In mathematics, function application is the act of applying a function to an argument from its domain so as to obtain the corresponding value from its range.
[1] In this sense, function application can be thought of as the opposite of function abstraction.
Function application is usually depicted by juxtaposing the variable representing the function with its argument encompassed in parentheses.
For example, the following expression represents the application of the function ƒ to its argument x.
In some instances, a different notation is used where the parentheses aren't required, and function application can be expressed just by juxtaposition.
For example, the following expression can be considered the same as the previous one: The latter notation is especially useful in combination with the currying isomorphism.
Given a function
, its application is represented as
written with the less common angle brackets) by the latter.
However, functions in curried form
can be represented by juxtaposing their arguments:
This relies on function application being left-associative.
U+2061 FUNCTION APPLICATION (, ) — a contiguity operator indicating application of a function; that is an invisible zero width character intended to distinguish concatenation meaning function application from concatenation meaning multiplication.
In axiomatic set theory, especially Zermelo–Fraenkel set theory, a function
is often defined as a relation (
One is usually not content to write "
, and usually wishes for the more common function notation "
", thus function application, or more specifically, the notation "
", is defined by an axiom schema.
Function application can be trivially defined as an operator, called apply or
, by the following definition: The operator may also be denoted by a backtick (`).
If the operator is understood to be of low precedence and right-associative, the application operator can be used to cut down on the number of parentheses needed in an expression.
For example; can be rewritten as: However, this is perhaps more clearly expressed by using function composition instead: or even: if one considers
to be a constant function returning
Function application in the lambda calculus is expressed by β-reduction.
The Curry–Howard correspondence relates function application to the logical rule of modus ponens.
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