Function 8 Key

[Jordan Tucker]

In mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y.[1] The set X is called the domain of the function[2] and the set Y is called the codomain of the function.[3]

Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a function of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity). The concept of a function was formalized at the end of the 19th century in terms of set theory, and this greatly enlarged the domains of application of the concept.

A function f from a set X to a set Y is an assignment of one element of Y to each element of X. The set X is called the domain of the function and the set Y is called the codomain of the function.

The set of the elements of X such that f ( x ) \displaystyle f(x) is defined and belongs to Y is called the domain of definition of the function. A partial function from X to Y is thus a ordinary function that has as its domain a subset of X called the domain of definition of the function. If the domain of definition equals X, one often says that the partial function is a total function.

In several areas of mathematics the term "function" refers to partial functions rather than to ordinary functions. This is typically the case when functions may be specified in a way that makes difficult or even impossible to determine their domain.

In computability theory, a general recursive function is a partial function from the integers to the integers whose values can be computed by an algorithm (roughly speaking). The domain of definition of such a function is the set of inputs for which the algorithm does not run forever. A fundamental theorem of computability theory is that there cannot exist an algorithm that takes an arbitrary general recursive function as input and tests whether 0 belongs to its domain of definition (see Halting problem).

A multivariate function, multivariable function, or function of several variables is a function that depends on several arguments. Such functions are commonly encountered. For example, the position of a car on a road is a function of the time travelled and its average speed.

If all the X i \displaystyle X_i are equal to the set R \displaystyle \mathbb R of the real numbers or to the set C \displaystyle \mathbb C of the complex numbers, one talks respectively of a function of several real variables or of a function of several complex variables.

The functional notation requires that a name is given to the function, which, in the case of a unspecified function is often the letter f. Then, the application of the function to an argument is denoted by its name followed by its argument (or, in the case of a multivariate functions, its arguments) enclosed between parentheses, such as in

When the symbol denoting the function consists of several characters and no ambiguity may arise, the parentheses of functional notation might be omitted. For example, it is common to write sin x instead of sin(x).

Functional notation was first used by Leonhard Euler in 1734.[10] Some widely used functions are represented by a symbol consisting of several letters (usually two or three, generally an abbreviation of their name). In this case, a roman type is customarily used instead, such as "sin" for the sine function, in contrast to italic font for single-letter symbols.

The functional notation is often used colloqually for referring to a function and simultaneously naming its argument, such as in "let f ( x ) \displaystyle f(x) be a function". This is an abuse of notation that is useful for a simpler formulation.

This is typically the case for functions whose domain is the set of the natural numbers. Such a function is called a sequence, and, in this case the element f n \displaystyle f_n is called the nth element of the sequence.

In some cases the argument of a function may be an ordered pair of elements taken from some set or sets. For example, a function f can be defined as mapping any pair of real numbers ( x , y ) \displaystyle (x,y) to the sum of their squares, x 2 + y 2 \displaystyle x^2+y^2 . Such a function is commonly written as f ( x , y ) = x 2 + y 2 \displaystyle f(x,y)=x^2+y^2 and referred to as "a function of two variables". Likewise one can have a function of three or more variables, with notations such as f ( w , x , y ) \displaystyle f(w,x,y) , f ( w , x , y , z ) \displaystyle f(w,x,y,z) .

A function may also be called a map or a mapping, but some authors make a distinction between the term "map" and "function". For example, the term "map" is often reserved for a "function" with some sort of special structure (e.g. maps of manifolds). In particular map may be used in place of homomorphism for the sake of succinctness (e.g., linear map or map from G to H instead of group homomorphism from G to H). Some authors[13] reserve the word mapping for the case where the structure of the codomain belongs explicitly to the definition of the function.

Some authors, such as Serge Lang,[12] use "function" only to refer to maps for which the codomain is a subset of the real or complex numbers, and use the term mapping for more general functions.

Given a function f \displaystyle f , by definition, to each element x \displaystyle x of the domain of the function f \displaystyle f , there is a unique element associated to it, the value f ( x ) \displaystyle f(x) of f \displaystyle f at x \displaystyle x . There are several ways to specify or describe how x \displaystyle x is related to f ( x ) \displaystyle f(x) , both explicitly and implicitly. Sometimes, a theorem or an axiom asserts the existence of a function having some properties, without describing it more precisely. Often, the specification or description is referred to as the definition of the function f \displaystyle f .

When a function is defined this way, the determination of its domain is sometimes difficult. If the formula that defines the function contains divisions, the values of the variable for which a denominator is zero must be excluded from the domain; thus, for a complicated function, the determination of the domain passes through the computation of the zeros of auxiliary functions. Similarly, if square roots occur in the definition of a function from R \displaystyle \mathbb R to R , \displaystyle \mathbb R , the domain is included in the set of the values of the variable for which the arguments of the square roots are nonnegative.

Many functions can be defined as the antiderivative of another function. This is the case of the natural logarithm, which is the antiderivative of 1/x that is 0 for x = 1. Another common example is the error function.

More generally, many functions, including most special functions, can be defined as solutions of differential equations. The simplest example is probably the exponential function, which can be defined as the unique function that is equal to its derivative and takes the value 1 for x = 0.

A graph is commonly used to give an intuitive picture of a function. As an example of how a graph helps to understand a function, it is easy to see from its graph whether a function is increasing or decreasing. Some functions may also be represented by bar charts.

On the other hand, if a function's domain is continuous, a table can give the values of the function at specific values of the domain. If an intermediate value is needed, interpolation can be used to estimate the value of the function. For example, a portion of a table for the sine function might be given as follows, with values rounded to 6 decimal places:

The image of f is the image of the whole domain, that is, f(X).[16] It is also called the range of f,[6][7][8][9] although the term range may also refer to the codomain.[9][16][17]

"One-to-one" and "onto" are terms that were more common in the older English language literature; "injective", "surjective", and "bijective" were originally coined as French words in the second quarter of the 20th century by the Bourbaki group and imported into English.[21] As a word of caution, "a one-to-one function" is one that is injective, while a "one-to-one correspondence" refers to a bijective function. Also, the statement "f maps X onto Y" differs from "f maps X into B", in that the former implies that f is surjective, while the latter makes no assertion about the nature of f. In a complicated reasoning, the one letter difference can easily be missed. Due to the confusing nature of this older terminology, these terms have declined in popularity relative to the Bourbakian terms, which have also the advantage of being more symmetrical.

An extension of a function f is a function g such that f is a restriction of g. A typical use of this concept is the process of analytic continuation, that allows extending functions whose domain is a small part of the complex plane to functions whose domain is almost the whole complex plane.

The idea of function, starting in the 17th century, was fundamental to the new infinitesimal calculus. At that time, only real-valued functions of a real variable were considered, and all functions were assumed to be smooth. But the definition was soon extended to functions of several variables and to functions of a complex variable. In the second half of the 19th century, the mathematically rigorous definition of a function was introduced, and functions with arbitrary domains and codomains were defined.

Functions are now used throughout all areas of mathematics. In introductory calculus, when the word function is used without qualification, it means a real-valued function of a single real variable. The more general definition of a function is usually introduced to second or third year college students with STEM majors, and in their senior year they are introduced to calculus in a larger, more rigorous setting in courses such as real analysis and complex analysis.

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