= Functions = '''Functions''' are mappings or assignments. <> ---- == Description == 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''. 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))''. ---- CategoryRicottone