sudden interest now? You are late for the "party", and not fashionably late!

Correct an egregious technical error, or genuinely improve the exposition, and I'd support the change in spite of the risk. (Specifically, come up with a great solution to the ƒ⁻¹ notation issue and I'll be delighted.) But these fidgety fiddles do neither. --KSmrq^T 14:09, 29 October 2007 (UTC)

Heh. :) Then add the word "informally" to the first sentence, saying:

In mathematics, the inverse of a function ƒ is, informally, a function which "reverses" ƒ.

Still better than the current "f from A to B" wording. I think trying too hard to be perfectly exact is just making the text hard to read. Oleg Alexandrov (talk) 15:07, 29 October 2007 (UTC)

You throw in a useless misleading extra word, complete with scare quotes suggesting you know it carries no real content, then you want to pile on "informally"? That makes it worse! Pray tell, what does "reverses" mean, that is not already contained in "opposite direction" along with the round trip property? Pretend you are a reader who has no idea what an inverse function may be; you might as well be reading 'the inverse is a function which "ploons" ƒ', using a nonsense verb instead of "reverses". Skip the windup; deliver the pitch. --KSmrq^T 20:48, 29 October 2007 (UTC)

Yes, adding "informally" makes it worse. What I like about "reverses" is that it's short and to the point; add more words and we lose this. Against "reverses" is that it's not clear what it means. For instance, Jim said above that the distinction between "reverse" and "inverse" is important, but it's not immediately obvious what this difference is.

What I don't like about the current first sentence is that it mentions the domain and range/codomain before the round-trip property. I think that mentioning those sets acts as a barrier: in the most naive view, functions are expressions like x² + 2 without any domain attached to them. The round-trip property (for some reason I'm not so keen on the word "round-trip") is the defining property.

As I understood Jim, he wants to cover two situations: where functions come with a codomain associated to them and where they do not. I don't recall him saying that we should cover one-sided inverses from the start. Indeed, one-sided inverses are in the generalizations section, so I don't think we need to accommodate them in the lead section.

I would thus write something like

"In mathematics, the inverse function of a function ƒ is a function which maps y to x whenever ƒ maps x to y. (Thus, if ƒ is a function from A to B, then its inverse goes from B to A.) Not every function has an inverse; those that do are called invertible."

The sentence between parentheses can be added if we decide that the "opposite direction" property is important. I removed "the set" as I agree with KSmrq's reasoning above.

PS: I just discovered that this is in fact almost the same as what Jim wrote before. And I also see that there is a lot going on at function (mathematics) which I have missed. -- Jitse Niesen (talk) 13:34, 30 October 2007 (UTC)

I find your analysis interesting, but your proposal disappointing. For example, you make a mistake I deliberately avoided, when you say "the" inverse function, implying uniqueness. (Likewise, "its inverse" wrongly implies uniqueness.) You complain about A and B, then introduce even more notation and the unfamiliar techical term "maps". We have no need to discard the generality I achieved, and I have yet to see any benefit in doing so. Here are variants the present language covers:

1. g(ƒ(x)) = x for all x∈dom(ƒ)
2. ƒ(g(y)) = y for all y∈dom(g)
3. g(ƒ(x)) = x for all x∈dom(ƒ) and ƒ(g(y)) = y for all y∈rng(ƒ)
4. additionally, rng(ƒ) = cod(ƒ) = dom(g)

We found important instances of all these in common mathematical literature. Is "round trip" so odious as to discard all this, especially when it is formally glossed as "composition", and when it properly highlights the essence (composition produces identity)? I say no. And, of course, the round trip property is precisely the difference between reverses (A→B becomes B→A) and inverts. (Consider any non-injective function.) Nor is mention of A and B so unnatural, as demonstrated by the temperature conversion example that immediately follows, where we have A as Celsius and B as Fahrenheit. And, by the way, I consider these two examples a vital part of the opening; in multiple subtle ways they properly frame important issues using everyday language and familiar relations. I have mentioned some of the issues elsewhere on this page.

Please stop picking at it! It ain't broke, don't "fix" it. Fiddle with the body; fiddle with another article. We have so many basic mathematics articles in such bad shape this an inexcusable squandering of editorial energy — especially mine. --KSmrq^T 14:52, 31 October 2007 (UTC)

Dear KSmrq, have you read Wikipedia:Ownership_of_articles? You have no right to tell people not to discuss this, and no right to tell people not to try to improve the article. For what it's worth, I think that Jitse Niesen's suggestion, including the parenthetical remark, is the clearest one yet and is a small improvement on the current introduction. Yes, it might be better to say "an inverse" rather than "the inverse". But the current first sentence preassumes a knowledge (or intuition) of function composition, which is not appropriate. "Maps" is not a techincal term, it is a fundamental part of the concept of "function". Sam Staton 15:40, 31 October 2007 (UTC)

Oleg, Jitse, and I have a history of editing articles together which is roughly three times as long as you have been seriously contributing to Wikipedia; I don't think ownership is an issue. I do have the right to say "That's not an improvement; here's why." And if you stick around long enough you will discover that basic articles are the most difficult, and that their openings routinely attract hundreds more fidgety edits than the rest of the article. So I also have the right to say "This is not the best use of our time." I'm sure Oleg and Jitse appreciate your willingness to defend them, as do I; but they're both experienced editors — admins, in fact — and I have full confidence that if I get my head wedged in an uncomfortable posture they'll help me get it unstuck.

Here is my take on your opinions. Clear to you is ducky, but not a good predictor of how a general reader — especially a young one — will react. You clearly lack "beginner's mind"; for example, you think that "maps" is not a technical term. Wrong. For the man-in-the-street, mapping is what a cartographer does to create a document showing streets and landmarks. And I am hard pressed to understand why you think "round trip" requires prior exposure to composition; the woman-in-the-street (equal time!) has made round trips to the office and back on a daily basis. If I were teaching inverse functions, I would certainly introduce function composition first; but we don't have that luxury in the context of the opening paragraph.

The reason openings of basic articles attract so many edits is because everybody — whether they know a little or a lot — sees what's there (usually ignoring the rest of the article) and thinks "I can say it better". Over 95% of the time, they're wrong for an article whose opening is the result of a number of prior editors finding their best compromise. These drive-by editors proudly think they've done their good deed for the day, with no idea that they just overlooked a mathematical detail like uniqueness, or unwittingly snowed an innocent with technical language. We can't criticize the impulse: without it there would be no Wikipedia! We can only hope to divert it into more productive channels. And opportunities abound; we have a myriad of basic articles that beg for a total rewrite — including their openings. --KSmrq^T 17:37, 31 October 2007 (UTC)

Thanks KSmrq. I'm not sure that I "clearly lack "beginner's mind"". I learnt about "maps" at school when I was 6 years old. Later, when I learnt what a function was, I learnt again what "map" meant, and indeed "map" is often used as another word for function. (This may be a cultural thing; I am from England.)

There are two ways of introducing inverses. The first way says that an "inverse of a function is the inverse relation, provided that is functional". This is the one that I learnt first, a long time ago. It is what Jitse proposed to use. The second way presupposes composition and talks about what you call "round-trips". I think you prefer this one. I'm indifferent, to be honest. I don't plan to get too involved in this discussion.

(Thank you for explaining your phrasing. I was worried that you were sounding intimidating - "don't fix it" is quite strong. I wasn't aware that you knew Oleg and Jitse quite well, and hence that they would know not to feel intimidated.) All the best. Sam Staton 18:42, 31 October 2007 (UTC)

Use of "maps" is a cultural thing, but I believe the relevant cultural difference is not cross-pond but immersion in mathematics or not. I would be very surprised if any English speaker in the world primarily thinks of maps as a synonym for functions unless they deal with a fair amount of mathematics.

Jim Belk and I discovered a fascinating diversity of approaches in various places we looked. Jitse asserts that it is quite common to think of a function as a formula, not as a functional relation (and I agree), so I'm not sure he's thinking in terms of reversing the ordered pairs of a relation. (Notice that Jim's picture essentially does that, though it would be more obvious if he ever got around to swapping the sides for ƒ⁻¹ as he planned. Sometimes we do better saying things with pictures instead of words alone.) I know in some quarters it became fashionable to define functions as a special kind of relation, but historically relations are a generalization of functions. The Bourbaki influence was, I fear, not so good for mathematics teaching. Composition is both easier and in some ways more sophisticated. It's easier because we don't need new concepts, we just use one function after another. It's more sophisticated because it transfers directly to category theory, and thence to most of mathematics (whereas relation reversal is highly specialized). If we choose to write the composition of ƒ then g as gƒ, then we also have a natural justification for the notation ƒ⁻¹, and a way to remember which is a left inverse and which a right. When we learn about matrix multiplication, we've already got the right mental model. So I claim that the composition approach has a number of advantages. Still, I cannot hope to convince you that the way you learned it and think of it is less natural; that's like trying to convince someone the way their mother cooks isn't the best in the world. :-) --KSmrq^T 19:45, 31 October 2007 (UTC)

I appreciate when KSmrq explains his rationale. But he should accept the existance of other editors who dare to disagree with him, and respect them. Sometimes he even refuses to explain the reason of his reverts (this happened both in Talk:Integral and Talk:Function (mathematics)), totally disregarding Wikipedia policies and guidelines. For instance, we discussed about the word "reverse" several times before, and only now he gave to Oleg a complete answer about the reason why he removed it from the introduction. As I explained above, I believe that this article urgently needs a revision in the definition section. Right now, although the current text by Jim Belk contains precious information, the readability of the section for laypersons is worse than it was before Jim's edits. Only by reading this talk page a layperson can fully understand. Paolo.dL 12:29, 1 November 2007 (UTC)

KSmrq, explaining your edits to other editors is not an "inexcusable squandering of editorial energy". Nobody needs to be excused for selecting the introduction of this article as a perfectible text worth of their attention. Also, you wrote to Sam Staton, with a smile: "I cannot hope to convince you that the way you learned it and think of it is less natural". The smile is not enough to hide your bias. You should not assume that you are the only one able to rationally evaluate the advantages and drawbacks of an approach. Based on my past experience, it appears evident to me that you sometimes manage to treat as enemies even those who hold you in high esteem and respect your opinion. I write this because I think that discussions on talk pages would be more productive and less time consuming if we could just accept others as peers, as suggested in some Wikipedia guideline. Paolo.dL 18:17, 1 November 2007 (UTC)

[edit] Unnecessary reference

The statement "Not every function has an inverse" is supplemented with a reference to "Smith, William K. Inverse Functions, MacMillan, 1966 (p. 60)." I removed this reference, and so did someone else in the past, but it keeps getting reinserted.

Let's keep this out of the article. The statement is trivial and one needn't consult a book called "Inverse Functions" to find that out. Any introductory math book will state it, and anyone, upon hearing of the statement, should be able to come up with an example of one (how's a constant function on any set of at least two elements?). Pointing the reader to an offline reference is intimidating and indicates the statement is difficult to prove.

If you realy want to justify the statement, give an example, or use a more proper reference. Surely there's an example at Mathworld, PlanetMath or Wikibooks? 213.114.200.31 (talk) 12:10, 12 April 2008 (UTC)

I am spit about this. But references do no harm. So stop being a stubborn revert warrior, limit yourself to making your case here. Let's see what others have to say. Oleg Alexandrov (talk) 12:25, 12 April 2008 (UTC)

I'm not a revert warrior, as I made a comment here after my third revert, because I don't intend to revert again. As I've said, I think it does do harm. No reference at all or an explicit counter-example is much more helpful to the reader. (By the way, did you mean "split"?) 213.114.200.31 (talk) 13:24, 12 April 2008 (UTC)

I don't think the citation is helpful here. No-one should have to go to a library to find this 1966 book, just for a counterexample. In fact, there are plenty of examples later in the article (even in the "Definition" section). In general, I don't think it is necessary source all the statements in the introduction, when they are explained in the body. Sam Staton (talk) 14:53, 12 April 2008 (UTC)

I moved the citation to the bottom of the page. There it is still useful, without being tied to any particular statement. Oleg Alexandrov (talk) 17:10, 12 April 2008 (UTC)

Great, yes, I think that's the best thing to do. Sam Staton (talk) 21:51, 12 April 2008 (UTC)