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Inmathematics,function application is the act of applying afunction to an argument from itsdomain so as to obtain the corresponding value from itsrange.^[1] In this sense, function application can be thought of as the opposite of functionabstraction.

Function application is usually depicted by juxtaposing the variable representing the function with its argument encompassed inparentheses. For example, the following expression represents the application of the functionƒ to its argumentx.

${\displaystyle f(x)}$

In some instances, a different notation is used where the parentheses aren't required, and function application can be expressed just byjuxtaposition. For example, the following expression can be considered the same as the previous one:

${\displaystyle fx}$

The latter notation is especially useful in combination with thecurrying isomorphism. Given a function${\displaystyle f:(X\times Y)\to Z}$, its application is represented as${\displaystyle f(x,y)}$ by the former notation and${\displaystyle f(x,y)}$ (or${\displaystyle f⟨x,y⟩}$ with the argument${\displaystyle ⟨x,y⟩\in X\times Y}$ written with the less common angle brackets) by the latter. However, functions in curried form${\displaystyle f:X\to (Y\to Z)}$ can be represented by juxtaposing their arguments:${\displaystyle fxy}$, rather than${\displaystyle f(x)(y)}$. This relies on function application beingleft-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.

Inaxiomatic set theory, especiallyZermelo–Fraenkel set theory, a function${\displaystyle f:X\mapsto Y}$ is often defined as arelation (${\displaystyle f\subseteq X\times Y}$) having the property that, for any${\displaystyle x\in X}$there is a unique${\displaystyle y\in Y}$ such that${\displaystyle (x,y)\in f}$.

One is usually not content to write "${\displaystyle (x,y)\in f}$" to specify that${\displaystyle y}$, and usually wishes for the more common function notation "${\displaystyle f(x)=y}$", thus function application, or more specifically, the notation "${\displaystyle f(x)}$", is defined by anaxiom schema. Given any function${\displaystyle f}$ with a givendomain${\displaystyle X}$ andcodomain${\displaystyle Y}$:^[2][3]

${\displaystyle \mathrm{\forall }x\in X,\mathrm{\forall }y\in Y(f(x)=y⟺}$${\displaystyle \mathrm{\exists }!z\in Y((x,z)\in f)\wedge (x,y)\in f)}$

Stating "For all${\displaystyle x}$ in${\displaystyle X}$ and${\displaystyle y}$ in${\displaystyle Y}$,${\displaystyle f(x)}$ is equal to${\displaystyle y}$if and only if there is a unique${\displaystyle z}$ in${\displaystyle Y}$ such that${\displaystyle (x,z)}$ is in${\displaystyle f}$ and${\displaystyle (x,y)}$ is in${\displaystyle f}$". The notation${\displaystyle f(x)}$ here being defined is a newfunctional predicate from the underlying logic, where each y is aterm in x.^[4] Since${\displaystyle f}$, as a functional predicate, must map every object in the language, objects not in the specified domain are chosen to map to an arbitrary object, suct as theempty set.^[5]

Function application can be defined as anoperator, calledapply or${\displaystyle \mathrm{\$}}$, by the following definition:

${\displaystyle f\mathrm{\$}x=f(x)}$

The operator may also be denoted by abacktick (`).

If the operator is understood to be oflow precedence andright-associative, the application operator can be used to cut down on the number of parentheses needed in an expression. For example;

${\displaystyle f(g(h(j(x))))}$

can be rewritten as:

${\displaystyle f\mathrm{\$}g\mathrm{\$}h\mathrm{\$}j\mathrm{\$}x}$

However, this is perhaps more clearly expressed by usingfunction composition instead:

${\displaystyle (f\circ g\circ h\circ j)(x)}$

or even:

${\displaystyle (f\circ g\circ h\circ j\circ x)()}$

if one considers${\displaystyle x}$ to be aconstant function returning${\displaystyle x}$.

Function application in thelambda calculus is expressed byβ-reduction.

TheCurry–Howard correspondence relates function application to the logical rule ofmodus ponens.

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