Function Composition and Partial Application

Function composition and partial application are two powerful techniques in functional programming that allow you to build complex functions from simpler ones. Both are key A-Level topics.

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Function Composition

Function composition is the process of combining two or more functions to create a new function. The output of one function becomes the input of the next.

Mathematical Definition

If f : B → C and g : A → B, then the composition (f . g) : A → C is defined as:

(f . g)(x) = f(g(x))

First apply g, then apply f to the result.

Haskell

In Haskell, the composition operator is the dot (.).

-- Two simple functions
double x = x * 2
addOne x = x + 1

-- Compose them: first double, then add one
doubleAndAddOne = addOne . double
doubleAndAddOne 5    -- double 5 = 10, addOne 10 = 11

-- Compose in the other order: first add one, then double
addOneAndDouble = double . addOne
addOneAndDouble 5    -- addOne 5 = 6, double 6 = 12