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A function [ilmath]f[/ilmath] 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 [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.


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