Editing Function (mathematics) (section)

== Representing a function ==

A [[Graph of a function|graph]] is commonly used to give an intuitive picture of a function.  As an example of how a graph helps to understand a function, it is easy to see from its graph whether a function is increasing or decreasing. Some functions may also be represented by [[bar chart]]s.

=== Graphs and plots ===
{{main|Graph of a function}}
[[File:Motor vehicle deaths in the US.svg|thumb|The function mapping each year to its US motor vehicle death count, shown as a [[line chart]]]]
[[File:Motor vehicle deaths in the US histogram.svg|thumb|The same function, shown as a bar chart]]

Given a function <math>f : X\to Y,</math> its ''graph'' is, formally, the set

<math display="block">G=\{(x,f(x))\mid x\in X\}.</math>

In the frequent case where {{mvar|X}} and {{mvar|Y}} are subsets of the [[real number]]s (or may be identified with such subsets, e.g. [[interval (mathematics)|intervals]]), an element <math>(x,y)\in G</math> may be identified with a point having coordinates {{math|''x'', ''y''}} in a 2-dimensional coordinate system, e.g. the [[Cartesian plane]]. Parts of this may create a [[Plot (graphics)|plot]] that represents (parts of) the function. The use of plots is so ubiquitous that they too are called the ''graph of the function''. Graphic representations of functions are also possible in other coordinate systems. For example, the graph of the [[square function]]

<math display="block">x\mapsto x^2,</math>

consisting of all points with coordinates <math>(x, x^2)</math> for <math>x\in \R,</math> yields, when depicted in Cartesian coordinates, the well known [[parabola]]. If the same quadratic function <math>x\mapsto x^2,</math> with the same formal graph, consisting of pairs of numbers, is plotted instead in [[polar coordinates]] <math>(r,\theta) =(x,x^2),</math> the plot obtained is [[Fermat's spiral]].

=== Tables ===
{{Main|Mathematical table}}
A function can be represented as a table of values.  If the domain of a function is finite, then the function can be completely specified in this way.  For example, the multiplication function <math>f:\{1,\ldots,5\}^2 \to \mathbb{R}</math> defined as <math>f(x,y)=xy</math> can be represented by the familiar [[multiplication table]]

{| class="wikitable" style="text-align: center;"
! {{diagonal split header|{{mvar|x}}|{{mvar|y}}}}
! 1  !! 2  !! 3  !! 4  !! 5
|-
! 1
| 1  || 2  || 3  || 4 || 5
|-
! 2
| 2  || 4  ||6 || 8  || 10
|-
! 3
| 3  || 6  || 9  || 12 || 15
|-
! 4
| 4  || 8  || 12 || 16 || 20
|-
! 5
| 5  || 10 || 15 || 20 || 25
|}

On the other hand, if a function's domain is continuous, a table can give the values of the function at specific values of the domain.  If an intermediate value is needed, [[interpolation]] can be used to estimate the value of the function.  For example, a portion of a table for the sine function might be given as follows, with values rounded to 6 decimal places:

{| class="wikitable" style="text-align: center;"
! {{mvar|x}} !! {{math|sin ''x''}}
|-
|1.289 || 0.960557
|-
|1.290 || 0.960835
|-
|1.291 || 0.961112
|-
|1.292 || 0.961387
|-
|1.293 || 0.961662
|}

Before the advent of handheld calculators and personal computers, such tables were often compiled and published for functions such as logarithms and trigonometric functions.

=== Bar chart ===
{{main|Bar chart}}
A bar chart can represent a function whose domain is a finite set, the [[natural number]]s, or the [[integer]]s. In this case, an element {{mvar|x}} of the domain is represented by an [[interval (mathematics)|interval]] of the {{mvar|x}}-axis, and the corresponding value of the function, {{math|''f''(''x'')}}, is represented by a [[rectangle]] whose base is the interval corresponding to {{mvar|x}} and whose height is {{math|''f''(''x'')}} (possibly negative, in which case the bar extends below the {{mvar|x}}-axis).