> [!NOTE] Definition (Inverse & Left-Right Inverses) > Given [[Function|functions]] $f:X\to Y$ and $g:Y\to X$ such that $g \circ f = \text{Id}_{X},$ the [[Identity Function|identity function]] on $X,$ we say that $g$ is the *left inverse* of $f$ or $f$ is the *right inverse* of $g.$ > > If $g \circ f=\text{Id}_{X}$ and $f \circ g=\text{Id}_{Y}$ then $g$ is an *inverse* for $y$ and vice versa.