A Partial [[Function]] is a [[Function]] $f$ from the [[Domain]] $X$ to the [[Codomain]] $Y$, but modified restriction that every element in the [[Codomain]] $Y$ may have *at most one* [[Pre-Image]]. A [[Partial Function]] must have one element in $Y$ for every element in the domain $X$, but is not required to have an element $X$ for every $Y$ (not [[Surjective]]). >[!example] Defining the function $f: x \mapsto x^2$ as $f: \R \to \R$ is a partial function, as [[Universal Quantifier|For All]] $x < 0$, there is no [[Pre-Image]] through $f$.