Methods for specifying a function. Examples. Function and methods for specifying it Definition of the concept of a function, methods for specifying a function

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.

Cons:

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

Figure 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.

Cons:

Most often, there is no complete function specification;
Low visibility.

A function is a law according to which a number x from a given set X is associated with only one number y, written , while x is called the argument of the function, y is called the value of the function.
There are different ways to define functions.

1. Analytical method.
Analytical method - This is the most common way to specify a function.
It consists in the fact that the function is given by a formula that establishes what operations need to be performed on x in order to find y. For example .
Let's look at the first example - . Here the value x = 1 corresponds to , the value x = 3 corresponds, etc.
A function can be defined on different parts of the set X by different functions.
For example:

In all the previously given examples of the analytical method of setting, the function was specified explicitly. That is, on the right was the variable y, and on the right was the formula for the variable x. However, with the analytical method of setting, the function can also be specified implicitly.
For example . Here, if we give the variable x a value, then to find the value of the variable y (the value of the function), we have to solve the equation. For example, for the first given function at x = 3, we will solve the equation:
. That is, the value of the function at x = 3 is -4/3.
With the analytical method of setting, the function can be specified parametrically - this is when x and y are expressed through some parameter t. For example,

Here at t = 2, x = 2, y = 4. That is, the value of the function at x = 2 is 4.
2. Graphic method.
With the graphical method, a rectangular coordinate system is introduced and a set of points with coordinates (x,y) is depicted in this coordinate system. At the same time. Example:
3. Verbal method.
The function is specified using a verbal formulation. A classic example is the Dirichlet function.
“The function is equal to 1 if x is a rational number; function equals 0 if x is an irrational number.”
4. Tabular method.
The tabular method is most convenient when the set X is finite. With this method, a table is compiled in which each element from the set X is assigned a number Y.
Example.

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).

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 certain number of values of the first decimal place, so we can talk about a function, albeit a somewhat unusual one. D (f) = . = 2 [" title="A function that is defined by the conditions: f (x) is an integer; f (x) x;x; f + 1 > x,x, the integer part of a number is called the integer part of a number. D (f) = (-;+), E (f) = Z (set of integers) For the integer part of the number x, use the notation [ x = 2 [ ]." class="link_thumb"> 7 !} A function that is determined by the following conditions: f (x) – integer; f(x)x;x; f + 1 > x,x, the integer part of a number is called the integer part of the number. D (f) = (-;+), E (f) = Z (set of integers) For the integer part of the number x, use the notation [x]. = 2 = 47 [ - 0.23] = - 1 x,x, the integer part of a number is called the integer part of a number. D (f) = (-;+), E (f) = Z (set of integers) For the integer part of the number x, use the notation [x]. = 2 ["> x,x, the integer part of a number is called the integer part of the number. D (f) = (-;+), E (f) = Z (set of integers) For the integer part of the number x, the notation [ x ] is used. = 2 = 47 [ - 0.23] = - 1"> x,x, the integer part of a number is called the integer part of the number. D (f) = (-;+), E (f) = Z (set of integers) For the integer part of the number x, use the notation [x]. = 2 [" title="A function that is defined by the conditions: f (x) is an integer; f (x) x;x; f + 1 > x,x, the integer part of a number is called the integer part of a number. D (f) = (-;+), E (f) = Z (set of integers) For the integer part of the number x, use the notation [ x = 2 [ ]."> title="A function that is determined by the following conditions: f (x) – integer; f(x)x;x; f + 1 > x,x, the integer part of a number is called the integer part of the number. D (f) = (-;+), E (f) = Z (set of integers) For the integer part of the number x, use the notation [x]. = 2 ["> !}

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 the graphs of these functions are also examined in parallel.

X y=x

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 investigate 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).

>>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).

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Lecture: Concept of function. Basic properties of the function.

Teacher: Goryacheva A.O.

ABOUT. : The rule (law) of correspondence between sets X and Y, according to which for each element from set X one and only one element from set Y can be found, is calledfunction .

A function is considered defined if:

The domain of definition of the function X is given;

The range of values of the function Y is specified;

The rule (law) of correspondence is known, and such that for each value of the argument only one value of the function can be found. This requirement of uniqueness of the function is mandatory.

ABOUT. : The set X of all valid real values of the argument x for which the function y = f (x) is defined is calleddomain of the function .

The set Y of all real values y that a function takes is calledfunction range .

Let's look at some ways to specify functions.

Tabular method . A fairly common one is to specify a table of individual argument values and their corresponding function values. This method of defining a function is used when the domain of definition of the function is a discrete finite set.

Graphic method . The graph of the function y = f(x) is the set of all points on the plane whose coordinates satisfy the given equation.

The graphical method of specifying a function does not always make it possible to accurately determine the numerical values of the argument. However, it has a big advantage over other methods - visibility. In engineering and physics, a graphical method of specifying a function is often used, and a graph is the only way available for this.

Analytical method . Most often, the law that establishes the connection between an argument and a function is specified through formulas. This method of specifying a function is called analytical.

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.

Verbal method . This method consists in expressing functional dependence in words.

Example 1: function E(x) is the integer part of x. In general, E(x) = [x] denotes the largest integer that does not exceed x. In other words, if x = r + q, where r is an integer (can be negative) and q belongs to the interval = r. The function E(x) = [x] is constant on the interval = r.

Example 2: function y = (x) is the fractional part of a number. More precisely, y =(x) = x - [x], where [x] is the integer part of the number x. This function is defined for all x. If x is an arbitrary number, then represent it in the form x = r + q (r = [x]), where r is an integer and q lies in the interval ; 2) (-;-2] ; 4) [-2;0]

5. Find all values of x at which the function takes negative values (Fig. e):

1) (-2;0); 2) [-6;6]; 3) (- ;0); 4) (- ;0) (0;+ )