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| A function maps every member of ''A'' to a member of ''B''. Or, for each ''x ∈ A'' a function assigns a unique ''f(x) ∈ B''. Such a function is notated as ''f : A -> B''. | A function is a [[Analysis/Relations|relation]] that satisfies an additional property; for each ''x ∈ A'' it assigns a unique ''f(x) ∈ B''. This can be considered a map for every ''x ∈ A'', each mapping to a singular ''f(x) ∈ B''. Or, it can be considered an assignment of a singular ''f(x) ∈ B'' to each ''x ∈ A''. A function is notated as ''f : A -> B''. |
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Note that there is a difference between inverse images and a true inverse. A unique inverse of a function only exists if it is [[Analysis/Injectivity|bijective]]. An inverse satisfies ''f(f ^-1^(x)) = x''. Functions can be '''composed'''. For two functions as ''f : A -> B'' and ''g : B -> C'', ''g ∘ f'' corresponds to ''(g ∘ f)(x) = g(f(x))''. |
Functions
Functions are mappings or assignments.
Contents
Description
A function is a relation that satisfies an additional property; for each x ∈ A it assigns a unique f(x) ∈ B. This can be considered a map for every x ∈ A, each mapping to a singular f(x) ∈ B. Or, it can be considered an assignment of a singular f(x) ∈ B to each x ∈ A. A function is notated as f : A -> B.
The image of a function f is the subset of B that corresponds to the entire domain of A: f(A) = {f(x) | x ∈ A}. The inverse image or pre-image is the subset of A that corresponds to the entire domain of B: f -1(B) = {x | f(x) ∈ B}.
Note that there is a difference between inverse images and a true inverse. A unique inverse of a function only exists if it is bijective. An inverse satisfies f(f -1(x)) = x.
Functions can be composed. For two functions as f : A -> B and g : B -> C, g ∘ f corresponds to (g ∘ f)(x) = g(f(x)).