One of the classic definitions of the concept “function” are those based on correspondences. Let us present a number of such definitions.
Definition 1
A relationship in which each value of the independent variable corresponds to a single value of the dependent variable is called function.
Definition 2
Let two non-empty sets $X$ and $Y$ be given. A correspondence $f$ that matches each $x\in X$ with one and only one $y\in Y$ Is called function($f:X → Y$).
Definition 3
Let $M$ and $N$ be two arbitrary number sets. A function $f$ is said to be defined on $M$, taking values from $N$, if each element $x\in X$ is associated with one and only one element from $N$.
The following definition is given through the concept of a variable quantity. A variable quantity is a quantity that takes on different numerical values in a given study.
Definition 4
Let $M$ be the set of values of the variable $x$. Then, if each value $x\in M$ corresponds to one specific value of another variable $y$ is a function of the value $x$ defined on the set $M$.
Definition 5
Let $X$ and $Y$ be some number sets. A function is a set $f$ of ordered pairs of numbers $(x,\ y)$ such that $x\in X$, $y\in Y$ and each $x$ is included in one and only one pair of this set, and each $y$ is in at least one pair.
Definition 6
Any set $f=\(\left(x,\ y\right)\)$ of ordered pairs $\left(x,\ y\right)$ such that for any pairs $\left(x",\ y" \right)\in f$ and $\left(x"",\ y""\right)\in f$ from the condition $y"≠ y""$ it follows that $x"≠x""$ is called a function or display.
Definition 7
A function $f:X → Y$ is a set of $f$ ordered pairs $\left(x,\ y\right)\in X\times Y$ such that for any element $x\in X$ there is a unique element $y\in Y$ such that $\left(x,\ y\right)\in f$, that is, the function is a tuple of objects $\left(f,\ X,\ Y\right)$.
In these definitions
$x$ is the independent variable.
$y$ is the dependent variable.
All possible values of the variable $x$ are called the domain of the function, and all possible values of the variable $y$ are called the domain of the function.
Analytical method of specifying a function
For this method, we need the concept of an analytical expression.
Definition 8
An analytical expression is the product of all possible mathematical operations on any numbers and variables.
The analytical way to specify a function is to specify it using an analytical expression.
Example 1
$y=x^2+7x-3$, $y=\frac(x+5)(x+2)$, $y=cos5x$.
Pros:
- Using formulas, we can determine the value of the function for any specific value of the variable $x$;
- Functions defined in this way can be studied using the apparatus of mathematical analysis.
Minuses:
- Low visibility.
- Sometimes you have to make very cumbersome calculations.
Tabular method of specifying a function
This method of assignment consists of writing down the values of the dependent variable for several values of the independent variable. All this is entered into the table.
Example 2
Picture 1.
Plus: For any value of the independent variable $x$, which is entered into the table, the corresponding value of the function $y$ is immediately known.
Minuses:
- Most often, there is no complete function specification;
- Low visibility.
To define a function means to establish a rule (law) with the help of which, based on the given values of the independent variable, we find the corresponding function values. Let's look at different ways to define a function.
This entry defines the temperature T as a function of time t:T=f(t). The advantages of the tabular method of specifying a function are that it makes it possible to determine certain specific values of the function immediately, without additional changes or calculations. Disadvantages: does not define the function completely, but only for some argument values; does not provide a visual representation of the nature of the change in the function with a change in the argument.
2. Graphic method.Schedule function y=f(x) is the set of all points of the plane whose coordinates satisfy this equation. This can be some curve, in particular a straight line, or a set of points on a plane.
The advantage is clarity, the disadvantage is that it is not possible to accurately determine the values of the argument. In engineering and physics, it is often the only available way to specify a function, for example, when using recording instruments that automatically record changes in one quantity relative to another (barograph, thermograph, etc.).
3. Analytical method. Using this method, the function is specified analytically, using a formula. This method makes it possible for each numerical value of the argument x to find the corresponding numerical value of the function y exactly or with some accuracy.
In the analytical method, a function can be specified by several different formulas. For example, the function
defined in the domain [- 
, 15] using three formulas.
If the relationship between x and y is given by a formula resolved with respect to y, i.e. has the form y = f(x), then they say that the function of x is given explicitly, for example. If the values of x and y are related by some equation of the form F(x,y) = 0, i.e. the formula is not resolved with respect to y, then the function is said to be specified implicitly. For example,. Note that not every implicit function can be represented in the form y =f(x); on the contrary, any explicit function can always be represented in the form of an implicit one: 
. Another type of analytical specification of a function is parametric, when the argument x and function y are functions of a third quantity - parameter t: 
, Where 
, T – some interval. This method is widely used in mechanics and geometry.
The analytical method is the most common way to define a function. Compactness, the ability to apply mathematical analysis to a given function, and the ability to calculate function values for any argument values are its main advantages.
4. Verbal method. This method consists in expressing functional dependence in words. For example, the function E(x) is the integer part of the number x, the Dirichlet function, the Riemann function, n!, r(n) is the number of divisors of the natural number n.
5. Semi-graphic method. Here, the function values are represented as segments, and the argument values are represented as numbers placed at the ends of the segments indicating the function values. So, for example, a thermometer has a scale with equal divisions with numbers on them. These numbers are the values of the argument (temperature). They stand in the place that determines the graphical elongation of the mercury column (function value) due to its volumetric expansion as a result of temperature changes.
>>Mathematics: Methods of specifying a function
Methods for specifying a function
By giving various examples of functions in the previous paragraph, we have somewhat impoverished the very concept of function.
After all, defining a function means specifying a rule that allows you to calculate the corresponding value y from an arbitrarily chosen value x from B(0. Most often, this rule is associated with a formula or several formulas - this method of specifying a function is usually called analytical. All functions discussed in § 7, were given analytically. Meanwhile, there are other ways to define a function, which will be discussed in this section.
If the function was specified analytically and we managed to construct a graph of the function, then we have actually moved from the analytical method of specifying the function to the graphical one. The reverse transition is not always possible. As a rule, this is a rather difficult but interesting task.
Not every line on the coordinate plane can be considered as a graph of some function. For example, a circle defined by the equation x 2 + y 2 - 9 (Fig. 51) is not a graph of a function, since any straight line x = a, where | a |<3, пересекает эту линию в д в у х точках (а для задания функции таких точек должно быть не более одной, т.е. прямая х = а должна пересекать линию F только в одной точке либо вообще не должна ее пересекать).
At the same time, if this circle is cut into two parts - the upper semicircle (Fig. 52) and the lower semicircle (Fig. 53), then each of the semicircles can be considered a graph of some function, and in both cases it is easy to switch from the graphical method of specifying the function to analytical.
From the equation x 2 + y 2 = 9 we find y 2 = 9 - x 2 and further 
The graph of the function is the upper semicircle of the circle x 2 + y 2 = 9 (Fig. 52), and the graph of the function is the lower semicircle of the circle x 2 + y 2 = 9 (Fig. 53).
This example allows us to draw attention to one significant circumstance. Look at the graph of the function (Fig. 52). It is immediately clear that D(f) = [-3, 3]. And if we were talking about finding the domain of definition of an analytically given function, then we would have to, as we did in § 7, spend time and effort on solving the inequality. That is why they usually try to work simultaneously with both analytical and graphical methods of specifying functions. However, after two years of studying algebra at school, you have already become accustomed to this.
In addition to analytical and graphical, in practice, a tabular method of specifying a function is used. With this method, a table is provided that indicates the values of the function (sometimes exact, sometimes approximate) for a finite set of argument values. Examples of tabular functions can be tables of squares of numbers, cubes of numbers, square roots, etc.
In many cases, table specification of a function is convenient. It allows you to find the value of a function for the argument values available in the table without any calculations.
Analytical, graphical, tabular - naitabular, simpler, and therefore the most popular verbal task functions, these methods are quite sufficient for our needs. In fact, in mathematics there are quite a few different ways to define a function, but we will introduce you to only one more method, which is used in very peculiar situations. We are talking about the verbal method, when the rule for specifying a function is described in words. Let's give examples.
Example 1.
The function y = f(x) is defined on the set of all non-negative numbers using the following rule: each number x > 0 is assigned the first decimal place in the decimal notation of the number x. If, say, x = 2.534, then f(x) = 5 (the first decimal place is the number 5); if x = 13.002, then f(x) = 0; if then, writing 0.6666... as an infinite decimal fraction, we find f(x) = 6. What is the value of f(15)? It is equal to 0, since 15 = 15,000..., and we see that the first decimal place after the decimal point is 0 (in fact, the equality 15 = 14,999... is also true, but mathematicians have agreed not to consider infinite periodic decimal fractions with a period 9).
Any non-negative number x can be written as a decimal fraction (finite or infinite), and therefore for each value of x we can find a specific value for the first decimal place, so we can talk about a function, albeit a somewhat unusual one. This function 
Example 2.
The function y = f(x) is defined on the set of all real numbers using the following rule: each number x is associated with the largest of all integers that do not exceed x. In other words, the function y = f(x) is determined by the following conditions:
a) f(x) - an integer;
b) f(x)< х (поскольку f(х) не превосходит х);
c) f(x) + 1 > x (since f(x) is the largest integer not exceeding x, which means f(x) + 1 is already greater than r). If, say, x = 2.534, then f(x) = 2, since, firstly, 2 is an integer, and secondly, 2< 2,534 и, в-третьих, следующее целое число 3 уже больше, чем 2,534. Если х = 47, то /(х) = 47, поскольку, во-первых, 47 - целое число, во-вторых, 47< 47 (точнее, 47 = 47) и, в-третьих, следующее за числом 47 целое число 48 уже больше, чем 47. А чему равно значение f(-0,(23))? Оно равно -1. Проверяйте: -1 - наибольшее из всех целых чисел, которые не превосходят числа -0,232323....
This function has (set of integers).
The function discussed in example 2 is called the integer part of a number; for the integer part of the number x, use the notation [x]. For example, = 2, = 47, [-0,(23)] = -1. The graph of the function y = [x] looks very peculiar (Fig. 54).
Functions can be specified in a variety of ways. However, the most common are the following three ways of specifying functions: analytical, tabular and graphical.
Analytical method of specifying a function. With the analytical method of specifying, a function is determined using an analytical expression, that is, using a formula indicating what actions must be performed on the value of the argument in order to obtain the corresponding value of the function.
In paragraphs 2 and 3, we have already encountered functions defined using formulas, i.e. analytically. Moreover, in step 2 for the function the domain of definition ) was established based on geometric considerations, and for the function the domain of definition was indicated in the condition. In step 3 for the function, the domain of definition was also specified by condition. However, very often a function is specified only using an analytical expression (formula), without any additional conditions. In such cases, by the domain of definition of a function we will understand the totality of all those values of the argument for which this expression makes sense and leads to the actual values of the function.
Example 1. Find the domain of a function
Solution. The function is specified only by a formula, its domain of definition is not specified and there are no additional conditions. Therefore, by the domain of definition of this function we must understand the totality of all those argument values for which the expression has real values. For this there must be . Solving this inequality, we come to the conclusion that the domain of definition of this function is the segment [-1.1].
Example 2. Find the domain of definition of the function.
Solution. The domain of definition obviously consists of two infinite intervals, since the expression does not make sense when and is defined for all other values.
The reader can now easily see that for a function the domain of definition will be the entire numerical axis, and for a function it will be an infinite interval
It should be noted that it is impossible to identify a function and the formula with which this function is specified. Using the same formula, you can define different functions. In fact, in paragraph 2 we considered a function with a domain of definition; in paragraph 3, a graph was built for a function with a domain of definition. And finally, we have just looked at a function defined only by a formula without any additional conditions. The domain of this function is the entire number line. These three functions are different because they have different scopes of definition. But they are specified using the same formula.
The opposite case is also possible, when one function in different parts of its domain of definition is given by different formulas. For example, consider a function y defined for all non-negative values as follows: for for i.e.
This function is defined by two analytical expressions that operate in different parts of its domain of definition. The graph of this function is shown in Fig. 18.
Tabular method of specifying a function. When specifying a function in a table, a table is compiled in which a number of argument values and corresponding function values are indicated. Logarithmic tables, tables of values of trigonometric functions and many others are widely known. Quite often it is necessary to use tables of function values obtained directly from experience. The following table shows the experimentally obtained resistivities of copper (in cm - centimeters) at different temperatures t (in degrees):
Graphical way to specify a function. In a graphical task, a graph of a function is given, and its values corresponding to certain values of the argument are directly found from this graph. In many cases, such graphs are drawn using recording devices.
Various ways of specifying a function Analytical, graphical, tabular are the simplest, and therefore the most popular ways of specifying a function; for our needs, these methods are quite sufficient. Analyticalgraphictabular In fact, in mathematics there are quite a few different ways of specifying a function, and one of them is verbal, which is used in very peculiar situations.
Verbal way of specifying a function A function can also be specified verbally, i.e. descriptively. For example, the so-called Dirichlet function is defined as follows: the function y is equal to 0 for all rational and 1 for all irrational values of the argument x. Such a function cannot be specified by a table, since it is defined on the entire numerical axis and the set of values for its argument is infinite. This function cannot be specified graphically either. An analytical expression for this function was nevertheless found, but it is so complex that it has no practical significance. The verbal method gives a brief and clear definition of it.
Example 1 The function y = f (x) is defined on the set of all non-negative numbers using the following rule: each number x 0 is assigned the first decimal place in the decimal notation of the number x. If, say, x = 2.534, then f(x) = 5 (the first decimal place is the number 5); if x = 13.002, then f(x) = 0; if x = 2/3, then, writing 2/3 as an infinite decimal fraction 0.6666..., we find f(x) = 6. What is the value of f(15)? It is equal to 0, since 15 = 15,000..., and we see that the first decimal place after the decimal point is 0 (in general, the equality 15 = 14,999... is true, but mathematicians have agreed not to consider infinite periodic decimal fractions with a period of 9).
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Of all the indicated methods of specifying a function, the greatest opportunities for using the apparatus of mathematical analysis are provided by the analytical method, and the graphical one has the greatest clarity. That is why mathematical analysis is based on a deep synthesis of analytical and geometric methods. The study of functions defined analytically is much easier and becomes clearer if we simultaneously consider the graphs of these functions.
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The great mathematician - Dirichlet B, professor at Berlin, and from 1855 at the University of Göttingen. Main works on number theory and mathematical analysis. In the field of mathematical analysis, Dirichlet was the first to precisely formulate and study the concept of conditional convergence of a series, established a test for the convergence of a series (the so-called Dirichlet test, 1862), and gave (1829) a rigorous proof of the possibility of expanding a function having a finite number of maxima and minima into a Fourier series. Dirichlet's significant works are devoted to mechanics and mathematical physics (Dirichlet's principle in the theory of harmonic functions). Dirichlet Peter Gustav Lejeune () German mathematician, foreign corresponding member. Petersburg Academy of Sciences (c), member of the Royal Society of London (1855), Paris Academy of Sciences (1854), Berlin Academy of Sciences. Dirichlet proved the theorem on the existence of an infinitely large number of prime numbers in any arithmetic progression of integers, the first term and the difference of which are mutually prime numbers, and studied (1837) the law of distribution of prime numbers in arithmetic progressions, and therefore introduced functional series of a special form ( so-called Dirichlet series).
