Encyclopedia > Graph of a function

  Article Content

Graph of a function

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

The graph of the function

<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,x3-9x) : x is a real number}. If the set is plotted on a Cartesian plane, the result is

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., the closed 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 numbers but it is not if its codomain is the complex field.



All Wikipedia text is available under the terms of the GNU Free Documentation License

 
  Search Encyclopedia

Search over one million articles, find something about almost anything!
 
 
  
  Featured Article
Scripps Institution of Oceanography

... recognition of the financial support given to the institution by the Scripps family. The institution conducts several oceanographic research vessels and has served as ...