Function (mathematics) Guide, Meaning , Facts, Information and Description
In mathematics, a function is a relation, such that each element of a set is associated with a unique element of another (possibly the same) set. The concept of a function is fundamental to virtually every branch of mathematics and every quantitative science.The terms function, mapping, map and transformation are usually used synonymously. The term operator is frequently used for dyadic functions expressed using infix notation.
Intuitive introduction
Essentially, a function is a "rule" that assigns a unique output to each given input. Here are some examples of functions:
- Each person has a favorite colour (red, orange, yellow, green, cyan, blue, indigo, or violet). The favorite colour is a function of the person. For example, John has favorite colour red, while Kim has favorite colour violet. Here, the input is the person, and the output is one of the 8 colours.
- Some children are selling lemonade in the summer. The number of lemonades they sell is a function of the temperature outside. For example, if it is 85 degrees outside, they sell 10 lemonades, but if it is 95 degrees outside, they sell 25 lemonades. Here, the input is the temperature, and the output is the number of lemonades they sell.
- A stone is dropped from different stories of a tall building. The dropped stone may take 2 seconds to fall from the second story, and (only) 4 seconds to fall from the 8th story. Here, the input is the story, and the output is the number of seconds. The relevant function describes the relationship between the time it takes the stone to reach the ground and the story. (See acceleration)
A very common type of function occurs when the argument and the function value are both numbers, the functional relationship is expressed by a formula, and the value of the function is obtained by direct substitution of the argument into the formula. Consider for example
A straightforward generalization is to allow functions depending on several arguments. For instance,
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 every element of the domain is associated to precisely one element of the codomain. Functions are abstractly defined as certain relations, as will be seen below. Because of this generality, the function concept is fundamental to virtually every branch of mathematics.
As a mathematical term, "function" was coined by Leibniz in 1694, to describe a quantity related to a curve, such as a curve's slope or a specific point of a curve. The functions Leibniz considered are today called differentiable functions, and they are the type of function most frequently encountered by nonmathematicians. For this type of function, one can talk about limitss and derivatives; both are measurements of the change of output values associated to a change of input values, and these measurements are the basis of calculus.
The word function was later used by Euler during the mid-18th century to describe an expression or formula involving various argumentss, e.g. f(x) = sin(x) + x3.
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 continuous functions that are nowhere differentiable. These functions were first thought to be only theoretical curiosities, and they were collectively called "monsters" as late as the turn of the 20th century. However, powerful techniques from functional analysis has shown that these functions are actually more common than differentiable functions. Such functions have since been applied to 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 to define every mathematical object as a set. Dirichlet and Lobachevcky independently and almost simultaneously gave the modern "formal" definition of function (see formal definition below).
In this 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.
The notion of function as a rule for computing, rather than a special kind of relation, has been formalized in mathematical logic by means of the lambda calculus.
Formally, a function f from a set X of input values to a set Y of possible output values (written as f : X → Y) is a relation between X and Y which satisfies:
A more concise expression of the above definition is the following: a function from X to Y is a subset f of the cartesian product X × Y, such that for each x in X, there is a unique y in Y such that the ordered pair (x, y) is in f.
The set of all functions f : X → Y is denoted by YX. Note that |YX| = |Y||X| (refer to Cardinal numbers).
A relation between X and Y that satisfies condition (1) is a multivalued function. Every function is a multivalued function, but not every multivalued function is a function. A relation between X and Y that satisfies condition (2) is a partial function. Every function is a partial function, but not every partial function is a function. In this encyclopedia, the term "function" will mean a relation satisfying both conditions (1) and (2), unless otherwise stated.
Consider the following three examples:
History
Formal definition
For each input value x in the domain, the corresponding unique output value y in the codomain is denoted by f(x).
| This relation is total but not many-to-one; the element 3 in X is related to two elements b and c in Y. Therefore, this is a multivalued function, but not a function. | ||
| This relation is many-to-one but not total; the element 1 in X is not related to any element of Y. Therefore, this is a partial function, but not a function. | ||
| This relation is both total and many-to-one, and so it is a function from X to Y. Note that the emphasis is on "-to-one" as "many" may actually mean "one". The function can be given explicitly as f = {(1, d), (2, d), (3, c)} or as
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X, the set of input values, is called the domain of f, and Y, the set of possible output values, is called the codomain. The range of f is the set of all actual outputs {f(x) : x in the domain}. Beware that sometimes the codomain is incorrectly called the range because of a failure to distinguish between possible and actual values.
Functions are named after their ranges, for example real functions and complex functions.
An endofunction is a function whose domain and range are identical.
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.
Several types of functions that are very useful have special names:
The image of an element x∈X under f is the output f(x).
The image of a subset A⊂X under f is the subset of Y defined by
The preimage (or inverse image) of a set B ⊂ Y under f is the subset of X defined by
The graph of a function f is the set of all ordered pairs(x, f(x)), for all x in the domain X. There are theorems formulated or proved most easily in terms of the graph, such as the closed graph theorem. If X and Y are real lines, then this definition coincides with the familiar sense of graph.
, This function is surjective but not injective.]]
Note that since a relation on the two sets X and Y is usually formalized as a subset of X×Y, the formal definition of function actually identifies the function f with its graph.
(More can be found at List of functions.)
Functions in applications are often functions of several variables: the values they take depend on a number of different factors. From a mathematical point of view all the variables must be made explicit in order to have a functional relationship - no 'hidden' factors are allowed. Then again, from the mathematical point of view, there is no qualitative difference between functions of one and of several variables. A function of three real variables is just a function that applies to triples of real numbers. The following paragraph says this in more formal language.
If the domain of a function is a subset of the Cartesian product of n sets then the function is called an n-ary function.
For example, the relation dist has the domain R × R and is therefore a binary function.
In that case dist((x,y)) is simply written as dist(x,y).
Another name applied to some types of functions of several variables is operation. In abstract algebra, operators such as "*" are defined as binary functions; when we write a formula such as x*y in this context, we are implicitly invoking the function *(x,y), but writing it in a convenient infix notation.
An important theoretical paradigm, functional programming, takes the function concept as central. In that setting, the handling of functions of several variables becomes an operational matter, for which the lambda calculus provides the basic syntax. The composition of functions (see under composing functions immediately below) becomes a question of explicit forms of substitution, as used in the substitution rule of calculus. In particular, a formalism called currying can be used to reduce n-ary functions to functions of a single variable.
The functions f: X → Y and g: Y → Z can be composed by first applying f to an argument x and then applying g to the result.
Thus one obtains a composite function g o f: X → Z defined by (g o f)(x) = g(f(x)) for all x in X.
As an example, suppose that an airplane's height at time t is given by the function h(t) and that the oxygen concentration at height x is given by the function c(x).
Then (c o h)(t) describes the oxygen concentration around the plane at time t.
An example of an inverse function, for f(x) = 2x, is f(x)−1 = x/2. The inverse function is the function that "undoes" its original. See also inverse image.
Inverses are sometimes difficult or impossible to find. Consider f(x) = x2. The function f(x) = √x is not an inverse when the domain of f is R. (As -22 is 4, but √4 is either 2 or -2).
The restriction of to is then the function . Intuitively, this is the same function as except that we restrict the domain of to .
An extension of a function is a function defined on a superset of such that . Provided the domain of is not the universal set, always has lots of extensions.
If f: X → R and g: X → R are functions with common domain X and codomain is a ring R, then one can define the sum function f + g: X → R and the product function f × g: X → R as follows:
This turns the set of all such functions into a ring. The binary operations in that ring have as domain ordered pairs of functions, and as codomain functions. This is an example of climbing up in abstraction, to functions of more complex types.
By taking some other algebraic structure A in the place of R, we can turn the set of all functions from X to A into an algebraic structure of the same type in an analogous way.
The number of computable functions from integers to integers is countable, because the number of possible algorithms is. The number of all functions from integers to integers is higher: the same as the cardinality of the real numbers. This argument shows that there are functions from integers to integers that are not computable. For examples of noncomputable functions, see the articles on the halting problem and Rice's theorem.
In the context of category theory, a function no longer represents a rule for taking an input to an output, but instead represents a relationship between its domain and its codomain. Since these functions are no longer functions in the usual sense, they are usually referred to as morphisms. A morphism is then an ordered triple (X, '\'Y, f), where f is a "function" with domain X and codomain Y. Since X and Y'' do not necessarily correspond to a set of objects, however, morphisms do not always behave like functions, and, for example, enlarging the codomain (which does nothing to a function) gives a different morphism which you cannot identify with the original one.
Ordinary functions are sometimes referred to as morphisms when they are morphisms in a concrete category.
This is an Article on Function (mathematics). Page Contains Information, Facts Details or Explanation Guide About Function (mathematics) Domains, codomains, and ranges
Injective, surjective and bijective functions
Images and preimages
Notice that the range of f is the image f(X) of its domain. In our function above, the image of {2,3} under f is f({2, 3}) = {c, d} and the range of f is {a, c, d}.
For our function above, the preimage of {a, b} is f −1({a, b}) = {1}.Graph of a function
Examples of functions
Most commonly used types of mathematical functions involving addition, division, exponents, logarithms, multiplication, polynomials, radicals, rationalss, subtraction, and trigonometric expressions. They are sometimes collectively referred as elementary functions -- but the meaning of this term varies among different branches of mathematics. Example of non-elementary functions (or special functions) are Bessel functions and gamma functions.Properties of functions
Functions can be
n-ary function: function of several variables
Composing functions
Inverse function
If a function f:X→Y is bijective then preimages of any element y in the codomain Y is a singleton. A function taking y∈Y to its preimage f−1(y) is a well-defined function called the inverse of f and is denoted by f−1.Restrictions and extensions
Suppose that is a subset of and that
is a function. Let
be the inclusion function
for x ∈ X.Pointwise operations
for all x in X.Computable and non-computable functions
Functions from the categorical viewpoint
References
See also
External links
a compendium of formulae for and visualizations of mathematical functions