Talk:Graph of a function

Topics missing

I was missing these kind of topics. We should also make a list of famous curves if it isn't yet somewhere in Wikipedia. For instance the Watt's curve in spherical polar coordinates: r² = b² - (a sin φ ± √(c² - a cos² φ))² and many more ... --XJamRastafire 19:15 Sep 18, 2002 (UTC)

---

Another Topic Missing: Graphing Functions. Example: How do you graph the function -3 if x≤-4?

---

—Preceding unsigned comment added by 99.37.114.196 (talk) 22:42, 19 October 2010 (UTC)

Please, I have read "We can approximate a function --by mean of several methods-- given a functional dependence of adequate size". It seems to me that "Graph of a function" and "Functional dependence" are very closed concepts and clearly represented by a two columns table with a picture like the following:

x | y
--+---
5 | 11
2 |  5
1 |  3

Please, let me know if you know such synonym and if so, where is (and who wrote) the original definition of such a type of "functional dependence"? For my part, I know that E. F. Codd in 1972 applied the concept and used the term as a mean of database design verification/normalization. Dr. Amstrong axiomatized this kind of dependences in 1974. I try to found the original mathematical concept before its computer application (if really such thing existed before Codd/Amstrong). Thank you. [Enrique Villar; mailto:evillarm@capgemini.es]

Graph of a function equals the function?

Article says "In mathematics, the graph of a function f is the collection of all ordered pairs (x,f(x))". The definition given is the definition of a function (so it says that the graph of a function is exactly equal to the function (by set equality)). 24.84.213.237 07:38, 7 May 2007 (UTC)

That's true if you define functions that way. But I don't think that's a good way to define functions - you really need the codomain as part of the definition, otherwise how can you tell whether or not the function is surjective? See also Function (mathematics)#Is a function more than its graph?. --Zundark 08:56, 7 May 2007 (UTC)

The article is now self-contradictory on this point, saying both "Note that although a function is always identified with its graph, they are not the same because it will happen that two functions with different codomain could have the same graph." and "In the modern foundation of mathematics known as set theory, a function and its graph are essentially the same thing." Which is it? The second statement is sourced, while the first is not, but if the concept of a codomain is part of the definition of a function, the first has to be correct. Also, the section "Is a function more than its graph?" referenced above has disappeared some time in the last decade.

Although it does seem peculiar that $\exp :\mathbb {R} \to \mathbb {R}$ and $\exp ':\mathbb {R} \to \mathbb {R} ^{+}$ would be considered different functions, but if codomains are part of the definition of a function then this would seem to be necessary. As a non-mathematician, I'm kind of surprised the standard formulation isn't that "a function is surjective with respect to a codomain", rather than considering surjection, injection, etc. part of the intrinsic properties of a function, but if this really is the standard formalism then this point should be unambiguous in the article here. Geoff (talk) 19:21, 6 January 2018 (UTC)

Actually, the section Binary relation#Is a relation more than its graph? seems to address this point fairly well. Perhaps this article should be reworded to parallel the discussion there. — Preceding unsigned comment added by Geoff (talk • contribs) 21:29, 6 January 2018 (UTC)

I have edited the article for removing some considerations that are WP:OR (editors opinions), and avoiding technical details that are interesting only when considering the logical formalization of the foundations of mathematics.

About subjectivity, ..., the domain and the codomain are, normally, a part of the definition of a function. This explain the standard formulation "a function is surjective", without specifying (again) the codomain. D.Lazard (talk) 22:30, 6 January 2018 (UTC)

Definitions in mathematics, and everywhere really, are context dependent. There is the set-theoretic definition, used in foundations of mathematics, and found in pretty much all introductory texts, in which a function is a special type of relation, a set of pairs. In this definition, as you noted, the graph of a function is exactly the same as the function. Also, the co-domain is not determined by the function in this definition. Now, there is the notion of morphism in the category of sets. These are functions (in the set-theoretic sense), but seen as arrows from its domain to its co-domain. As such, the co-domain is part of what the morphism is, as is the domain, and the relation itself. In contexts in which there is interest in studying the properties of functions as morphisms, emphasis is made on making the co-domain part of the definition of the function. For example, if you want to say what it means for a morphism (function!?) to be surjective, you need to look at the morphism, and not just the relation (the set of pairs, the graph alone). Category theory is relatively modern compared to the (set-theoretic) definition of function. It takes some time for language to homogenize. In many introductory books you can even find these two definitions (set-theoretic function and morphism of the category of sets) conflated. They would give the set-theoretic definition as a set of pairs, but then claim, contradictorily that the co-domain is part of the definition of the function. But as of now, both ways of looking at functions are widely used. One only need to make sure in each case, which is being used. Meanwhile, Wikipedia would have to also present all the aspects of the current use of the terms 'function', 'graph', 'morphism', etc. Cactus0192837465 (talk) 18:54, 8 September 2018 (UTC)

A quote from Halmos' A Hilber Space Problem Book, annotation in brackets from me:

   The terminology is standard [referring to graph]. It is curious that it should be so, but it is. According to a widely adopted approach to the foundations of mathematics, a function, by definition, is a set of ordered pairs satisfying a certain univalence condition. According to that approach, the graph of A is A, and it is hard to see what is accomplished by giving it another name. Nevertheless most mathematicians cheerfully accept the unnecessary word; at the very least it serves as a warning that the same object is about to be viewed from a different angle.

Cactus0192837465 (talk) 19:33, 8 September 2018 (UTC)

See also Apostol's Mathematical analysis, which doesn't define graph, but does define (page 35) function, and after that defines mapping, making the distinction clear between these two different concepts. See Bridge's Foundations of Real and Abstract Analysis, in which (page 285) it defines graph of a mapping, not graph of a function, and clarifies (just like Halmos) that the graph is the same as the function. Cactus0192837465 (talk) 21:29, 8 September 2018 (UTC)

Is it typically y vs x or x vs y?

Is it typically y vs x or x vs y? —Preceding unsigned comment added by 64.180.160.235 (talk) 05:23, 16 January 2009 (UTC)

What is typically used is y vs. x, such that x is horizontal and y is vertical. However, when specifically talking about plotting a function vs. its input, it is more clear and intuitive to plot f(x) vs. x (or f(y) vs. y or whatever), since the variables x and y are just placeholders. EmergencyBackupChicken (talk) 17:00, 7 May 2009 (UTC)

Graph vs. Plot?

The term `graph' should really be restricted to use when referring to actual graphs: nodes and edges. The `graph' of a function as described here is really its plot. This is a common misconception that leads to much confusion, and it irks me that it shows up a lot, even in academia. What would be a good way to incorporate this information while still allowing people to find what they are looking for after being told the wrong term? EmergencyBackupChicken (talk) 17:00, 7 May 2009 (UTC)

The word graph is used in both senses in mathematics. I don't think I've ever seen plot used to refer to the graph of a function (as opposed to a graphical representation of the graph of a function). --Zundark (talk) 09:13, 8 May 2009 (UTC)

From the linked Wolfram Mathworld article, "A graph is sometimes also called a plot." I strongly agree with EmergencyBackupChicken. As far as I can tell, the only purpose this page serves is to perpetuate the misconception that a plot of a function is a graph. I have never seen nor heard that the graph of f is the set {x, y | f(x) = y}. I can't imagine how this definition is useful. Conveniently, I can't check the references. Metaquanta (talk) 04:28, 1 November 2018 (UTC)

This discussion is eternal, and almost unfixable at this point. If talking formally I prefer to refer to 'graph of a function' as a chart (both a set of elements in $R^{n+m}$ and its graphical representation on carthesian plane for n=1 and m=1), even if often chart is actually a graphical representation of the graph (nodes and edges), especially in terms like organizational chart, where you have some directed acyclic graph and nodes are represented as boxes or ovals. In my native language, Polish, we refer to 'graph of a function' as wykres, and to graph (nodes and edges) as graf'. So, I think it is less confusing (we still call portable graphic calculator that makes graphical function charts, as kalkulator graifczny). That is why I like to use term chart for graphical representation of the function (its 'graph'). For completeness, the image is called obraz in Poland, which is basically direct naive translation (one can say image is really wizerunek and obraz is picture, but wizerunek can also be a picture, especially if it is a portratit / picture of a person. It also make sense that wizerunek can refer to methaphorical sense of picture, i.e. I picture him as a serious person., but image and picture, are essentially synonymous, similar to obraz, portrait, wizerunek and obrazek, last one usually a small picture in a picture frame on a wall). Unfortunately chart doesn't play well with more than 2 dimensions, and formally in English, nobody uses chart of a function in formal definitions. To add to the confusion geometria wykreslna, is usually a term used for 2d and 3d drawings used in engineering and CAD, and involves things like projections, intersections and possibly more complex curves, all usually in 2D. And as a last point, formally most graph theory people refer to graphical representation of a graph on 2-d plane as a drawing or sometimes embedding. 81.6.34.246 (talk) 21:11, 8 November 2019 (UTC)

Merge epigraph and hypograph

Someone else proposed that Hypograph (mathematics) and Epigraph (mathematics) be merged with each other, but I think they should both be merged here. As far as I can tell from a textbook I looked at briefly, not much can be said about these two notions besides their definition. So I propose to merge them here as derived concepts. JMP EAX (talk) 08:27, 1 August 2014 (UTC)

I would prefer to merge them with each other but not here. For one thing they do have some properties that are different from the graph of a function (e.g. the application to the definition of convex functions). For another, the graph of a function is a topic of great importance in elementary and secondary-school mathematics education and I think adding more advanced concepts such as epigraphs to our article will violate WP:TECHNICAL. —David Eppstein (talk) 01:47, 3 December 2014 (UTC)

I agree with Eppstein. The notion of epigraph is quite unrelated with what is discussed in this page. --Txebixev (talk) 22:25, 25 January 2015 (UTC)

"Someone else proposed"? I have not found this someone else in the corresponding talk pages. --Txebixev (talk) 22:27, 25 January 2015 (UTC)

Links to pages about graph theory in the 'See also' section

In the 'See also' section of this article, there are links to the articles Graph (mathematics) and Graph theory. Mathematically, these have nothing to with the graph of a function, so I think there's a case for removing them. I suppose they might possibly be useful for someone who was confused about the two uses of the word 'graph' in mathematics, but it seems to suggest a link between two areas of mathematics that isn't there.  J.Gowers  19:53, 1 May 2015 (UTC)

 Fixed by removing these links and expanding the hatnote. D.Lazard (talk) 20:38, 1 May 2015 (UTC)

Requested merge and move 14 January 2016

The following is a closed discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a move review. No further edits should be made to this section.

The result of the move request was: no consensus. Jenks24 (talk) 08:50, 1 February 2016 (UTC)

---

Graph of a function + Plot (graphics) → Graph (plot) – The scope of this article should be widened to cover the graph of a relation, of an equation, etc., which do not have their own articles. At the same time, I am proposing to merge it with Plot (graphics), because their scopes are strongly overlapping; note that it has been "proposed" to move this article to Plot. Petr Matas 11:42, 14 January 2016 (UTC) Relisted. Jenks24 (talk) 05:53, 24 January 2016 (UTC)

• Strongly oppose to both move and merge. The move proposed for Graph of a function is confusing: firstly, "graph of a function'" shows clearly that it is about mathematics, which is not the case of Graph (plot). Secondly, the proposed title suggest wrongly that the article is about plotting of the graphs of graph theory, which is an interesting and difficult problem that deserves its own article. Thirdly, the subject of Plot (graphics) is, or should be, much wider than plotting functions and relations, it includes the plot of any figure such as triangles, and many other things. Also, the concept of "graph of an equation" seems original research, and "graph of a relation" is an ambiguous concept, as a relation on a set S is nothing else than a directed graph that has S as a set of vertices. D.Lazard (talk) 13:16, 14 January 2016 (UTC)

  Your objections seem valid except for the equation (it is quite common to display graphs (or plots?) of x² + y² = 1 and the like) and the relation (on R; for example, inverse of sin x is a relation, but not a function). Is there a better title, which would include these concepts? Petr Matas 13:55, 14 January 2016 (UTC)

  Comment: There is a confusion here between the common meaning and the mathematical meaning of "graph", and this is this confusion that I consider as original research: the graph of a function is a mathematical object (a set of pairs of numbers) which is defined independently of any plotting. On the contrary, it is not usual in mathematics to call "graph of an equation" the curve of the solutions of a (bivariate) equation; again, this curve is defined independently of any plotting (or graph). The distinction appears clearly in Curve sketching. "Graph of an equation" is further confusing, as it is not the equation that is graphed, but the curve of the solutions. D.Lazard (talk) 14:46, 14 January 2016 (UTC)

---

The above discussion is preserved as an archive of a requested move. Please do not modify it. Subsequent comments should be made in a new section on this talk page or in a move review. No further edits should be made to this section.

"Graph of a function"; ???? of a curve

The article seems to imply that one can only graph functions. Many curves are not functions!, but I think I see graphs or plots of various curves every day. In math, I see ellipses, spirals, etc. In physics, I see foldback (power supply design), negative resistance, and their I-V curves, pressure–volume diagrams, etc. But I don't know what to call them. Wikipedia shows many curves, but only seems to call them "curves", never naming the 2-D graphic representation thereof (even though functions have graphs). Is the representation of a curve a graph? a plot (like a scatter plot or "x-y chart with connecting lines" that every spreadsheet program can "chart")? a diagram (which article only includes graph of a function)? a chart (though the article omits x-y chart)? Or else what is it? An ellipse and a spiral can be polar functions, but an off-center circle has two (or zero) radii per angle, so is not a polar function either. For fun, some named functions have "curve" in their name (yield curve, ...).

The term might be "graph of a relation", which redirects to one paragraph in the intro of graph of a function: "The concept of the graph of a function is generalized to the graph of a relation." (The bolding there implies that "graph of a relation" is a co-subject of the "graph of a function" article.) "Relation (mathematics)" redirects to "Binary relation", which talks "functional relations" and "non-functional relations" but doesn't show any non-functions, and doesn't mention the word "curve", or state that a "relation" can be a "curve" of any kind.

Maybe this article should be "Graph", after bumping the existing "Graph" to "Graph (disambiguation)". Then this article can cover "graph of a function" and "graph of a relation" without implying that you can't graph a Lissajous curve. - A876 (talk) 05:35, 6 June 2019 (UTC)

Here, we have a conflict with the common meaning (graphical representation) of "graph", and one of its mathematical meanings. This is rather common, and is generally solved by using, in a mathematical context, a synonymous for the common meaning, here, "plot" instead of "graph". Here, the problem is enforced by the fact that curves have a very long history (more than 2000 years), and that the need of distinguishing between a curve and its graphical representation is very recent (about 100 years). It follows that the disambiguation between the different meanings of a graph is often left to the context. Moreover, using a fully unambiguous wording would often appear as pedantic, which is not really useful.

Nevertheless, as I understand your concern, I have added "For graphical representation see Plot (graphics)" in the hatnote. D.Lazard (talk) 09:45, 6 June 2019 (UTC)

(When I browse for info on "graphs" here, I have to be careful not to sidestep into that otherworld of "graph theory", where the connectedness of those lines, arrows, and dots or circles means something, but their coordinates and curvatures don't mean anything.)

I woke up thinking of this: "The graph of an equation".

Equations and parametric equations don't always define functions. x² + ^y2 = 1 "looks like" a unit circle.

In apparent popular usage (Google hits), one can "plot the function" (6,640,000), "graph the function" (2,710,000), "plot a function" (853,000), or "graph a function" (723,000). "Plot" is most-often the verb. One can "plot the equation" (6,600,000) or "graph the equation" (2,060,000), but it's the opposite balance with the indefinite article: "graph an equation" (72,300) or "plot an equation" (19,300). "Plot" is still most-often the verb. But "plot" could mean "make a plot" or "make a graph", and "graph" could mean "make a graph" or "make a plot".

One can "plot the graph" (4,010,000) or "plot a graph" (481,000); one is much less likely to "graph the plot" (293,000) or "graph a plot" (74,500). So "plot" is what you do - the action, and "graph" is what you made (plotted) - the picture. (See Plotter.)

Incongruously, "plot of a function" (44,000,000) outnumbers "graph of a function" (13,500,000), but "plot of a function" seems to include "graph of a function" and other results.

Finally, "plot the graph of a function" (250,000) outnumbers "graph the plot of a function" (0), and "plot the graph of an equation" (98,500) outnumbers "graph the plot of an equation" (0).

I withdraw my prior renaming suggestion. Maybe this article could or should generalize by being renamed "Graph of an equation".

Equations and parametric equations can be said to define "binary relations". Thus another possible title is "Graph of a binary relation". But there are also ternary relations, so "Graph of a relation"? (No, those seem too abstract.)

A graphing calculator "is capable of plotting graphs..." (wherein "graphs" links to this article (Graph of a function)). So a graphing calculator "plots the graph of a function". But the calculators plot ellipses etc. too, from their equations.

I didn't find much else contradicting. A previous version of Plotter (disambiguation) had a clunker, but I replaced it. - A876 (talk) 23:37, 6 June 2019 (UTC)

I also withdraw my previous "proposal" (above). This one might be ready to play:

I think the concept embodied in THIS article covers:

• graph of a [binary] relation
  • graph of a function y = x ÷ x
  • graph of an equation x² + y² = 1; x² = y²
  • graph of an inequality ("a relation" (not wikilinked)) y ≤ x + 5; x² + y² ≤ 1
  • graph of a curve
  • graph of a parametric equation x = cos t and y = sin t; same as (x,y) = (cos t, sin t)
• graph of a point in two-dimensional Euclidean space
  • graph of an ordered pair, or 2- tuple (looks like a point in a plane)
  • graph of a data set, where x = value in column A and y = value in column B
• graph of a sequence
  • graph of an integer sequence
  • graph of an array data structure
  • graph of a list (abstract data type)
  • graph of a data set, where x = row # and y = value in column (scatter plot; bar graph, etc.; histogram)

Therefore its best title is, simply, Graph. THIS is the "graph" that everyone learns first, circa 7th grade. So why is "Graph" a disambiguation page? THIS should be "Graph"; THAT should be "Graph (disambiguation)".

SOME OTHER THING, that other kind of graph, which maybe 10% of the population learns about, after high school; and maybe 2% are aware of; that thing with the circles, lines, arrows, and labels, but no coordinates; that thing that looks like (I said it!) a diagram IS NOT called, simply, a graph. It is "Graph (discrete mathematics)", apparently part of "Discrete mathematics". As a "graph", it is secondary, or at best parallel to the "continuous" graph – possibly an underlying generalization thereof, but more like a different concept with a common ancestor. What it has in common is pen, ink, lines, and paper. It SHAREs the name (it is graphical), but "graph theory" isn't primary over number lines and graphing. (Likewise YET ANOTHER THING, "Graph (topology)".)

A quick look at middle-school texts should confirm which use of "graph" is primary. (Who claims it first? algebra or discrete mathematics?) Bert: "Bring me that graph." Ernie: "What graph? Did you hide it under these charts and diagrams?" Ernie: "Look between the maps and map graphs." A move or move-request is coming soon... - A876 (talk) 22:56, 12 June 2019 (UTC)

(I edited the above.) Also:

Cousins: Mathematical diagram; Diagram; Chart

• Chart recorder continuously plots one or more functions (of time)
• Plotter continuously plots a parametric function (defined by two functions of time), much like an Etch A Sketch; much like a vector monitor or oscilloscope#X-Y mode. - A876 (talk) 16:26, 24 June 2019 (UTC)

Graph is definitively a things with edges and vertices, its graphical representation usually called a drawing. In my language (Polish) we do have a separate word for graph of a function (or some other relation) and its graphical representation on 2-d carthesian plane (or even polar cordinates), wykres (close to drawing, but not exactly), and it is pretty exclusively used for that. Kreślarz is a person who makes sketches (drawings), usually engineering / mechanical / architectural ones on a paper, and possibly in CAD these days. It plays well with geometria wykreślna, a theory of 3D and 2D drawing on paper and in 2D space (things like perspective and such), related to affine geometry of course.

History?

The Mathologer mentioned graphs being invented by Nicole Oresme in his latest video so I came here to see what it said and there is nothing about it here. I'm sure that there is a lot more to it than just that but it makes me think that it would be nice to have a short history section about this subject, or to link to any existing coverage if it is already covered somewhere else that I didn't find. --DanielRigal (talk) 15:20, 21 October 2023 (UTC)

History of the function concept is a specific article, where the contribution of Oresme is mentioned. This article was linked twice in Function (mathematics), but not in a visible way. So I added a link to the history article at the top of the infobox {{Functions}}, and I have added this template at the beginning of Graph of a function. D.Lazard (talk) 16:58, 21 October 2023 (UTC)