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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, the graph of a
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
$f$ is the set of
ordered pair In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

s $\left(x, y\right)$, where $f\left(x\right) = y.$ In the common case where $x$ and $f\left(x\right)$ are
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s, these pairs are
Cartesian coordinates A Cartesian coordinate system (, ) in a plane Plane or planes may refer to: * Airplane or aeroplane or informally plane, a powered, fixed-wing aircraft Arts, entertainment and media *Plane (Dungeons & Dragons), Plane (''Dungeons & Dragons'') ...

of points in
two-dimensional space Two-dimensional space (also known as 2D space, 2-space, or bi-dimensional space) is a geometric setting in which two values (called parameter A parameter (), generally, is any characteristic that can help in defining or classifying a particul ...
and thus form a subset of this plane. In the case of functions of two variables, that is functions whose
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function *Doma ...
consists of pairs $\left(x, y\right),$ the graph usually refers to the set of
ordered triple In mathematics, a tuple is a finite ordered list (sequence) of Element (mathematics), elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, referred to as ''the empty tupl ...
s $\left(x, y, z\right)$ where $f\left(x,y\right) = z,$ instead of the pairs $\left(\left(x, y\right), z\right)$ as in the definition above. This set is a subset of
three-dimensional space Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameters) are required to determine the position of an element (i.e., point). This is the informal meaning of the ...
; for a continuous
real-valued function Mass measured in grams is a function from this collection of weight to positive number">positive Positive is a property of Positivity (disambiguation), positivity and may refer to: Mathematics and science * Converging lens or positive lens, i ...
of two real variables, it is a
surface File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to prevent floating below the textile. A surface, as the term is most generally used, is the outermost or uppermost layer of a physical obje ...
. A graph of a function is a special case of a relation. In
science Science () is a systematic enterprise that Scientific method, builds and organizes knowledge in the form of Testability, testable explanations and predictions about the universe."... modern science is a discovery as well as an invention. ...

,
engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...

,
technology Technology ("science of craft", from Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. I ...

,
finance Finance is a term for the management, creation, and study of money In a 1786 James Gillray caricature, the plentiful money bags handed to King George III are contrasted with the beggar whose legs and arms were amputated, in the left corn ...

, and other areas, graphs are tools used for many purposes. In the simplest case one variable is plotted as a function of another, typically using rectangular axes; see ''
Plot (graphics) A plot is a graphical technique for representing a data set, usually as a Graph of a function, graph showing the relationship between two or more variables. The plot can be drawn by hand or by a computer. In the past, sometimes mechanical or elect ...
'' for details. In the modern
foundations of mathematics Foundations of mathematics is the study of the philosophical Philosophy (from , ) is the study of general and fundamental questions, such as those about existence Existence is the ability of an entity to interact with physical or mental r ...
, and, typically, in
set theory Set theory is the branch of that studies , which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of , is mostly concerned with those that are relevant to ...
, a function is actually equal to its graph. However, it is often useful to see functions as mappings, which consist not only of the relation between input and output, but also which set is the domain, and which set is the
codomain In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

. For example, to say that a function is onto (
surjective In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

) or not the codomain should be taken into account. The graph of a function on its own doesn't determine the codomain. It is common to use both terms ''function'' and ''graph of a function'' since even if considered the same object, they indicate viewing it from a different perspective.

# Definition

Given a mapping $f : X \to Y,$ in other words a function $f$ together with its domain $X$ and codomain $Y,$ the graph of the mapping is the set $G(f) = \,$ which is a subset of $X\times Y$. In the abstract definition of a function, $G\left(f\right)$ is actually equal to $f.$ One can observe that, if, $f : \R^n \to \R^m,$ then the graph $G\left(f\right)$ is a subset of $\R^$ (strictly speaking it is $\R^n \times \R^m,$ but one can embed it with the natural isomorphism).

# Examples

## Functions of one variable

The graph of the function $f : \ \to \$ defined by $f(x)= \begin a, & \textx=1, \\ d, & \textx=2, \\ c, & \textx=3, \end$ is the subset of the set $\ \times \$ $G(f) = \.$ From the graph, the domain $\$ is recovered as the set of first component of each pair in the graph $\ = \$. Similarly, the
range Range may refer to: Geography * Range (geographic)A range, in geography, is a chain of hill A hill is a landform A landform is a natural or artificial feature of the solid surface of the Earth or other planetary body. Landforms together ...
can be recovered as $\ = \$. The codomain $\$, however, cannot be determined from the graph alone. The graph of the cubic polynomial on the
real line In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
$f(x) = x^3 - 9x$ is $\.$ If this set is plotted on a
Cartesian plane A Cartesian coordinate system (, ) in a plane Plane or planes may refer to: * Airplane or aeroplane or informally plane, a powered, fixed-wing aircraft Arts, entertainment and media *Plane (Dungeons & Dragons), Plane (''Dungeons & Dragons'') ...

, the result is a curve (see figure).

## Functions of two variables

The graph of the
trigonometric function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

$f(x,y) = \sin(x^2)\cos(y^2)$ is $\.$ If this set is plotted on a three dimensional Cartesian coordinate system, the result is a surface (see figure). Oftentimes it is helpful to show with the graph, the gradient of the function and several level curves. The level curves can be mapped on the function surface or can be projected on the bottom plane. The second figure shows such a drawing of the graph of the function: $f(x, y) = -(\cos(x^2) + \cos(y^2))^2.$

# Generalizations

The graph of a function is contained in a
Cartesian product In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
of sets. An $X$$Y$ plane is a Cartesian product of two lines, called $X$ and $Y,$ while a cylinder is a cartesian product of a line and a circle, whose height, radius, and angle assign precise locations of the points.
Fibre bundle In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...
s are not Cartesian products, but appear to be up close. There is a corresponding notion of a graph on a fibre bundle called a
section Section, Sectioning or Sectioned may refer to: Arts, entertainment and media * Section (music), a complete, but not independent, musical idea * Section (typography), a subdivision, especially of a chapter, in books and documents ** Section sign ...
. The , say the multifunction $\mathcal : X \rightrightarrows Y,$ is the set $\operatorname \mathcal := \left\.$

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Asymptote In analytic geometry In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέ ...

*
Chart A chart is a graphical representation Graphic communication as the name suggests is communication using graphic elements. These elements include symbols such as glyphs and icon (computing), icons, images such as drawings and photographs, and ca ...

*
Concave function In , a concave function is the of a . A concave function is also ously called concave downwards, concave down, convex upwards, convex cap, or upper convex. Definition A real-valued f on an (or, more generally, a in ) is said to be ''concave' ...

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Convex function (in green) is a convex set File:Convex polygon illustration2.svg, Illustration of a non-convex set. Illustrated by the above line segment whereby it changes from the black color to the red color. Exemplifying why this above set, illustrated in gr ...

* Contour plot * Critical point *
Derivative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

*
Epigraph Epigraph may refer to: * An inscription, as studied in the archeological sub-discipline of epigraphy * Epigraph (literature), a phrase, quotation, or poem that is set at the beginning of a document or component * Epigraph (mathematics), the set of ...
* Normal to a graph *
Slope In mathematics, the slope or gradient of a line Line, lines, The Line, or LINE may refer to: Arts, entertainment, and media Films * ''Lines'' (film), a 2016 Greek film * ''The Line'' (2017 film) * ''The Line'' (2009 film) * ''The Line'', ...

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Stationary point In mathematics, particularly in calculus, a stationary point of a differentiable function of one variable is a point on the graph of a function, graph of the function where the function's derivative is zero. Informally, it is a point where the ...
* Tetraview * Vertical translation *
y-intercept Image:Y-intercept.svg, 300px, Graph ''y''=''ƒ''(''x'') with the ''x''-axis as the horizontal axis and the ''y''-axis as the vertical axis. The ''y''-intercept of ''ƒ''(''x'') is indicated by the red dot at (''x''=0, ''y''=1). In analytic ...

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