Talk:Inverse function

bijectivity would be more sensible

For the sake of generality, the article mainly considers injective functions. Yet, other articles (see for example Bijection#Inverses and Injective function#Injections_can_be_undone) consider that a function is “invertible if and only if it is a bijection”, and they link to the present article whenever mention is made of a complete inverse (i.e., a bijection). To consider injective functions instead of bijective functions, when speaking of inverses, creates much confusion. Even in the present article, in the section #Definitions, in the last paragraph, it is said that “If $f$ and $f^{-1}$ are functions on $X$ and $Y$ respectively, then both are bijections”, even though $f$ and $f^{-1}$ were actually defined above as having $X$ and $Y$ as their respective domains.

In section #Definitions, the article links to Inverse element; this implies that an inverse function is an inverse element with regard to function composition. Yet, as defined in the present article, an inverse function is just a right inverse, not necessarily a (two-sided) inverse. For the sake of both consistency and clarity, then, it seems crucial to make the article center not on injections, but on bijections.--Anareth (talk) 15:11, 28 August 2016 (UTC)

I think the confusion is in the subject mater in that some uses od invertible functions require bijective and some only injective. It would be wrong limit this article to only the bijective case. Yes, this causes confusion, but that's the nature of the subject.David Sherwood (talk) 21:36, 4 October 2016 (UTC)

A function is invertible if and only if it is a bijection. Those who hold that only injectiveness is required are the same set of folk who believe that all functions are onto (that is, they reject the concept of a codomain). Unfortunately, this definitely minority point of view has infected several articles and when not pointed out it certainly creates confusion. The article (especially the lead) needs to be rewritten from the general viewpoint and, if necessary, a section devoted to this minority view should be included much later in the article. I'll try to rewrite the lead with as little jargon as possible and concentrate on the meaning of an inverse function rather than its uniqueness. --Bill Cherowitzo (talk) 04:22, 5 October 2016 (UTC)

What I wrote above reflects my own point of view, but my edits will be much more neutral on this issue. As I started to rewrite I realized that because different editors with different POV's had been editing this article for a while, the result was a confusing mess with references to things that no longer existed and conflicting terminology. I can smooth this out with appropriate references, but it will take a little more time than I originally thought it would. --Bill Cherowitzo (talk) 23:27, 5 October 2016 (UTC)

Multivariate inverse function

Please read 'Multivariate inverse function', Is it easy to be understand? Can you accept it?

For multivariate function $x_{0}=f(x_{1},x_{2},\cdots ,x_{i},\cdots ,x_{n})$,$f_{i}:x_{i}\mapsto f(x_{1},x_{2},\cdots ,x_{i},\cdots ,x_{n})$.

If $f_{i}$ is bijection for any $x_{j}(j=1,2,\cdots ,n,j\neq i)$ we call $f_{i}^{-1}$ an multivariate inverse function about $x_{i}$. Introduce unary operator $I_{i}$ and denote :

$f_{i}^{-1}=I_{i}(f)$.

For example,$f(x_{1},x_{2})=x_{1}^{3}+x_{2}^{2}$ is invertible about variable $x_{1}$ and is not invertible about variable $x_{2}$.

Partial inverses can be extend to multivariate functions too. We can define multivariate inverse function for an irreversible function if we can divide it into r partial functions $f_{(k)},k=1,\cdots ,r$ and denote its inverses as :

$f_{i,(k)}^{-1}=I_{i}(f_{(k)}),k=1,\cdots ,r$.

For example,$f(x_{1},x_{2},x_{3})=x_{1}^{2}+x_{2}^{3}+x_{3}^{4}$

$x_{0}=f_{(1)}(x_{1},x_{2},x_{3})=x_{1}^{2}+x_{2}^{3}+x_{3}^{4},x_{1}\geq 0,x_{3}\geq 0$

$x_{0}=f_{(2)}(x_{1},x_{2},x_{3})=x_{1}^{2}+x_{2}^{3}+x_{3}^{4},x_{1}\geq 0,x_{3}<0$

$x_{0}=f_{(3)}(x_{1},x_{2},x_{3})=x_{1}^{2}+x_{2}^{3}+x_{3}^{4},x_{1}<0,x_{3}\geq 0$

$x_{0}=f_{(4)}(x_{1},x_{2},x_{3})=x_{1}^{2}+x_{2}^{3}+x_{3}^{4},x_{1}<0,x_{3}<0$

$x_{1}=f_{1,(1)}^{-1}(x_{0},x_{2},x_{3})=I_{i}(f_{(1)})(x_{0},x_{2},x_{3})=[x_{0}-x_{2}^{3}-x_{3}^{4}]^{1/2},x_{0}\geq 0,x_{3}\geq 0$

$x_{3}=f_{3,(3)}^{-1}(x_{0},x_{1},x_{2})=I_{3}(f_{(3)})(x_{0},x_{1},x_{2})=-[x_{0}-x_{1}^{2}-x_{2}^{3}]^{1/4},x_{0}\geq 0,x_{0}\geq 0$

The concept of multivariate inverse function is useful to express the solution of a transcendental equation. — Preceding unsigned comment added by Woodschain175 (talk • contribs) 22:37, 25 June 2017 (UTC)

Notation

Hello all! I believe I read somewhere that the inverse of a function f(x) can be denoted as inv(f(x)). I think it was in a computer science context, rather than pure math. Does this sound familiar to anyone? I'm going to look around to see if I can find any references for this notation eventually before putting it in the article. If anyone knows a reference, I'd appreciate you putting it in! Thanks for reading. :) JonathanHopeThisIsUnique (talk) 04:33, 30 November 2017 (UTC)

This is computer code, not mathematical notation. For instance Mathematica uses Inversefunction[f][x] but translates back into the conventional mathematical expression when asked to present in "TraditionalForm".Limit-theorem (talk) 11:18, 30 November 2017 (UTC)

Thanks for replying! I don't exactly understand your distinction between computer code and mathematical notation though. Isn't it just a specific form of mathematical notation used in a computer science context? I would agree that it's probably not widely used enough to mention in the article if that's your point. Either way, I appreciate your help. JonathanHopeThisIsUnique (talk) 01:40, 3 December 2017 (UTC)

Not necessarily strictly increasing or decreasing

@Dr Greg: recently reverted my edit, thereby restoring the wording

A continuous function f is invertible on its range (image) if and only if it is either strictly increasing or decreasing (with no local maxima or minima).

with the edit summary

X^3 is strictly monotonic.

I never denied that, and the restored version does not say anything about “strictly monotonic”. Instead, it says the function must be “strictly increasing or decreasing”, which X^3 is not (since its slope is 0 at X=0). See Monotonic function#Monotonicity in calculus and analysis for the definition of this: If the order ≤ in the definition of monotonicity is replaced by the strict order <, then one obtains a stronger requirement. A function with this property is called strictly increasing. [my bolding]

So right now our passage is incorrect. As for “strictly monotonic”, I don’t think the monotonic function article specifically mentions that wording, but it would be very surprising if anyone uses “strictly monotonic” when permitting weak inequalities ≤ or ≥. Loraof (talk) 19:34, 4 January 2018 (UTC)

The function $f(x)=x^{3}$ is strictly increasing (or strictly monotonic, which means the same thing), because if $x<y$ then $x^{3}<y^{3}$ for all real values of $x$ and $y$, even if one of them is zero. The fact that $f'(0)=0$ doesn't stop it from being strictly increasing. -- Dr Greg  talk  20:12, 4 January 2018 (UTC)

Okay, thanks! Loraof (talk) 20:53, 4 January 2018 (UTC)

Include phrase in the leading section

something as "all functions that have an inverse must be bijective", to be simple and encyclopedic.