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| 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}''. | 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 [[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 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.
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)).