Talk:Inverse function/Archive 1

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Inverse function theorem

"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)

Is the inverse of a function ever equal to that function to the power minus one?

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)

Yes: { (1,1) } is such a function. manczura@ccccd.edu

Definition is incorrect

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).

Existence of an inverse

Why must f be bijective? Shouldn't it be enough for f to be injective?

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 $Y$, $f\colon X\to Y$, 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:

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 $f^{-1}\colon Y\to X$)

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)

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)

Equivalent definitions

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)

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)

Left and right inverses, and Equivalent definitions

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)

User:Wahrmund, I am sure you did that unintentionally, but yesterday by copying and pasting a long block of old text you destroied many of my recent edits in different sections of the article. As you see in the "history" page associated with this article (just click on the tag at the top of the page), I provide the reason of each of my edits separately and carefully. Please see my "edit summaries" in the history page before undoing them. Please edit different sections separately (by clicking the respective "edit" link), and explain each change separately in the relevant edit summary, or in this talk page. Thanks, Paolo.dL 10:04, 7 September 2007 (UTC)

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)

Morris, please see this comparison between your latest edit and a previous version of the article. It appears evident that you opened an old version (by means of the history page), then you edited that old version and saved it, ignoring the following two warnings appearing (within frames with orange background) immediately above the editing window:

This is an old revision of this page, as edited by Paolo.dL at 08:49, 6 Sept 2007. It may differ significantly from the current revision.
You are editing an old revision of this page. If you save it, any changes made since then will be removed.

Please carefully read these warnings. Any other hypothesis about what happened is, in my opinion, almost as unlikely as the occurrence described by the infinite monkey theorem. I perfectly know that a newbie, when passionately editing an article, may not see warnings (something similar happened to me some time ago), but actually your single click on that "Save page" button was sufficient to remove from the article 13 changes done after 08:49, 6 September 2007! And they included 4 edits by Sam Staton and 9 by me! That's a lot of work. Please never do that again, unless you really want to delete all changes made after a given date and time. Thanks, Paolo.dL 13:13, 9 September 2007 (UTC)

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