# Difference between revisions of "Function"

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A function '''ought''' be defined for everything in its domain, that's for every point in the domain the function maps the point to something. | A function '''ought''' be defined for everything in its domain, that's for every point in the domain the function maps the point to something. | ||

===Examples=== | ===Examples=== | ||

− | (See | + | (See notation below if you're not sure what the <math>f:X\rightarrow Y</math> notation means) |

* <math>f:\mathbb{R}\rightarrow\mathbb{R}</math> given by <math>f(x)=\frac{1}{x}</math> isn't defined at <math>0</math> | * <math>f:\mathbb{R}\rightarrow\mathbb{R}</math> given by <math>f(x)=\frac{1}{x}</math> isn't defined at <math>0</math> | ||

* <math>f:\mathbb{R}\rightarrow\mathbb{R}</math> given by <math>f(x)=x^2</math> is correct, it is not [[Surjection|surjective]] though, because nothing maps onto the negative numbers, however <math>f:\mathbb{R}\rightarrow\mathbb{R}_{\ge 0}</math> with <math>f(x)=x^2</math> is a surjection. It is not an [[Injection|injective function]] as only <math>0</math> maps to one point. | * <math>f:\mathbb{R}\rightarrow\mathbb{R}</math> given by <math>f(x)=x^2</math> is correct, it is not [[Surjection|surjective]] though, because nothing maps onto the negative numbers, however <math>f:\mathbb{R}\rightarrow\mathbb{R}_{\ge 0}</math> with <math>f(x)=x^2</math> is a surjection. It is not an [[Injection|injective function]] as only <math>0</math> maps to one point. |

## Revision as of 20:01, 28 February 2015

A function [ilmath]f[/ilmath] is a special kind of relation

## Domain

A function **ought** be defined for everything in its domain, that's for every point in the domain the function maps the point to something.

### Examples

(See notation below if you're not sure what the [math]f:X\rightarrow Y[/math] notation means)

- [math]f:\mathbb{R}\rightarrow\mathbb{R}[/math] given by [math]f(x)=\frac{1}{x}[/math] isn't defined at [math]0[/math]
- [math]f:\mathbb{R}\rightarrow\mathbb{R}[/math] given by [math]f(x)=x^2[/math] is correct, it is not surjective though, because nothing maps onto the negative numbers, however [math]f:\mathbb{R}\rightarrow\mathbb{R}_{\ge 0}[/math] with [math]f(x)=x^2[/math] is a surjection. It is not an injective function as only [math]0[/math] maps to one point.

## Notation

A function [ilmath]f[/ilmath] from a domain [ilmath]X[/ilmath] to a set [ilmath]Y[/ilmath] is denoted [ilmath]f:X\rightarrow Y[/ilmath]

TODO: Come back after the relation page and fill this out