Function application

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.

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 ${\displaystyle f:(X\times Y)\to Z}$, its application is represented as ${\displaystyle f(x,y)\!}$ by the former notation and ${\displaystyle f\;\langle x,y\rangle }$ by the latter. However, functions in curried form ${\displaystyle f:X\to (Y\to Z)}$ can be represented by juxtaposing their arguments: ${\displaystyle f\;x\;y}$, rather than ${\displaystyle f(x)(y)}$. This relies on function application being left-associative.

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