Graph of a function: Difference between revisions

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Mathematically, the '''graph of a [[function]]''' is the collection of all pairs (''x'', ''f''(''x)'')) of the function.

The graph of the function

In a technical rather than theoretical context, the term is usual used for a graphical representation of the function. This is formed by creating a [[Cartesian coordinate system|Cartesian plane]], and on this marking all points ''(x,y)'' for which ''y=f(x)'' holds. For a [[continuous function]], this will result in some type of [[curve]].

:<math>f(x)=\left\{\begin{matrix} a, & \mbox{if }x=1 \\ d, & \mbox{if }x=2 \\ c, & \mbox{if }x=3. \end{matrix}\right.</math>

is {(1,a), (2,d), (3,c)}.

The graph of the cubic polynomial on the real line

:<math>f(x)=x^3-9x</math>

is {(''x'',''x''³-9''x'') : ''x'' is a real number}. If the set is plotted on a [[Cartesian coordinate system|Cartesian plane]], the result is

[[Image::cubicpoly.png]]

Therefore 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. [[Close graph theorem]] in [[functional analysis]].

The concept of the graph of a function is generalised to the graph of a [[relation]]. Note that althrough a function is always identified with its graph, they are not the same because it will happen two functions with different [[codomain]] could have the same graph. For example, the cubic polynomial mentioned above is a [[surjection]] if its codomain is the [[real number]]s but it is not if its codomain is the [[complex number|complex field]].