In this journal describes a function in mathematics. One of the precise definition of a function is that it consists of three ordered sets, which can be written as (X, Y, F). X is the domain of the function, Y is the codomain, and F is a set of ordered pairs.
Here I only explain the meaning and function for example. The journal is compiled based on a book about mathematical functions.
Example graphs of functions,
Both domain and range of the image is the set of real numbers between -1 and 1.5. In mathematics, a function is a relation between the set of elements called the domain and a set of elements called the codomain. Associate the function of each element in the domain with exactly one element in the codomain. The elements can be linked so that any thing (the words, objects, attributes) but usually the amount of mathematics, such as real numbers.
Examples of functions with domain (A, B, C) and (1,2,3) with codomain associates A 1, B 2, and C with 3. Examples of functions with both real numbers as its domain and codomain is a function f (x) = 2x, which associates with each real number real number is twice as large. In this case, we can write f (5) = 10.
There are many ways to represent or imagine functions: functions can be described by a formula, by a plot or graph, by an algorithm that calculates it, the arrow between objects, or information about its properties. Sometimes, a function which is described through the relationship with other functions (eg, inverse function). In this discipline, the function is often determined by a table of values or formulas.
In an environment where the output of the function of the number, functions can be added and multiplied, producing a new function. Collection of functions with certain properties, such as the function is continuous and differentiable functions, usually closed under certain operations, called function space and studied as objects in their own right, in disciplines such as real analysis and complex analysis. An operation on important functions, which distinguish them from numbers, is the composition of functions. Composite function is obtained by using the output of one function as input from the others. This operation provides functional theory with the most powerful structure.
In pure mathematics, a function which is defined using set theory, and there are arguments which indicate the existence of uncountably many functions, most of which can not be expressed with the formula or algorithm.
Functions that are used in every field of mathematics that takes a function as the primary object of study is called analysis.
Because the function so widely used, many traditions have grown around them use. The symbol for input to the function are often called independent variables or arguments, and often represented by the letter X, if the input of time, with the letter t. The symbol for output is called the dependent variable or value and is often represented by the letter y. The function itself is most often called f, and thus the notation y = f (x) indicates that the function named f has an input called x and an output named y.
A function f takes an input x, and return the output f (x). A metaphor describes the function as a “machine” or “black box” that turn inputs into outputs.
The set of all inputs are allowed for a particular function is called domain of the function. The set of all output produced is called an image or range of functions. Images often a subset of a larger set, called the codomain functions.
So, if the function f (x) = x2 can take as domain the set of all real numbers, as the image of the set of all non-negative real numbers, and as codomain the set of all real numbers. In this case, we will describe f as a real-valued functions of real variables. Sometimes, especially in computer science, the term “distance” refers to the codomain than images, so care needs to be taken when using the word.
This is a common practice in mathematics to introduce the functions with names like ƒ temporary. For example, f (x) = 2x +1, then f (3) = 7; when the name is not necessary for the function, the form y = 2x 1 can be used. If this function is often used, may be given a name that more permanent.example:
A function of two or more variables are considered in the formal mathematics have domains consisting of ordered pairs or tuples of argument values. For example Sum (x, y) = x + y operating on integers Sum is a function with domain consisting of pairs of integers. Sum then have a domain which consists of elements such as (3.4), an integer codomain, and the relationship between the two which can be explained by a set of ordered pairs like ((3.4), 7). Evaluating Sum (3.4) then gives the value associated with 7 pairs (3.4).
An object is indexed by a set equal to one function. For example, the sequence 1, 1 / 2, 1 / 3, …, 1 / n, … can be written as an ordered sequence where n is a natural number, or as a function f (n) = 1 / n of the set of natural numbers into the set of rational numbers.
A partition function to describe the domain surjective set indexed by the codomain. This partition is known as a core function, and this section called the fiber or the level set function on each element codomain. (Non-surjective functions to describe the domain divide and may set the blank).
One of the precise definition of a function is that it consists of three ordered sets, which can be written as (X, Y, F). X is the domain of the function, Y is the codomain, and F is a set of ordered pairs. In each ordered pair (a, b), a is the first element of the domain, the second b is an element of the codomain, and every element in the domain name is the first element in one and only one ordered pair. The set of all b known as the image of the function. Some authors use the term “range” means an image, others to mean codomain.
For example, the function defined by f (x) = x2 is considered a tight three-set. Domain and codomain are real numbers, and ordered the pair include a pair (3, 9).
The notation f: X → Y indicates that f is a function with domain X and Y. codomain, example:
In most practical situations, the domain and range is understood from context, and only the relationship between inputs and outputs are provided.
Usually written: y = x2
The graph of a function is a set of ordered pairs. As a set can be described in a pair of coordinate axes for example, (3, 9) is the point of intersection of the lines x = 3 and y = 9.
A function is a special case of a mathematical concept is more general, relationships, in which restriction that each element of the domain that appears as the first element in the one and only one ordered pair removed (or, in other words, the restriction that each input will be associated with exactly one output). A relation is “single-value” or “functional” as to each element of the domain set, containing the graph at most one ordered pair (and probably does not exist) with the first element. A relation is called “left-total” or just “total” as to each element of the domain, the graph contains at least one ordered pair as the first element (and perhaps more than one). A good relationship left-total and single-valued function.
In some parts of mathematics, including recursion theory and functional analysis, it is easier to study partial functions where some of the value of the domain does not have an association in the graph, ie, single valued relationship. For example, a function f such that f (x) = 1 / x does not define values for x = 0, and similarly only partially true function of the real line to line. The term total function can be used to emphasize the fact that every element of the domain does not appear as the first element of ordered pairs in the graph. In other parts of mathematics, the relationship is not single valued equally combined with the function: this is known as a multivalued function, with the appropriate term single-valued function to normal function.
Some authors (especially in set theory) defines it simply as a function of the graph f, with the restriction that the graph can not contain two different ordered pairs with first element.
Many operations in set theory such as power-set-in a class of all groups in its domain, therefore, although they described as an informal function, they do not fit with the set-theoretical definitions described above.
Specific input in a function called the function arguments. For each value of the argument x, y in the appropriate unique codomain is called the value function at x, the output of f for argument x, or the image of x under ƒ. The image x can be written as f (x) or as y.
The graph of the function f is the set of all ordered pairs (x, f (x)), for all x in the domain X. If X and Y are subsets of R, real numbers, then this definition coincides with the familiar sense of “graph” as an image or a plot function, the ordered pair into Cartesian coordinates.
A function can also be called a map or mapping. Some authors, using the term “function” and “map” to refer to various types of functions. Other types of specific functions including functionals and operators.
Formal description of a function usually involves the function name, its domain, the codomain, and rules of correspondence. Thus, we often see two parts of the notation, for example:
It can be concluded:
• “ƒ is a function from N to R” (people often write informally “Let ƒ: X → Y” means “Let ƒ be a function from X to Y”), or
• “ƒ is a function of N to R”, or
• “f is R-valued function of the N-value of the variable”,
Here the function named “ƒ” have natural numbers as domain, real numbers as codomain, and maps n to itself divided by π. Less formal, long form may be abbreviated
Where f (n) is read as “f as a function of n” or “fn”. There is some loss of information: we are no longer explicitly given the domain codomain N and R.
It is common to remove the parentheses around the argument when there is little chance of confusion, thus: sin x; is known as prefix notation. Write a function after the arguments, as in x ƒ, known as postfix notation, as usual factorial function written n!, Although generalization, the gamma function, written Γ (n). Brackets are used to resolve ambiguity and stated priority, although in some formal setting of consistent good use prefix or postfix notation, eliminating the need for any brackets.
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