All Topics  
Graph of a function

 

 
   Email Print
   Bookmark   Link
 

 

Graph of a function
 
 
  Topics Graph of a function Topic Home

In mathematics, the graph of a function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
 f is the collection of all ordered pair
Ordered pair

In mathematics, an ordered pair is a collection of two distinguishable objects, one being the first coordinate system , and the other being the second coordinate ....
s (x, f(x)). In particular, if x is a real number
Real number

In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
, graph means the graphical representation of this collection, in the form of a curve
Curve

In mathematics, a curve consists of the points through which a continuous function moving point passes. This notion captures the intuitive idea of a geometrical dimension object, which furthermore is connectedness in the sense of having no continuous function or continuum ....
 on a Cartesian plane
Cartesian coordinate system

In mathematics, the Cartesian coordinate system is used to determine each Point uniquely in a Plane through two numbers, usually called the x-coordinate or abscissa and the y-coordinate or ordinate of the point....
, together with Cartesian axes, etc. Graphing on a Cartesian plane is sometimes referred to as curve sketching. If the function input x is an ordered pair (x1, x2) of real numbers, the graph is the collection of all ordered triples (x1, x2, f(x1, x2)), and its graphical representation is a surface
Surface

In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space E3....
 (see three dimensional graph).

The graph of a function on real numbers is identical to the graphic representation of the function.






Discussion
Ask a question about 'Graph of a function'
Start a new discussion about 'Graph of a function'
Answer questions from other users
Full Discussion Forum



Encyclopedia


In mathematics, the graph of a function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
 f is the collection of all ordered pair
Ordered pair

In mathematics, an ordered pair is a collection of two distinguishable objects, one being the first coordinate system , and the other being the second coordinate ....
s (x, f(x)). In particular, if x is a real number
Real number

In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
, graph means the graphical representation of this collection, in the form of a curve
Curve

In mathematics, a curve consists of the points through which a continuous function moving point passes. This notion captures the intuitive idea of a geometrical dimension object, which furthermore is connectedness in the sense of having no continuous function or continuum ....
 on a Cartesian plane
Cartesian coordinate system

In mathematics, the Cartesian coordinate system is used to determine each Point uniquely in a Plane through two numbers, usually called the x-coordinate or abscissa and the y-coordinate or ordinate of the point....
, together with Cartesian axes, etc. Graphing on a Cartesian plane is sometimes referred to as curve sketching. If the function input x is an ordered pair (x1, x2) of real numbers, the graph is the collection of all ordered triples (x1, x2, f(x1, x2)), and its graphical representation is a surface
Surface

In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space E3....
 (see three dimensional graph).

The graph of a function on real numbers is identical to the graphic representation of the function. For general functions, the graphic representation cannot be applied and the formal definition of the graph of a function suits the need of mathematical statements, e.g., the closed graph theorem
Closed graph theorem

In mathematics, the closed graph theorem is a basic result in functional analysis which characterizes continuous linear operators between Banach spaces in terms of the operator graph of a function....
 in functional analysis
Functional analysis

Functional analysis is the branch of mathematics, and specifically of mathematical analysis, concerned with the study of vector spaces and operators acting upon them....
.

The concept of the graph of a function is generalised to the graph of a relation
Relation (mathematics)

In mathematics , a relation is a property that assigns truth values to combinations of k first-order logic. Typically, the property describes a possible connection between the components of a k-tuple....
. 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
Codomain

In mathematics, the codomain, range or target set, of a function , described symbolically as ' : ' ? ', is the set ' into which all of the output of the function is constrained to fall....
 could have the same graph. For example, the cubic polynomial mentioned below is a surjection if its codomain is the real number
Real number

In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
s but it is not if its codomain is the complex field
Complex number

In mathematics, the complex numbers are an extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies:...
.

Examples


Functions of one variable

The graph of the function
is
.


The graph of the cubic polynomial on the real line
Real line

In mathematics, the real line is simply the set R of singleton real numbers.However, this term is usually used when R is to be treated as a space of some sort, such as a topological space or a vector space....
is
.
If this set is plotted on a Cartesian plane, the result is a curve (see figure).

Function of two variables


The graph of the trigonometric function on the real line
f(x, y) = sin
Siné

Maurice Sinet, known as Sin? is a France cartoonist.As a young man he studied drawing and graphic arts, earning his life as a cabaret singer....
(x2)·tan
COT

Cot, COT or CoT may refer to:...
(y2)
is
.
If this set is plotted on a three dimensional Cartesian coordinate system, the result is a surface (see figure).

Tools for plotting function graphs


Hardware
Hardware

Hardware is a general term that refers to the physical cultural artifacts of a technology. It may also mean the physical components of a computer system, in the form of computer hardware....
 


  • Graphing calculator
    Graphing calculator

    A graphing calculator typically refers to a class of handheld calculators that are capable of plotting graph of a function, solving simultaneous equations, and performing numerous other tasks with variables....
  • Oscilloscope
    Oscilloscope

    An oscilloscope is a type of electronic test instrument that allows signal voltages to be viewed, usually as a two-dimensional graph of one or more electrical potential differences plotted as a function of time or of some other voltage ....


Software

See List of graphing software

See also


External links

  • Weisstein, Eric W. "." From MathWorld--A Wolfram Web Resource.