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## FunctionThis article covers mathematics. Other uses of the word function include: - In sociology, social functions are the basis of functionalism.
- In computer science, a function is a subprogram or subroutine, commonly one intended to directly return a value to its caller. See also functional programming.
The concept of
## Introduction
Intuitively, a function is a way to assign to each value of the argument The most familiar kind of function is that where the argument and the function's value are both numbers, and the functional relationship is expressed by a formula, and the value of the function is obtained from the arguments by direct substitution. Consider for example x its square.A straightforward generalization is to allow functions depending not on a single number, but on several. For instance, x and y and assigns to them their product, xy. In the sciences, we often encounter functions that are not given by (known) formulas. Consider for instance the temperature distribution on Earth over time: this is a function which takes location and time as arguments and gives as output the temperature at that location at that time. We have seen that the intuitive notion of function is not limited to computations using single numbers and not even limited to computations; the mathematical notion of function is still more general and is not limited to situations involving numbers. Rather, a function links a "domain" (set of inputs) to a "codomain" (set of possible outputs) in such a way that to every element of the domain is associated precisely one element of the codomain. Functions are abstractly defined as certain relations, as will be seen below. Because of this generality, functions appear in a wide variety of mathematical contexts, and several mathematical fields are based on the study of functions. ## History
As a mathematical term, "
The word function was later used by Euler during the mid-18th Century to describe an expression or formula involving various arguments; ie: During the 19th Century, mathematicians started to formalize all the different branches of mathematics. Weierstrass advocated building calculus on arithmetic rather than on geometry, which favoured Euler's definition over Leibniz's (see arithmetization of analysis). By broadening the definition of functions, mathematicians were then able to study "strange" mathematical objects such as functions which are nowhere differentiable. Those functions, first thought as purely imaginary and called collectively "monsters" as late as the turn of the 20th century, were later found to be important in the modelling of physical phenomena such as Brownian motion. Towards the end of the 19th century, mathematicians started trying to formalize all of mathematics using set theory and they sought definitions of every mathematical object as a set. It was Dirichlet that gave the modern "formal" definition of function (see #Formal Definition below). In Dirichlet's definition, a function is a special case of a relation. In most cases of practical interest, however, the differences between the modern definition and Euler's definition are negligible. ## Formal Definition
Formally, a function *f*is*functional*: if*x f y*(*x*is*f*-related to*y*) and*x f z*, then*y*=*z*. i.e., for each input value, there should only be one possible output value.*f*is*total*: for all*x*in*X*, there exists a*y*in*Y*such that*x f y*. i.e. for each input value, the formula should produce at least one output value within*Y*.
x in the domain, the corresponding unique output value y in the codomain is denoted by f(x).Consider the following three examples:
Occasionally, all three relations above are called functions. In this case, the function satisfies Conditions (1) and (2) is said to be a "well-defined function" or "total function". In this encyclopedia, the terms "well-defined function", "total function" and "function" are synonymous. ## Domains, Codomains, and Ranges
In computer science, the datatypes of the arguments and return values specify the domain and codomain (respectively) of a subprogram. So the domain and codomain are constraints imposed initially on a function; on the other hand the range has to do with how things turn out in practice. ## Graph of a functions
The graph of a function
If
Note that since a relation on the two sets ## Images and preimages
The
The image of a subset f is the image f(X) of its domain. In our example of discrete function, the image of {2,3} under f is f({2,3})={c,d} and the range of f is {a,c,d}.
The *f*^{ −1}(`B`) := {`x`in*X*:*f*(`x`)∈`B`}.
f^{ −1}({a,b})={1}.
Note that with this definiton, Some consequences that follow immediately from these definitions are: `f`(`A`_{1}∪`A`_{2}) =`f`(`A`_{1}) ∪`f`(`A`_{2}).`f`(`A`_{1}∩`A`_{2}) ⊆`f`(`A`_{1}) ∩`f`(`A`_{2}).`f`^{ −1}(`B`_{1}∪`B`_{2}) =`f`^{ −1}(`B`_{1}) ∪`f`^{ −1}(`B`_{2}).`f`^{ −1}(`B`_{1}∩`B`_{2}) =`f`^{ −1}(`B`_{1}) ∩`f`^{ −1}(`B`_{2}).`f`(`f`^{ −1}(`B`)) ⊆`B`.`f`^{ −1}(`f`(`A`)) ⊇`A`.
A, A_{1} and A_{2} of the domain and arbitrary subsets B, B_{1} and B_{2} of the codomain.
The results relating images and preimages to the algebra of intersection and union work for any number of sets, not just for 2.## Injective, surjective and bijective functionsSeveral types of functions are very useful, deserve special names: - injective (one-to-one) functions send different arguments to different values; in other words, if
*x*and*y*are members of the domain of*f*, then*f*(*x*) =*f*(*y*) if and only if*x*=*y*. Our example is an injective function. - surjective (onto) functions have their range equal to their codomain; in other words, if
*y*is any member of the codomain of*f*, then there exists at least one*x*such that*f*(*x*) =*y*. - bijective functions are both injective and surjective; they are often used to show that the sets
*X*and*Y*are "the same" in some sense.
## Examples of functions(More can be found at List of functions.)
- The relation
*wght*between persons in the United States and their weights. - The relation between nations and their capitals.
- The relation
*sqr*between natural numbers`n`and their squares`n`^{2}. - The relation
*ln*between*positive*real numbers`x`and their natural logarithms ln(`x`). Note that the relation between real numbers and their natural logarithms is not a function because not every real number has a natural logarithm; that is, this relation is not total and is therefore only a partial function. - The relation
*dist*between points in the plane**R**^{2}and their distances from the origin (0,0). - The relation
*grav*between a point in the punctured plane**R**^{2}\\ {(0,0)} and the vector describing the gravitational force that a certain mass at that point would experience from a certain other mass at the origin (0,0).
Elementary functions -- but the meaning of this term varies among different branches of mathematics. Example of non-elementary functions are Bessel functions and gamma functions.
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