In mathematics, the restriction of a function ${\displaystyle f}$ is a new function, denoted ${\displaystyle f\vert _{A}}$ or ${\displaystyle f{\upharpoonright _{A}},}$ obtained by choosing a smaller domain ${\displaystyle A}$ for the original function ${\displaystyle f.}$ The function ${\displaystyle f}$ is then said to extend ${\displaystyle f\vert _{A}.}$

Let ${\displaystyle f:E\to F}$ be a function from a set ${\displaystyle E}$ to a set ${\displaystyle F.}$ If a set ${\displaystyle A}$ is a subset of ${\displaystyle E,}$ then the restriction of ${\displaystyle f}$ to ${\displaystyle A}$ is the function $${\displaystyle {f|}_{A}:A\to F}$$ given by ${\displaystyle {f|}_{A}(x)=f(x)}$ for ${\displaystyle x\in A.}$ Informally, the restriction of ${\displaystyle f}$ to ${\displaystyle A}$ is the same function as ${\displaystyle f,}$ but is only defined on ${\displaystyle A}$.

If the function ${\displaystyle f}$ is thought of as a relation ${\displaystyle (x,f(x))}$ on the Cartesian product ${\displaystyle E\times F,}$ then the restriction of ${\displaystyle f}$ to ${\displaystyle A}$ can be represented by its graph,

where the pairs ${\displaystyle (x,f(x))}$ represent ordered pairs in the graph ${\displaystyle G.}$

A function ${\displaystyle F}$ is said to be an extension of another function ${\displaystyle f}$ if whenever ${\displaystyle x}$ is in the domain of ${\displaystyle f}$ then ${\displaystyle x}$ is also in the domain of ${\displaystyle F}$ and ${\displaystyle f(x)=F(x).}$ That is, if ${\displaystyle \operatorname {domain} f\subseteq \operatorname {domain} F}$ and ${\displaystyle F{\big \vert }_{\operatorname {domain} f}=f.}$

A linear extension (respectively, continuous extension, etc.) of a function ${\displaystyle f}$ is an extension of ${\displaystyle f}$ that is also a linear map (respectively, a continuous map, etc.).

For a function to have an inverse, it must be one-to-one. If a function ${\displaystyle f}$ is not one-to-one, it may be possible to define a partial inverse of ${\displaystyle f}$ by restricting the domain. For example, the function $${\displaystyle f(x)=x^{2}}$$ defined on the whole of ${\displaystyle \mathbb {R} }$ is not one-to-one since ${\displaystyle x^{2}=(-x)^{2}}$ for any ${\displaystyle x\in \mathbb {R} .}$ However, the function becomes one-to-one if we restrict to the domain ${\displaystyle \mathbb {R} _{\geq 0}=[0,\infty ),}$ in which case $${\displaystyle f^{-1}(y)={\sqrt {y}}.}$$

(If we instead restrict to the domain ${\displaystyle (-\infty ,0],}$ then the inverse is the negative of the square root of ${\displaystyle y.}$) Alternatively, there is no need to restrict the domain if we allow the inverse to be a multivalued function.