Difference between revisions of "Function"

Revision as of 20:01, 28 February 2015

A function $f$ is a special kind of relation

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.

(See notation below if you're not sure what the

$f:X\rightarrow Y$ notation means)

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

A function $f$ from a domain $X$ to a set $Y$ is denoted $f:X\rightarrow Y$

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

@@ Line 4: / Line 4: @@
 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 domain below first if you do not know what the domain is)
+(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 [[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.