function composition is the application of one to the results of another. For instance, the functions and can be by computing the output of g when it has an f: X → Y g: Y → Z composed argument of instead of Intuitively, if is a function of and is a function of then f(x) x. z g y y f x, z is a function of x. Thus one obtains a function defined by (g = for all in composite g f: X Z f)(x) g(f(x)) x X. → ∘ ∘ The notation is read as "g circle or "g composed with "g after "g following or g f f", f", f", f", ∘ just "g of f". The composition of functions is always . That is, if and are functions with suitably f, g, h chosen and , then (g = (f where the parentheses serve to indicate that f h) g) h, ∘ ∘ ∘ ∘ composition is to be performed first for the parenthesized functions. Since there is no distinction between the choices of placement of parentheses, they may be safely left off.
function composition is the application of one to the results of another. For instance, the functions f: X → Y and g: Y → Z can be composed by computing the output of g when it has an argument of f(x) instead of x. Intuitively, if z is a function g of y and y is a function f of x, then z is a function of x. Thus one obtains a composite function g ∘ f: X → Z defined by (g ∘ f)(x) = g(f(x)) for all x in X. The notation g ∘ f is read as "g circle f", or "g composed with f", "g after f", "g following f", or just "g of f". The composition of functions is always . That is, if f, g, and h are functions with suitably chosen and , then f ∘ (g ∘ h) = (f ∘ g) ∘ h, where the parentheses serve to indicate that composition is to be performed first for the parenthesized functions. Since there is no distinction between the choices of placement of parentheses, they may be safely left off.
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