Miles per gallon gasoline equivalent: Difference between revisions

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In [[mathematics]], '''function application''' is the act of applying a [[function (mathematics)|function]] to an argument from its [[domain of a function|domain]] so as to obtain the corresponding value from its [[range (mathematics)|range]].
 
==Representation==
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''.
 
:<math>f(x) \, </math>
 
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:
 
:<math>f\; x</math>
 
The latter notation is especially useful in combination with the [[currying]] isomorphism. Given a function <math>f : (X \times Y) \to Z</math>, its application is represented as <math>f(x, y)\!</math> by the former notation and <math>f \; \langle x, y \rangle</math> by the latter.  However, functions in curried form <math>f : X \to (Y \to Z)</math> can be represented by juxtaposing their arguments: <math>f\; x \; y</math>, rather than <math>f(x)(y)</math>. This relies on function application being [[left-associative]].
 
==As an operator==
 
{{main|Apply}}
 
Function application can be trivially defined as an [[Operator (mathematics)|operator]], called [[Apply]] or <math>$</math>, by the following definition:
 
:<math>f \mathop{\,$\,} x = f(x)</math>
 
The operator may also be denoted by a [[backtick]] (`).
 
If the operator is understood to be of [[Order of operations|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;
 
:<math>f(g(h(j(x)))) \, </math>
 
can be rewritten as:
 
:<math>f \mathop{\,$\,} g \mathop{\,$\,} h \mathop{\,$\,} j \mathop{\,$\,} x</math>
 
However, this is perhaps more clearly expressed by using [[function composition]] instead:
 
:<math>(f \circ g \circ h \circ j)(x)</math>
 
==Other instances==
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]].
 
==See also==
* [[prefix notation]]
 
[[Category:Functions and mappings]]
 
 
{{mathanalysis-stub}}

Latest revision as of 06:01, 12 December 2014

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