Inverse (function)

From CS2800 wiki

The inverse of a function [math]f : A \href{/cs2800/wiki/index.php/%5Cto}{\to} B [/math] may refer to a left inverse, right inverse, or a two-sided inverse. If it is unclear from context, it usually refers to a two-sided inverse.

All three of these are functions from [math]B \href{/cs2800/wiki/index.php/%5Cto}{\to} A [/math] with some relationship to [math]f [/math]:

Definition: Left inverse
Given a function [math]f : A \href{/cs2800/wiki/index.php/%E2%86%92}{→} B [/math], a left inverse [math]g [/math] of [math]f [/math] is a function [math]g : B \href{/cs2800/wiki/index.php/%E2%86%92}{→} A [/math] satisfying [math]g \href{/cs2800/wiki/index.php/%5Ccirc}{\circ} f \href{/cs2800/wiki/index.php/Equality_(functions)}{=} \href{/cs2800/wiki/index.php?title=Id&action=edit&redlink=1}{id} [/math].
Definition: Right inverse
Given a function [math]f : A \href{/cs2800/wiki/index.php/%E2%86%92}{→} B [/math], a right inverse [math]g [/math] of [math]f [/math] is a function [math]g : B \href{/cs2800/wiki/index.php/%E2%86%92}{→} A [/math] satisfying [math]f \href{/cs2800/wiki/index.php/%5Ccirc}{\circ} g \href{/cs2800/wiki/index.php/Equality_(functions)}{=} \href{/cs2800/wiki/index.php?title=Id&action=edit&redlink=1}{id} [/math].
Definition: Two-sided inverse
If [math]f : A \href{/cs2800/wiki/index.php/%E2%86%92}{→} B [/math] is a function, then [math]g : B \href{/cs2800/wiki/index.php/%E2%86%92}{→} A [/math] is a two-sided inverse of [math]f [/math] if [math]f \href{/cs2800/wiki/index.php/%5Ccirc}{\circ} g \href{/cs2800/wiki/index.php/Equality_(functions)}{=} \href{/cs2800/wiki/index.php?title=Id&action=edit&redlink=1}{id}_B [/math] and [math]g \href{/cs2800/wiki/index.php/%5Ccirc}{\circ} f \href{/cs2800/wiki/index.php/Equality_(functions)}{=} \href{/cs2800/wiki/index.php?title=Id&action=edit&redlink=1}{id}_A [/math].