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)).