In mathematics, the restriction of a function f {\displaystyle f} is a new function, denoted f | A {\displaystyle f\vert _{A}} or f ↾ A , {\displaystyle f{\upharpoonright _{A}},} obtained by choosing a smaller domain A {\displaystyle A} for the original function f . {\displaystyle f.} The function f {\displaystyle f} is then said to extend f | A . {\displaystyle f\vert _{A}.}
Formal definition Let f : E → F {\displaystyle f:E\to F} be a function from a set E {\displaystyle E} to a set F . {\displaystyle F.} If a set A {\displaystyle A} is a subset of E , {\displaystyle E,} then the restriction of f {\displaystyle f} to A {\displaystyle A} is the function
f | A : A → F {\displaystyle {f|}_{A}:A\to F}
given by f | A ( x ) = f ( x ) {\displaystyle {f|}_{A}(x)=f(x)} for x ∈ A . {\displaystyle x\in A.} Informally, the restriction of f {\displaystyle f} to A {\displaystyle A} is the same function as f , {\displaystyle f,} but is only defined on A {\displaystyle A} . If the function f {\displaystyle f} is thought of as a relation ( x , f ( x ) ) {\displaystyle (x,f(x))} on the Cartesian product E × F , {\displaystyle E\times F,} then the restriction of f {\displaystyle f} to A {\displaystyle A} can be represented by its graph,
G ( f | A ) = { ( x , f ( x ) ) ∈ G ( f ) : x ∈ A } = G ( f ) ∩ ( A × F ) , {\displaystyle G({f|}_{A})=\{(x,f(x))\in G(f):x\in A\}=G(f)\cap (A\times F),}
where the pairs ( x , f ( x ) ) {\displaystyle (x,f(x))} represent ordered pairs in the graph G . {\displaystyle G.}
Extensions A function F {\displaystyle F} is said to be an extension of another function f {\displaystyle f} if whenever x {\displaystyle x} is in the domain of f {\displaystyle f} then x {\displaystyle x} is also in the domain of F {\displaystyle F} and f ( x ) = F ( x ) . {\displaystyle f(x)=F(x).} That is, if domain f ⊆ domain F {\displaystyle \operatorname {domain} f\subseteq \operatorname {domain} F} and F | domain f = f . {\displaystyle F{\big \vert }_{\operatorname {domain} f}=f.}
A linear extension (respectively, continuous extension, etc.) of a function f {\displaystyle f} is an extension of f {\displaystyle f} that is also a linear map (respectively, a continuous map, etc.).
Examples The restriction of the non-injective function f : R → R , x ↦ x 2 {\displaystyle f:\mathbb {R} \to \mathbb {R} ,\ x\mapsto x^{2}} to the domain R + = [ 0 , ∞ ) {\displaystyle \mathbb {R} _{+}=[0,\infty )} is the injection f : R + → R , x ↦ x 2 . {\displaystyle f:\mathbb {R} _{+}\to \mathbb {R} ,\ x\mapsto x^{2}.}
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