Talk:Function composition

(*/Archive: someone's homework problem)

I am surprised this article was not categorized so far. Thanks Paul.
Oleg Alexandrov 01:34, 16 Jun 2005 (UTC)

typography
Any better way to write the operator than the current f o g? $$f \circ g$$?


 * Now I saw someone convert small o in another article into Unicode U+2218 &#x2218; so I looked at this again. 2218 seems to be the right character, but on Talk:Category theory someone points out that it "doesn't display correctly on some browsers, most notably Internet Explorer." Anyway, I suppose other articles in Wikipedia should follow the choice made in this article. See also Template_talk:Unicode. --TuukkaH 18:05, 3 October 2005 (UTC)


 * I agree that the unicode character is best. It doesn't display because IE just doesn't support Unicode. I use Mozilla.He Who Is 21:41, 4 June 2006 (UTC)


 * A decade later, there is really no excuse not to use ∘ (Unicode U+2218, the unicode ring operator). Using the degree sign is a hack we should discard. Cerberus (talk) 19:51, 14 September 2016 (UTC)

negative functional powers
There is an error in the definition of negative functional powers. -- Wasseralm 19:51, 17 August 2005 (UTC)
 * I don't see it. Can you be more specific? Oleg Alexandrov 20:41, 17 August 2005 (UTC)


 * I fixed it. It basically said $$f^{k}=(f^{-1})^k$$. Rasmus (talk) 20:46, 17 August 2005 (UTC)
 * Looking at the timestamps, I had seen the article after the error was fixed. :) Thanks, Oleg Alexandrov 20:47, 17 August 2005 (UTC)


 * OK, this was the easy part. There is another intricate problem in the section, due to the fact that you consider f: X -> Y, where Y can be a proper subset of X. According to the definition of "composition" given at the beginning of the article, f o f cannot be formed (at this point). Intuitively (or with a slightly different definition of composition) f o f is defined and a function X -> Y. Thus all positive powers are defined. f^0 is a problem, becaus it has to be a function X -> X. It gets worse for the negative powers: If f: X -> Y is bijective, then f^(-1) : Y -> X. Thus, F^(-1) cannot composed with itself. To get rid of this problems, better take f : X -> X (compare the german article ("Deutsch"). Yours, -- Wasseralm 20:52, 19 August 2005 (UTC)


 * That X&sub;Y should be sufficient for a bijective function f:X->Y to admit negative function powers, right? I added that as a condition. Btw. feel free to make any changes yourself. That is what a wiki is for after all! Rasmus (talk) 21:57, 19 August 2005 (UTC)


 * You write in your edit summary : "Functional powers - actually, for negative power of function to make sence one would need f:X->X.".
 * If f:X->Y is bijective, its inverse function f -1:Y->X should be self-composable if and only if X&sub;Y. Of course in that case, unless X=Y, f is not itself self-composable, so it does not admit positive functional powers, but I do not see why it can't admit negative functional powers. Obviously it is not a necessary condition for negative functional powers that X=Y (quick counter-example: f(x)=2x, X=[0,1], Y=[0,2], f -1(x)=x/2 and f -2 is a nice function with domain [0,2] and codomain [0,0.5]). Rasmus (talk) 16:03, 20 August 2005 (UTC)

OK, you are right. But in this case I would claim we should not try to be too general. The condition f:X->X is in my opinion a very reasonable one to talk about negative functional powers. If the negative functional powers were a really important concept, then maybe it would be worth be general. But since it is just a curiosity, I would think it is not worth the trouble putting the most general condition.

But it is up to you. If you feel like going back to If f:X->Y with Y a subset X, be my guest. :) Oleg Alexandrov 17:34, 20 August 2005 (UTC)
 * It seems you must have Y = X. Otherwise, there is an element x in X &minus; Y. If f is a bijection onto its image, then x has nowhere to go without making the proposed "inverse" function no-longer 1-1. Revolver 16:47, 7 October 2005 (UTC)

Composition notation
We say:
 * "In the mid-20th century, some mathematicians decided that writing "g o f" to mean "first apply f, then apply g" was too confusing and decided to change notations. They wrote "xf" for "f(x)" and "xfg" for "g(f(x))".  However, this movement never caught on, and nowadays this notation is found only in old books."

Does anyone have a reference for this? I'm sure I recall the left-to-right notation being used in my undergrad maths degree, which was only three years ago. &mdash; Matt Crypto 11:55, 19 August 2005 (UTC)
 * I've seen John Baez, who is thoroughly modern, use the left to right notation for composition, especially in the context of category theory. -Lethe | Talk 18:45, 3 October 2005 (UTC)
 * Edit: actually, I've seen Baez write gf for first act with g and then with f, but I have never seen him write xf for f(x). That notation seems pretty rare to me. -Lethe | Talk 18:48, 3 October 2005 (UTC)
 * Maybe you just didn't realise it. In any case, it is typical in category theory to conceptualise "elements" of a set as a certain type of morphism, so in that case, it would be perfectly correct and consistent to write xfg for g(f(x)). Revolver 16:47, 7 October 2005 (UTC)
 * There's another notation, which I think I saw in Jacobson's textbook, is xf for f(x). -Lethe | Talk 17:06, 7 October 2005 (UTC)
 * Just a passing note from a Haskell programmer: xfg and fg cannot both be correct type-wise, assuming an unambiguous definition of the notation. That is, h = f  o  g is correct, and Haskell has $ for correctly expressing y = f $ g $ x as well, but o and $ are not interchangeable. Of course, mathematicians needn't fear type errors like Haskell programmers need. In any case, you can see some programmers and programming languages prefer a left-to-right notation, such as cat file | sort | uniq for composition in shell. --TuukkaH 20:36, 3 October 2005 (UTC)
 * Is it worth noting that most (if not all) concatenative programming languages use a direct equivalent of the xfg syntax to express their programs? --Piet Delport 10:59, 2 January 2006 (UTC)

Sometmes I wish god had only given us only one hand. I suffer from terrible left/right confusion. A kind of dyslexia I suppose, for example I'm always confusing east and west. Which reminds me of the old conundrum: "why does a mirror reverse left and right, but not top and bottom?" Paul August &#9742; 17:30, 7 October 2005 (UTC)

Having functions act on the right of their arguments is still quite common among algebraists. See, for instance, Smith's Postmodern Algebra for a recent book that does this. Since there are obviously some folks who have put a lot of time into this, I will not edit the paragraph myself, but I strongly recommend that it be rewritten to suggest that the other convention is still in common use.Mkinyon 21:22, 21 February 2006 (UTC)


 * Note the discussion of this convention at Group homomorphism. I will try to come up with a rewrite for the offending paragraph in the next few days. It is not correct as written, and the pseudo-historical "In the mid-20th century, some mathematicians..." is not up to encyclopedia standards. Michael Kinyon 21:37, 13 March 2006 (UTC)

Fun with composition signs
JA: As for typography, these tricks work in some settings:


 * undefined"&omicron;"undefined


 * fundefined&omicron;undefinedg


 * Fundefined&omicron;undefinedG


 * Lundefined&omicron;undefinedM

JA: Jon Awbrey 03:20, 27 January 2006 (UTC)

Derivative of composite function
Would a derivative of a composite function be a "composite derivative"? ~Kaimbridge ~ 20:23, 30 January 2006 (UTC)
 * I don't know what you mean by a "composite derivative". The derivative of a composition of two functions is given by the chain rule. Paul August &#9742; 20:47, 30 January 2006 (UTC)
 * The "composition of two functions" is technically a "composite function", so I was just inquiring if its derivative would be a "composite derivative"——okay, I'll inquire over at chain rule. ~Kaimbridge ~ 20:15, 3 February 2006 (UTC)


 * JA: A function that arises through the composition of functions is just a function. Once it's composed, you cannot say for sure where it came from, since the return decomposition is not in general unique.  Hence its derivation, oops, mode of arising, is not its essence, in other words, not a part of its "ontology".  Jon Awbrey 13:28, 14 March 2006 (UTC)

Upper Limit of the Composition of Tan and Cos
Consider the graph of the following function:

$$(tan\circ cos)^{\infty}(x).$$

As the exponent converges to infinity, the function assumes a quizzical shape. A cyclical zig-zag made up of boxes set side by side. Wach one peaks at about 1.5614, and has a width of about 1.52. The latter seems to is about pi/2, which makes sense, since that makes pi its period. As for the former, I have been trying to find a connection between it and other known constants, and have yet to find anything. Any suggestions? He Who Is 21:54, 4 June 2006 (UTC)


 * What are f and g? -lethe talk [ +] 21:57, 4 June 2006 (UTC)

Woops... Wasn't thinking when I wrote that. Tangent and cosine. Also, I looked at it more closely and realized that 1 is the maximum of cosx, and the peak of tancostancos...x is tan1. But I still think it is a rather interesting operation, since for everything between pi/4 + npi/2, for all integer n, it converges to tan1. Also, if one looks closely, you can see it has no zeroes, nor does it converge to zero. It actually grows to a value of about .002, shich I assume equals:

$$\lim_{n\to\infty} (\tan\circ\cos)^n(0)$$. He Who Is 22:00, 4 June 2006 (UTC)
 * Those are just the solutions of $$(\tan\circ\cos)^2(x) = x$$. Repeated iteration of $$x \to f(x)$$ will tend to attractors, such as fixed points of $$f$$. –EdC 17:08, 5 February 2007 (UTC)

$$(\sqrt x)^2$$ undefined?
I'm a bit confused by the line "For example, $$(\sqrt x)^2 = \sqrt{x^2}$$ only when $$x \ge 0$$; for all negative $$x$$, the first expression is undefined." For x < 0 don't we have $$(\sqrt x)^2 = (i \sqrt -x)^2 = i^2 (\sqrt -x)^2 = -x$$? TooMuchMath 05:06, 20 September 2006 (UTC)
 * Not if f and g are defined to be real-valued functions, which I think is being assumed in the context. Profzoom 22:38, 26 September 2006 (UTC)
 * I have to go with TooMuchMath on this. The domain was not specified and the commmutativity is invalid anyhow. Let's not say $$\sqrt x$$ is undefined for $$x < 0$$ since this is false.--134.117.28.234 18:54, 30 November 2006 (UTC)
 * I agree with TooMuchMath's point on the domain not being specified, but it is true that the square root function and the square function commute under composition only for non-negatives. The last step in TooMuchMath's equation string is wrong: the two negatives should cancel to be a positive. Basically, $$(\sqrt x)^2$$ always returns the input, regardless of its sign, but $$\sqrt{x^2}$$ returns the number with the same magnitude as the input but with a non-negative sign, i.e., it is equivalent to the absolute value function. So for the square root and square functions to commute under composition for a particular number, that number must be the same as its absolute value: $$x \ge 0$$. David815 (talk) 23:39, 24 May 2014 (UTC)


 * Whether the domain was specified or not, $$\sqrt x$$ is not defined for negative values of $$x$$. Indeed, how can it be defined, even if complex numbers are allowed?  For example, what is $$\sqrt{-1}$$?  There are two different complex $$z$$ such that $$z^2 = -1$$. --Alexey Muranov (talk) 13:17, 17 January 2019 (UTC)
 * See Square_root. Although it is true that $$z^2 = -1$$ has more than one solution, it is conventional to define $$\sqrt{-1}$$ uniquely. McKay (talk) 06:29, 18 January 2019 (UTC)


 * You can surely select a function from a "multivalued" relation, but there is no special reason to do it this way and to prefer, for example, i over -i. If one uses the "principal branch of the square root," this should be specified, as this is not the default meaning of the square root. --Alexey Muranov (talk) 19:04, 18 January 2019 (UTC)


 * I am unaware of the convention to define $$\sqrt{-1}$$ uniquely. I know that in number theory it is customary to extend the field of rationals with element like $$\sqrt{2}$$ or $$\sqrt{-3}$$, but I thought that in this case either only the field extension matters, and it does not depend on the chosen value, or the element $$\sqrt{-3}$$ is treated formally, and not as a particular complex number. --Alexey Muranov (talk) 19:10, 18 January 2019 (UTC)

Something more general than function composition?
I'm wondering if there is something more general than function composition.

Example, I have a function f that maps elements of X onto real numbers. I use this function to define another function g that maps subsets of X onto real numbers--perhaps g gives an average, median, total, minimum, maximum etc.

How can I describe the relationship between f and g? Clearly I cannot say g is composed of f.  I want to say something like g is 'based on' f.  Anyone know of anything in the literature? If so, I guess there should be a link to funciton composition... —The preceding unsigned comment was added by 220.253.86.44 (talk) 00:03, 24 April 2007 (UTC).


 * Perhaps what you are looking for is the idea of an operator. Operators take one function as input, and spew out another function (or spew out a number, or a set ...). Operators that take two functions as input, and spew something out, are called binary operators. There are many, many operators. The classic example is D, which, given any function, spews out its derivative. One can certainly find ways of defining avg so that it is an operator. linas 03:01, 24 April 2007 (UTC)

Thanks for the suggestion and links. So, in my example, g would be an operator and f its operand. In this article composition is an operator, as is the function g o f. My main problem with this is that "operators" appear to be poorly defined and have multiple--conflicting--meanings.220.253.85.77 03:35, 26 April 2007 (UTC)


 * Don't know how to help you. As the article on operator says, an operator is "just a function". That's all, nothing more. That's neither poorly defined, nor is it conflicting. But you have to think "outside the box" to understand why "composition" (and things like it) are operators (i.e. are "just functions"). You have to answer the questions: what is the domain? what is the range? The domain and range of operators are typically not numbers of any sort. You might be looking for the concept of a dual space. In particular, you might be intersted in the space that is dual to the space of all functions-- see functional analysis. Or perhaps you're just looking for the concept of an integral -- the average of a function is just its integral...linas 04:53, 26 April 2007 (UTC)


 * To be fair, if an operator were "just a function" then there would be no point to operators at all! We already have functions! For operators to be useful, they must be a _type_ of function.  There appears to be some ambiguity with precisely what type of function an operator is!InformationSpace 02:56, 27 April 2007 (UTC)


 * What you're talking may be related to a closure in computer science. It's a function that takes an argument and returns another function generated from that argument.  Karl Dickman talk 21:05, 26 November 2008 (UTC)

In the opening line...
...the functions f: X → Y and g: Y → Z can be composed by computing the output of f when it has an argument of g(x) instead of x.

Should this not be computing the output of g when it has an argument of f(x) ? Tobz1000 (talk) 17:01, 19 May 2009 (UTC)


 * They can be composed in either order, but the one you propose would match the second paragraph and the image better, so I swapped them. &mdash; Carl (CBM · talk) 17:32, 19 May 2009 (UTC)

For instance, the functions f : X → Y and g : Y → Z can be composed by computing the output of g when it has an argument of f(x) instead of x.

Shouldn't that last bit be instead of y, now that the example has been changed? I don't feel qualified to make an edit, but I conferred with a friend and we agreed that it seemed like the anecdote was g(f(x)), where f(x) replaces y in g(y). GeoffHadlington (talk) 03:27, 29 August 2013 (UTC)


 * I've rephrased it. The "instead of" was confusing to interpret.  — Quondum 06:52, 29 August 2013 (UTC)

Semicolon for functional composition
A further variation encountered in computer science is the Z notation: is used to denote the traditional (right) composition, but ⨾ (a fat semicolon with Unicode code point U+2A3E[2]) denotes left composition.

The unicode notation is left untranslated on my computer, even though I have Unicode Arial which works well most of the time. I suggest that someone with knowledge of this add an explanation about where to get the font that would render this symbol. Better yet, why not just refer to it as ";"? The details of Z code is a very special subject that may not belong in this article. SixWingedSeraph (talk) 14:58, 31 August 2009 (UTC)
 * Cambria (typeface) has it (because is M$'s math font). JMP EAX (talk) 06:24, 24 August 2014 (UTC)
 * You can find a [partial] list of fonts that support it here. The article now mentions the semicolon too. JMP EAX (talk) 06:38, 24 August 2014 (UTC)

Function composition always associative?
The statement that function composition is always associative is obviously false. The referenced page on associativity gives serveral examples of non-associative functions, including substraction over the integers and cross product of vectors. —Preceding unsigned comment added by 132.198.98.23 (talk) 22:54, 23 March 2010 (UTC)

Sorry, I was momentarily confused with terminology. Of course there is a difference between the order in which one collaspes a chain of maps (associativity of function composition) and the order in which one creates pairs in a sequence of operands for the application of a function of the form A X A -> A (associativity of a binary operator). The later is often but not alway associative. —Preceding unsigned comment added by 132.198.98.23 (talk) 23:55, 23 March 2010 (UTC)


 * There are many non-associative algebras, for instance octonions or hyperbolic quaternions. A product of four elements will depend on the order of products in the whole. Say we have abcd as the ambiguous product. One association is ((ab)c)d which can be considered the operation of function b, then c, then d. Since the algebra is non-associative other compositions of the three functions may fail to give the same result. Here we have a case where function composition is non-associative. — Rgdboer (talk) 02:51, 29 April 2017 (UTC)


 * I don't which functions are composed in your example. Isn't your "((ab)c)d" just an example of a non-associative multiplication, not of a non-associative function-composition, similar to the previous 2 comments (of 23 March 2010)? - Jochen Burghardt (talk) 06:08, 29 April 2017 (UTC)

A function may be defined by multiplication: f(x) = 2x for example. I prefer to apply products to the right for agreement with reading order. Now let’s take ((ab)c)d as a sequence of functions on hyperbolic quaternions using a = 1, b = i, c = j, d = j. The first function is "multiply on the right by i", and the second and third functions are "multiply on the right by j". Now if the second and third functions are composed first, then the final result is i, not the –i of the other composition order.

Comprehending such a difference requires entry into a non-associative structure. The lack of symmetry makes such structures unappealing except for mathematicians like those studying octonions or Lie algebras or some other structure with enough order to accommodate non-associativity. But these studies do not arise in secondary school, so hand waving grants "composition of functions is always associative". The Encyclopedia should not perpetuate a false promise.Rgdboer (talk) 00:42, 30 April 2017 (UTC)


 * Foolishness. Function composition is always associative. This is easily proved (even if the proof is long and boring) usually in a college sophomore level introduction to proofs class. The proof is a direct application of the appropriate definitions and does not depend on any properties of the functions or any structure of the sets these functions relate. The error in your example is easy to see with the proper notation. Let $f_{z}(x) = xz$ where $z$ is any element of the hyperbolic quaternions (or any other non-associative quasigroup). What you are implicitly assuming is that $f_{z}∘f_{w} = f_{wz}$ which is not valid in a non-associative quasigroup since applied to $x$ the LHS = $(xw)z$ while the RHS = $(x)wz$.--Bill Cherowitzo (talk) 04:02, 30 April 2017 (UTC)


 * Let me be more concrete. With respect to the hyperbolic quaternions you have
 * ((1i)j)j = (ij)j = kj = −i while (1i)(jj) = i1 = i so clearly ((1i)j)j ≠ (1i)(jj).
 * Now consider the function compositions $f_{j}∘(f_{j}∘f_{i})(1) = f_{j}(f_{j}(f_{i}(1))) = ((1i)j)j = −i$ and $(f_{j}∘f_{j})(f_{i}(1)) = x$. Now if $f_{j}∘f_{j}$ is the identity function (multiplication by 1) because $jj = 1$, then $x = i$ as you claimed in this example. However, this composition is not the identity function. In fact, $f_{j}∘f_{j}(i) = (ij)j = kj = −i$, so in fact, $x = −i$ in agreement with the other composition.--Bill Cherowitzo (talk) 18:51, 30 April 2017 (UTC)

Thank you for spelling out that $f_{z}∘f_{w} = f_{wz}$ presumption and its invalidity. — Rgdboer (talk) 02:40, 1 May 2017 (UTC)

Typography again
As noted above in about six years ago, the appropriate symbol appears to be Unicode U+2218: &#x2218;. I find it displays correctly on IE9 and Mozilla Firefox 8, and is used in List of mathematical symbols. The large circle symbol used in this article is a disconcertingly large workaround. Is there any reason (now that browsers may reasonably be expected to support the more common Unicode symbols) not to update this accordingly in the article? — Quondum☏ 18:42, 11 May 2012 (UTC)
 * It depends less on the browser and more on the fonts installed. A list is here: . Support for it is indeed much better than for (say) \fatsemi. JMP EAX (talk) 06:57, 24 August 2014 (UTC)
 * is indeed now used throughout the article, except where  markup is used. -- Beland (talk) 18:40, 24 January 2021 (UTC)

commutativity and function composition
Regarding function composition for which g ∘ f = f ∘ g holds: Do composed functions which have this property have a dedicated name? Can they be considered symmetric functions? --Abdull (talk) 11:59, 18 July 2013 (UTC)
 * No, they can’t. Also, there is no such thing as “functions which have this property”, there are pairs of functions that commute, and this relation is not transitive. The nearest match to your query is a commutative subgroup of a group of transformations; see group action. Incnis Mrsi (talk) 12:37, 18 July 2013 (UTC)
 * Actually there is. It's called center of a semigroup. JMP EAX (talk) 20:23, 23 August 2014 (UTC)
 * And for a chosen subset (rather than the whole group/semigroup) it's called a centralizer. JMP EAX (talk) 07:13, 24 August 2014 (UTC)
 * Of course the place where you should have learned that is the article on commutativity (in general), but that's the usual polished turd. JMP EAX (talk) 02:07, 26 August 2014 (UTC)

Should this page be linked to the page: "Piecewise"?
The term "Composite function" would often be interpreted in mathematics as a "Piecewise" function. The closest wikipedia page to the desired result that turns up in a search of the term "Composite function" is "Function composition". Should someone add something like "Not to be confused with piecewise functions (piecewise)"?

Micsthepick (talk) 01:12, 5 June 2016 (UTC)

New expression of multivariate function composition
Please read 'New expression of multivariate function composition', Is it easy to be understand? Can you accept it?

For multivariate function composition


 * $$f (x_1, \ldots, x_{i-1}, g(x_1, x_2, \ldots, x_n), x_{i+1}, \ldots, x_n).$$

We will give it three expressions like (f.g) for unary function composition. In the expression of (f.g), '.' can be considered as a binary operation taking f and g as its operands or a binary function taking f and g as its variables.

For multivariate function, the first expression is like an operation:
 * $$(fC_{i}g)(x_{1},\cdots,x_{n})=f[x_{1},x_{2},\ldots,x_{i-1},g(x_{1},\ldots,x_{n}),x_{i+1},\cdots,x_{n}].$$

The second one is like a function:
 * $$[C_{i}(f,g)](x_{1},\ldots,x_{n})=f[x_{1},x_{2},\cdots,x_{i-1},g(x_{1},\ldots,x_{n}),x_{i+1},\cdots,x_{n}].$$

The third one is like a fraction:
 * $$[C_{i}\frac{f}{g}](x_{1},\ldots,x_{n})=f[x_{1},x_{2},\cdots,x_{i-1},g(x_{1},\ldots,x_{n}),x_{i+1},\cdots,x_{n}].$$

Why do we use these forms? We can describe any expression in a fire-new way. For example,$$ x^a+x^b$$,first we denote it as $$ f_{a}[f_{p}(x,a),f_{p}(x,b)]$$, in which $$f_{a}(x_{1},x_{2})=x_{1}+x_{2}$$ and $$f_{p}(x_{1},x_{2})=x_{1}^{x_{2}}$$. In addition, we denote subtraction as $$f_{s}$$,multiplication as $$f_{m}$$, division as $$f_{d}$$, root as $$f_{r}$$ and logarithm as $$f_{l}$$ respectively. We want give an expression like $$ [W(f_{a},f_{p})(x,a,b)]$$ in which the left part is called bare function containing only symbolics of function and the right part contains only variables.

$$ f_{a}[f_{p}(x,a),f_{p}(x,b)]$$ is an expression of a function of three variables. We consider $$ f_{a}$$ and $$ f_{p}$$ as especial functions of three variables too and introduce unary operator $$P^n_{i,j}$$ to express these especial functions of three variables.

$$[A^3_{1,3}(f_{a})](x_{1},x_{2},x_{3})=x_{1}+x_{3}+O(x_{2})=f_{a}(x_{1},x_{3})+O(x_{2})$$

Here $$ x_{1}$$ or $$ x_{3}$$ is transitional variable and $$O(x)=0$$..

$$[P^3_{1,2}(f_{p})](x,a,b)=x^a+O(b)=f_{p}(x,a)+O(b)$$

$$[P^3_{1,3}(f_{p})](x,a,b)=x^b+O(a)=f_{p}(x,b)+O(a)$$

By these examples we know the meaning of superscript and subscript of $$ P^n_{i,j}$$ and we call it function promotion.

It is clear that we obtain $$ f_{a}[f_{p}(x,a),f_{p}(x,b)]$$ by substituting $$ x_{1}$$ and $$ x_{3}$$ in $$ f_{a}(x_{1},x_{3})$$ by $$f_{p}(x,a)$$ and $$f_{p}(x,b)$$ respectively. So $$ f_{a}[f_{p}(x,a),f_{p}(x,b)]$$ can be written in:

$${\{}[P^3_{1,3}(f_{a})]C_1[P^3_{1,2}(f_{p})]{\}}C_{3}[P^3_{1,3}(f_{p})](x,a,b)$$

or

$$C_{3}{\{}C_{1}[P^3_{1,3}(f_{a}),P^3_{1,2}(f_{p})],P^3_{1,3}(f_{p}){\}}(x,a,b)$$

or

$$C_{3}\frac{C_1[P^3_{1,3}(f_a),P^3_{1,2}(f_p)]}{P^3_{1,3}(f_p)}(x,a,b)$$

We never mind how complex they are. We consider them as multivariate functions being composition results of two other multivariate functions being composition results and or promotion results. These new expressions are different from $$x^{a}+x^{b}$$. Actually we had departed bare function from variables in these new expressions and there is only one "x" in them. This is what we want to do when we solve transcendental equations like $$x^a+x^b=c$$.

For an unary function promotion, $$ P^n_{j}(u)=u(x_{j})$$. In special,$$u=x_{j}$$,$$ P^n_{j}(e)=x_{j}$$ in which 'e' is the identity function.

In $$C_{i}(f,g)$$ if $$ g=P^n_{j}(e)=x_{j}$$ and $$i\neq j,$$
 * $$C_{i}[f,P^n_{j}(e)](x_{1},\ldots,x_{n})=f[x_{1},x_{2},\cdots,x_{i-1},x_{i+1},\cdots,x_{j-1},x_{j},x_{j+1},\cdots,x_{n}].$$

Note,there is no $$ x_{i}$$ in the expression.

$$C_{i}[f,P^n_{j}(e)]$$ is called oblique projection of f. Actually it is a function of n-1 variables and is dependent on only f and i,j so we denote it as $$C_{i,j}(f)$$. For example,$$C_{i,j}(f_{p})(x)=f_{p}(x,x)=x^x$$ — Preceding unsigned comment added by Woodschain175 (talk • contribs) 22:34, 25 June 2017 (UTC)

Pipe symbol in math
You're right, my edit was not needed. Background: lists all irregular parameters. Especially "="-sign and "|"-sign may cause trouble in regular parameter input. e.g. when entering  has both errors.

Now that list has listed these two instances unnecessarily (because, a pipe in a wikilink works fine). When cleaning up that list, I assumed this was a problem. And since the list is recreated every month, I did so twice ;-). THe only advantage of using | would be, that it does not show up on the list again, in January. -DePiep (talk) 10:24, 4 December 2017 (UTC)


 * If it is helpful for your editing, I have no objections against | . I just hadn't understood the reason of your edits (twice); now I do. - Jochen Burghardt (talk) 18:54, 4 December 2017 (UTC)

Generalized composition - what's it good for?
This article defines something called "generalized composition" which ... strikes me as bizarre. I'm widely read in math, but have never encountered this definition before -- normally, one does not (cannot) assume that the arity of the composed multivariate functions are all identical, like that. Normally, one has just $$f(g_1(x_1,\cdots,x_k), g_2(y_1,\cdots,y_m), \cdots, g_n(z_1,\cdots, z_p))$$ and I'm wondering where the definition here is used, and what it's used for. It feels very linear-algebra-ish, without the linear. The reference on it says "universal algebra" .. I've gone through Paul Cohen's book "Universal Algebra", and I can't even begin to imagine how such a definition of "generalized composition" would appear in there. (I looked: the index points at page 113 which states that the universal functor on the category of sets is the free composition of canonical morphisms. That's not only a mouthful, but is also like a totally different universe than the one here...)  Am I being stupid? What is this thing used for? It looks pretty... 67.198.37.16 (talk) 03:21, 21 December 2017 (UTC)
 * Oh, I think I get it, one can just mash up the arguments I use above into one big giant vector $$(x_1,\cdots,x_k, y_1,\cdots,y_m, \cdots, z_1,\cdots, z_p)$$ and adjust the defintion of the g's to work with that, and you get the "generalized" definition. Its still kind of bizarre do me, because it "hides" or makes invisible the use of a giant Diagonal morphism, pretending its not there when it is:  that is:
 * $$f(g_1(x_1,\cdots,x_m), \cdots, g_n(x_1,\cdots, x_m)) = (f \circ (g_1,\cdots,g_n) \circ \Delta_{nm})(x_1,\cdots, x_m)$$
 * where
 * $$\Delta_{nm}(x_1,\cdots, x_m)= ((x_1,\cdots,x_m), \cdots n\;times\cdots, (x_1,\cdots, x_m))$$
 * is the diagonal morphism that makes n copies of a vector of length m. Hiding the presence of the diagonal in the definition, pretending its not there when it is has already caused grief on an unrelated project I work on. Which is why I'm interest in my original question: what's it good for? 67.198.37.16 (talk) 04:16, 21 December 2017 (UTC)

"Pointwise" application
I agree to that "pointwise" doesn't make sense in the first lead sentence. Although the article pointwise is not very clear, I take from it that an operation o: Y × Y → Y on a set Y can be lifted pointwise to an operation O: (X→Y) × (X→Y) → (X→Y) on the set X→Y of all functions from X to Y, by defining (O(f,g))(x) = o(f(x),g(x)) for all x∈X. Commonly, o and O are denoted by the same symbol. The article gives addition and multiplication (apparently of real numbers) as examples. This agrees with what I'd learned as a student. In function composition, we needn't have two functions with same domain and range set, so the notion of "pointwise" isn't applicable. - Jochen Burghardt (talk) 08:31, 17 January 2019 (UTC)

The context for this discussion: currently, the first sentence reads "[...] function composition is the pointwise application of one function to the result of another [...]". This is not exact and makes no sense because the results of the second function are not necessary functions themselves, but can be numbers, and "pointwise application" of something to a number makes no sense.

Here is an example of the actual pointwise application: let $$f$$ be the function on real numbers that for each $$x$$ yields the operation of multiplication by $$x$$, and let $$g$$ be the identity function on the reals. Then the pointwise application of $$f$$ to $$g$$ yields the function $$h(x) = x^2$$.

The operation of pointwise application is basically the combinator S of lambda-calculus and combinatory logic: $$S(f)(g) = x\mapsto f(x)(g(x))$$.

Also, one should not say "poinwise sum of the result of $$f$$ and the result of $$g$$" when what one means is the poinwise sum of $$f$$ and $$g$$.--Alexey Muranov (talk) 09:19, 17 January 2019 (UTC)


 * I also oppose to the formulation of the current first sentence, however I think that it should be me made explicit how the definition of the "composition of two functions" (being again a function) relies on "pointwise application" of functions to elements of their domain. I do not know whether in this article "composition" should be viewed as operation in some algebra of functions. I do not experience a serious flaw in considering "pointwise results of a function", contrasting this with some "image of a subset" of this function's domain, while I shy away from using a "pointwise function". Purgy (talk) 10:44, 17 January 2019 (UTC)


 * Purgy, composition of functions makes no use of pointwise application. Application of a function to elements of its domain is an ordinary application, not pointwise.
 * If $$\theta$$ is an operation of two arguments (like addition, multiplication, application, composition, etc.), then pointwise $$\theta$$ is the operation $$\Theta$$ defined by $$\Theta(f,g) = x\mapsto\theta(f(x),g(x))$$. --Alexey Muranov (talk) 12:22, 17 January 2019 (UTC)

Yes, the use of 'pointwise' in that sentence is a clear conceptual error. Was first introduced here as the sentence In mathematics, function composition is the pointwise application of one function to another to produce a third function. which is still wrong, but at least one can reinterpret the meaning of 'pointwise' to be exactly what is done during composition and the sentence would be correct, although circular. It was made worse in as     In mathematics, function composition is the pointwise application of one function to the result of another to produce a third function. for which there is no salvaging. It doesn't look like there is any conflict here. Even the editor who introduced it agrees. Everyone can see that composition is not a result of any pointwise application of two functions, or worse of a function and an element, even if the latter is seen as the inclusion map. Cactus0192837465 (talk) 13:03, 17 January 2019 (UTC)


 * I am unaware of a specific, technical meaning of "pointwise", and rather convinced that "functional composition" is based on such a "pointwise property". I consider "pointwise" as contrasting to "local", which I take as an attribute fitting to the definition of "germs", or to "global", which I would use for "functional identity", or also, as I wrote above, "pointwise" contrasts to "image" of subsets of the domains. Assuming suitable domains, codomains and quantification, I consider
 * $$(f\circ g) (x)\overset{\text{def}}{=}f(g(x)),\quad h(x) \overset{\text{def}}{=}(f\circ g) (x),\quad h \overset{\text{def}}{=}f\circ g $$
 * to be "pointwise definitions" of a functional composition and a function $$h,$$ and a "global definition" of $$h,$$ respectively. I do not understand the motives for removing "pointwise". In the formulation $$\text{“}\Theta(f,g) = x\mapsto\theta(f(x),g(x))\text{”}$$ I recognize $$\Theta$$ as a binary operation on functions (without any referral to any algebraic structure), but I cannot associate a derivable meaning to $$\theta.$$ Also, I am not versed in lambda calculus $$(\text{“}\mathbf{S}(f)(g) = x\mapsto f(x)(g(x))\text{”}).$$
 * Now the lead does not contain any blatant errors I am aware of, but I think it still deserves improvement. Purgy (talk) 12:51, 18 January 2019 (UTC)


 * Because 'pointwise' is meaningless in 'pointwise application of one function to another' and because it is fundamentally wrong in 'pointwise application of one function to the result of another'. Pointwise is applied in mathematics exclusively as defined at the beginning of this section. You being unaware means next to nothing, really. Wikipedia wouldn't be very useful if it were based on what you are aware. The property $$\Theta(f,g)(x)=\Theta(f(x),g(x))$$ is a property not satisfied by all operations on functions $$f,g$$. That it is why it is given a special name, a pointwise operation. Convolution is an example of an operation of functions which is in general not pointwise. Local is unrelated to poitwise. Local is applied to properties, in topology. Someone could use, if they wanted to, the awkward term pointwise property. Perhaps to refer to a property that depends only on the value at a point. Sign could be called a pointwise property. Terminology is flexible, but any such verbal acrobatic that is not widely used should only be included if references exist using it. Not because some random guy online has the feeling that it sounds kind-of-not-wrong-I-don't-know-I-am-not-aware-of. Cactus0192837465 (talk) 15:04, 18 January 2019 (UTC)


 * , I am pretty used to the meaning of "pointwise" in the sense I described, as a characteristic of certain operations on functions, of which composition is not an example. I think I also understand what use of the word "pointwise" you could be referring to, but this is not how it was used in the edited sentence.  According to your description, I believe you are talking about pointwise definitions, not pointwise operations.
 * The composition of functions is indeed normally defined in a pointwise manner: $$(g\circ f)(x) = g(f(x))$$ (doing differently would be a bit tricky, and it might be necessary to start with some other operations and maybe even operations over operations...). This pointwise definition, however, does not use any pointwise operations.
 * The opposite of "pointwise" in relation to definitions or theories is "point-free" or "pointless". See, for example, Point-free topology or Point-free programming.  In point-free programming, sometimes a function is called "point-free", while in fact it is its definition that is point-free, but one can argue that in programming a function is inseparable from its definition. --Alexey Muranov (talk) 16:31, 18 January 2019 (UTC)

Ooops! When I started this talk section, I thought you'd all agree to my understanding of "pointwise" - but it seems there are many different understandings around. I think it would be helpful if some people could give a formal definition of their understanding of "pointwise opertion", "pointwise definition" etc. I think the page Talk:pointwise would be most appropriate for this, since the article "pointwise" could well be improved by adding such definitions.

As for the "function composition" article, I take from your above discussion that "pointwise" should be avoided in the lead. While it might be possible to define composition "in a pointwise way" (this vague notion reflecting my lack of understanding) by employing lambda calculus and higher-order functions, this would be far too complicated for the beginners' audience the article is intended for. - Jochen Burghardt (talk) 19:26, 19 January 2019 (UTC)
 * In every reference that you pick up 'pointwise' is used (either explicitly or implicitly) to mean an operation defined to commute with all point evaluations. Composition isn't. The only ones arguing are basing their rants on "I am unaware", or "I consider", or "I do not understand". Unless they can produce a reliable source, their rants can be simply ignored. Wikipedia is not for arguing what a random person believes things could be called. Given that this is not a forum, either they bring a reliable source, or scram. They can post their believes in a blog or something. Cactus0192837465 (talk) 20:16, 19 January 2019 (UTC)


 * , as I mentioned above, "pointwise" is quite a general term and can be used in different contexts, not only to mean a pointwise operation. Though, here, for example, the term "pointful" is used as the opposite of "point-free"/"pointless", while i think that "pointwise" would be appropriate as well. --Alexey Muranov (talk) 21:31, 19 January 2019 (UTC)

Here is a relevant opinion on math.SE. --Alexey Muranov (talk) 21:38, 19 January 2019 (UTC)
 * You don't need to mention it again. That is your original research and it doesn't belong in Wikipedia. You are simply conflating two words, pointwise and pointful, for the mere fact that they share the same root (point). In other words, the relation that you are saying, you are making it up. Their uses are unrelated, period. Likewise, math.SE is a forum, not a reliable source (even though the link agrees with what we are saying). Cactus0192837465 (talk) 21:46, 19 January 2019 (UTC)


 * This is not my research, this is about an English word, which means what it means. It is reasonable for the article "pointwise" to mention other possible meanings, besides operations on functions. Like pointwise reasoning. See also on DuckDuckGo and on Google --Alexey Muranov (talk) 23:12, 19 January 2019 (UTC)


 * StackExchange sites are not exactly forums IMO, answers are individually upvoted/downvoted and commented. --Alexey Muranov (talk) 23:19, 19 January 2019 (UTC)
 * This is the talk page of function composition, neither pointwise's, nor Wikipedia's definition of reliable sources. Dixi. Cactus0192837465 (talk) 00:07, 20 January 2019 (UTC)


 * Well, i was just reacting to your previous comments, where your were discussing the meaning of "pointwise" in general (on this talk page)--Alexey Muranov (talk) 00:27, 20 January 2019 (UTC).

While I appreciate your mathematical knowledge, in my opinion you could try harder to adhere to WP:POLITE. As for sources, I do consider it admissable on a talk page to contribute some statements without having a source at hand. Glancing through the current section, I don't find any source that you gave here; however you were far from scamming. Moreover, please note that I didn't just say "I don't know"! The only thing that comes close to a formal definition of "pointwise" here is mine. Your (unsourced) description "an operation defined to commute with all point evaluations" can probably turned into one, but you didn't do that. - Anyway, I think the issue of "pointwise" is settled as far as it is relevant here, but further clarification on that is useful for the "pointwise" article. - Jochen Burghardt (talk) 10:08, 20 January 2019 (UTC)
 * The definition that I gave in words is exactly the same definition that you wrote in formulas, which is exactly the same definition that is used (whether or not they make it explicit) in all texts in which 'pointwise' is used. I gave its translation into words to avoid repetition. There are references already in the Wikipedia page, and opening any book in algebra or analysis that uses the word will have the same use. All my impoliteness is directed to Purgy Purgatorio, since they worked enough to earn it. Cactus0192837465 (talk) 13:17, 20 January 2019 (UTC)
 * I inserted my above formal definition, slightly fleshed out, at the pointwise article. - Jochen Burghardt (talk) 18:49, 20 January 2019 (UTC)

Planned edits
I'm a big fan of the concept of providing real life examples to illustrate a point, so:

Seems initially promising. However, my understanding of oxygen concentration is that it is roughly constant by elevation. Perhaps it is not but I suggest that the average layperson doesn't have a good sense of the meaning of this function, and therefore gets distracted by issues that have nothing to do with the point being illustrated. (I can go into more details if desired). the good news is there is a very simple solution. I think the average person knows that temperature tends to vary by elevation, so the concept of a function describing temperature as a function of elevation is reasonably well understood. There is no problem with the concept of them airplanes elevation expressed as a function of time t, so this example would work better if we talked about:

Any disagreement?

The composition example:

Composition of functions on a finite set: If f = {(1, 3), (2, 1), (3, 4), (4, 6)}, and g = {(1, 5), (2, 3), (3, 4), (4, 1), (5, 3), (6, 2)}, then g ° f = {(1, 4), (2, 5), (3, 1), (4, 2)}} isn't wrong but it's hardly intuitive. I suggested people new to the concept of composition functions won't find this example insightful.

In contrast, is a wonderful example on the right side of the page, expressed as a graphic with the caption "concrete example for the composition of two functions". I suggest moving not that graphic up so it appears to the right of the composition example, then change the values in the composition example to match that graphic. Then readers who are not perfectly clear on what's going on in the text and formula portrayal can look at the graphic and gain insight.

This also a graphic with the caption g ∘ f, the composition of f and g. For example, (g ∘ f )(c) = #. I think someone was trying to be clever using symbols in Z, but in an introductory exposition this just adds a level of confusion that isn't matched by useful insight. while slightly more general than the concrete example, I don't see that it adds anything useful and propose simply removing it.

I don't believe this graphic is referred to in the text, so unless it had something that I'm missing, I don't think it deserves to remain.-- S Philbrick  (Talk)  20:24, 20 July 2020 (UTC)


 * I think these are good ideas, except that using "c" for "temperature" is confusing, as is using "t" for both "time" and "temperature". What about e.g. atmospheric pressure ("p") instead? - Jochen Burghardt (talk) 09:22, 21 July 2020 (UTC)
 * , Good point. Proposed wording:
 * S Philbrick (Talk)  22:53, 21 July 2020 (UTC)


 * What about either "an airplane's height" and "h(t)", or alternatively "an airplane's elevation" and "e(t)", to get the abbreviation matching the word here, too? I don't speak English sufficiently good to judge whther "an airplane's height" could be misunderstood as the distance of its ceiling from its floor. - Jochen Burghardt (talk) 12:12, 22 July 2020 (UTC)
 * , I'm going back and forth on this. Maybe just my upbringing, but I'm not a big fan of using "e" to designate a function. not completely opposed, just don't like it. I agree with you that elevation is less likely to be ambiguous then height. I do agree with the preference to have the letter of the function match the letter of the concept. It's obviously not required but it does add something. That leads me to two options:
 * I slightly prefer the first. S Philbrick  (Talk)  13:39, 22 July 2020 (UTC)
 * I slightly prefer the first. S Philbrick  (Talk)  13:39, 22 July 2020 (UTC)
 * I slightly prefer the first. S Philbrick  (Talk)  13:39, 22 July 2020 (UTC)
 * I slightly prefer the first. S Philbrick  (Talk)  13:39, 22 July 2020 (UTC)
 * I slightly prefer the first. S Philbrick  (Talk)  13:39, 22 July 2020 (UTC)
 * I slightly prefer the first. S Philbrick  (Talk)  13:39, 22 July 2020 (UTC)

I made the changes. I'll copy the graphic removed here, in case anyone is interested. -- S Philbrick (Talk)  14:15, 22 July 2020 (UTC)
 * An airplane's altitude, as measured by an altimeter, would be different from its elevation, which could be an angle, as measured by remote sensing equipment, such as a telescope or radar (a radar might also measure range to the object). Sorry to comment late. --Ancheta Wis   (talk  &#124; contribs) 14:26, 22 July 2020 (UTC)
 * So we could use "an airplane's altitude" and "a(t)", what about that? I feel "a" is a slightly better function name than "e", although not quite as good as "h". - Jochen Burghardt (talk) 15:17, 22 July 2020 (UTC)
 * , Sounds good to me. S Philbrick  (Talk)  22:27, 23 July 2020 (UTC)

Leading image
I propose to insert at the very beginning the present SVG image and its caption,  with two meanings of “composition”:  the binary operation, also a result of the operation. Arthur Baelde (talk) 13:55, 1 March 2023 (UTC)


 * I reverted on the grounds of wp:coatrack -- Ancheta Wis   (talk  &#124; contribs) 13:32, 13 May 2023 (UTC)