Talk:Inverse function/Archive 1

message: "For functions between Euclidean spaces, the inverse function theorem gives a sufficient and necessary condition for the inverse to exist."
I don't see why the Inverse function theorem is a necessary condition for the inverse to exist. (I've found the same claim on page PlanetMath.) Mozó 18:04, 9 October 2006 (UTC)
thread_title:  Inverse function theorem 

message: Simple question (perhaps badly phrased in the title): is f^-1 (x) = (f(x))^-1 ever true? In case I've written that wrong, that is to say that if f(x) is some function of x, g(x) is that function to the power minus one, and is also the inverse of f(x). I realise that f^-1 (x) does not indicate (f(x))^-1 usually, which can be confusing with trigonometric functions, but could it ever be the same thing?
:So g(x) = 1/f(x) = f-1(x) which implies that the functional inverse of 1/f(x) equals f(x). What you're asking, essentially, is if the multiplicative inverse has an inverse function. In fact, it is its own inverse. 59.112.51.89 19:36, 27 April 2007 (UTC)
thread_title:  Is the inverse of a function ever equal to that function to the power minus one? 

message: The definition is incorrect as X is not necessarily the domain of f^{-1}. —The preceding unsigned comment was added by 24.94.246.41 (talk) 22:35, 9 February 2007 (UTC)
thread_title:  Definition is incorrect 

message: The "equivalent definition" (that is obvious in the first place) is now really excessive. I propose to revert a bit. Sam Staton 09:09, 6 September 2007 (UTC)
replies: text: It is obvious only for those who know the notation used, and I believe that most readers know the notation f(x), but much less readers know function composition and symbolic logic. However, I only tried to interpret Wahrmund's suggestion (see his 5 September edit) in such a way as to avoid his equations with three members. See if you like the shorter version that I edited a few minutes ago. Paolo.dL 10:07, 6 September 2007 (UTC)
thread_title:  Equivalent definitions 

message: User:Wahrmund, you reverted my edits to the section on left and right inverses. Was it intentional -- can you explain? Otherwise I will redo them. I see you also undid a sensible edit by User:Paolo.dL. Why? Sam Staton 09:51, 7 September 2007 (UTC)
replies: text: User:Sam Staton  This has to be a software malfunction.  My last edit was confined exclsuviely to "Equivalent definitions". I made absolutely NO changes to Left and Right Inverses.  I didn't even read it.  And I never cut-and-paste long blocks of text. Please let me know which Paolo.dL edit you are referring to, as I think there were several of these.  Then I will attempt to address the issue.   FYI, I will be out of the country and unavailable from Sept. 12 to Oct. 10.  Morris K. 01:20, 8 September 2007 (UTC)
thread_title:  Left and right inverses, and Equivalent definitions 

message: I believe there's a subtle logic flaw in this definition:
:Formally, if f is a function with domain X and range Y, f\colon X\to Y, then f^{-1} is its inverse function if and only if for every x \in X we have:
:: f^{-1}(f(x))=x,\,
:: and for every y \in Y we have:
:: f(f^{-1}(y))=y.\,
Notice that Y is explicitly defined as the range of f, and at the same time used as codomain of f in f\colon X\to Y. Thus, actually the definition contains three conditions, in this order:
# the codomain of f must coincide with its range
# f must be "reversible" (it must be possible to undo it)
# f^{-1} must be also "reversible" (with f^{-1}\colon Y\to X)
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)
replies: text: 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)

text: 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)
thread_title:  Fixed subtle logic flaw in the definition 

message: 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)
replies: text: 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)
thread_title:  Section by section review 

message: 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)
thread_title:  Comparison between original and new definition 

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

message: How do we find the inverse of a function of two variables, or even if it is invertible or not?
For example, if
f(x,y)=z
what is
f^{-1}(z). I can show that z is unique, i.e for every combination of (x,y), there is only one z.
220.227.207.32 (talk) 06:47, 1 July 2008 (UTC)
replies: text: To determine if it is invertible, you may examine its Jacobian determinant. For more information see the "curve genus" discussion at Talk:Implicit_function_theorem EverGreg (talk) 10:37, 1 July 2008 (UTC)
replies: text: Or maybe your function is a function f: X×Y → Z, in which case the inverse will be a function f-1: Z→ X×Y. In other words, the inverse is a function that assigns a pair (x,y) to every z. Sam (talk) 10:49, 1 July 2008 (UTC)

text: Yes, to show that your function is invertible you must show that it is one-to-one. That is, not only that for every (x,y) there is only one z, like you say you've done, but also that for every z there is only one (x,y) pair. If this dosn't hold, you can use the Jacobian determinant to determine for what subsets of x,y, and z the function is invertible.  —Preceding unsigned comment added by EverGreg (talk • contribs) 11:35, 1 July 2008 (UTC)
thread_title:  Inverse of a function of two variables