number gaps. Numerical segments, intervals, half-intervals and rays are called numerical intervals Intervals and their types

Among the sets of numbers there are sets where the objects are numerical intervals. When specifying a set, it is easier to determine by the interval. Therefore, we write down the sets of solutions using numerical intervals.

This article gives answers to questions about numerical gaps, names, notation, images of gaps on the coordinate line, correspondence of inequalities. In conclusion, the table of gaps will be considered.

Each number span is characterized by:

• name;
• the presence of ordinary or double inequality;
• designation;
• geometric image on the coordinate line.

The numerical range is set using any 3 methods from the list above. That is, when using inequality, notation, images on the coordinate line. This method is the most applicable.

Let's make a description of the numerical intervals with the above indicated sides:

Definition 2

• Open number beam. The name is due to the fact that it is omitted, leaving it open.

This interval has the corresponding inequalities x< a или x >a , where a is some real number. That is, on such a ray there are all real numbers that are less than a - (x< a) или больше a - (x >a) .

The set of numbers that will satisfy an inequality of the form x< a обозначается виде промежутка (− ∞ , a) , а для x >a , like (a , + ∞) .

The geometric meaning of an open beam considers the presence of a numerical gap. There is a correspondence between the points of the coordinate line and its numbers, due to which the line is called the coordinate line. If it is necessary to compare numbers, then on the coordinate line, the larger number is to the right. Then an inequality of the form x< a включает в себя точки, которые расположены левее, а для x >a - points that are to the right. The number itself is not suitable for solving, therefore, in the drawing it is indicated by a punched out dot. The gap that is needed is highlighted by hatching. Consider the figure below.

From the above figure, it can be seen that the numerical gaps correspond to a part of a straight line, that is, rays starting at a. In other words, they are called rays without a beginning. Therefore, it was called the open number ray.

Let's look at a few examples.

Example 1

For a given strict inequality x > − 3, an open ray is given. This entry can be represented as coordinates (− 3 , ∞) . That is, these are all points lying to the right than - 3 .

Example 2

If we have an inequality of the form x< 2 , 3 , то запись (− ∞ , 2 , 3) является аналогичной при задании открытого числового луча.

Definition 3

• number beam. The geometric meaning is that the beginning is not discarded, in other words, the ray leaves behind its usefulness.

Its assignment goes with the help of non-strict inequalities of the form x ≤ a or x ≥ a . For this type, special notation of the form (− ∞ , a ] and [ a , + ∞) is accepted, and the presence of a square bracket means that the point is included in the solution or in the set. Consider the figure below.

For an illustrative example, let's set a numerical ray.

Example 3

An inequality of the form x ≥ 5 corresponds to the notation [ 5 , + ∞) , then we get a ray of this form:

Definition 4

• Interval. Setting using intervals is written using double inequalities a< x < b , где а и b являются некоторыми действительными числами, где a меньше b , а x является переменной. На таком интервале имеется множество точек и чисел, которые больше a , но меньше b . Обозначение такого интервала принято записывать в виде (a , b) . Наличие круглых скобок говорит о том, что число a и b не включены в это множество. Координатная прямая при изображении получает 2 выколотые точки.

Consider the figure below.

Example 4

Interval example - 1< x < 3 , 5 говорит о том, что его можно записать в виде интервала (− 1 , 3 , 5) . Изобразим на координатной прямой и рассмотрим.

Definition 5

• Numeric line. This interval differs in that it includes boundary points, then it has the form a ≤ x ≤ b . Such a non-strict inequality says that when writing as a numerical segment, square brackets [ a , b ] are used, which means that the points are included in the set and are shown as filled.

Example 5

Having considered the segment, we get that its specification is possible using the double inequality 2 ≤ x ≤ 3 , which is represented as 2 , 3 . On the coordinate line, the data points will be included in the solution and shaded.

Definition 6 Example 6

If there is a half-interval (1 , 3 ] , then its designation can be in the form of a double inequality 1< x ≤ 3 , при чем на координатной прямой изобразится с точками 1 и 3 , где 1 будет исключена, то есть выколота на прямой.

Gaps can be shown as:

• open number beam;
• number beam;
• interval;
• numerical segment;
• half-interval.

To simplify the calculation process, it is necessary to use a special table, where there are designations for all types of numerical intervals of a straight line.

Name	inequality	Designation	Image
Open number beam	x< a	- ∞ , a
	x > a	a , +∞
number beam	x ≤ a	(-∞, a]
	x ≥ a	[ a , +∞)
Interval	a< x < b	a , b
Numerical segment	a ≤ x ≤ b	a , b
Half interval

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Among the number sets, that is sets, whose objects are numbers, distinguish the so-called number gaps. Their value is that it is very easy to imagine a set corresponding to a specified numerical range, and vice versa. Therefore, with their help it is convenient to write down the set of solutions of the inequality.

In this article, we will analyze all types of numerical intervals. Here we give their names, introduce notation, draw numerical intervals on the coordinate line, and also show which simplest inequalities correspond to them. In conclusion, we will visually present all the information in the form of a table of numerical intervals.

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Types of numerical intervals

Each numerical interval has four inextricably linked things:

• the name of the number range,
• corresponding inequality or double inequality,
• designation,
• and its geometric image in the form of an image on a coordinate line.

Any numerical interval can be specified in any of the last three ways in the list: either by an inequality, or by a designation, or by its image on a coordinate line. Moreover, according to this method of assignment, for example, by inequality, others are easily restored (in our case, the designation and the geometric image).

Let's get down to specifics. Let us describe all the numerical intervals on the four sides indicated above.

Let's start with a description of the numerical interval, called open number beam. Note that the adjective "numerical" is often omitted, leaving the name open beam.

This numerical interval corresponds to the simplest inequalities with one variable of the form x a , where a is some real number. That is, according to the meaning of the written inequalities, the open number ray is made up of all that are less than the number a (in the case of the inequality x a).

The set of numbers satisfying the inequality x a , like (a, +∞) .

It remains to show the geometric image of the open beam, it will become clear from it that the considered numerical interval received such a name not by chance. Let's turn to. It is known that there is a one-to-one correspondence between its points and real numbers, which allows the coordinate line to be called the number line. And when talking about comparing numbers we noted that the larger number is located on the coordinate line to the right of the smaller one, and the smaller one is to the left of the larger one. Based on these considerations, the inequality x a - points lying to the right of point a . The number a itself does not satisfy these inequalities, in order to emphasize this in the drawing, it is depicted as a dot with an empty center. Above the points, which correspond to numbers that satisfy the inequality, depict oblique shading:

From the above drawings, it can be seen that these numerical intervals correspond to parts of the number line, which are rays starting at point a , but excluding point a itself. In other words, they are rays without a beginning. Hence the name - open number beam.

Let us give some concrete examples of open numerical rays. Thus, the strict inequality x>−3 defines an open number ray. It is also defined by the notation (−3, ∞) . And on the coordinate line, this numerical interval is a set of points lying to the right of the point with coordinate −3, not including this point itself. Another example: inequality x<2,3 , как и запись (−∞, 2,3) , задает открытый числовой луч, который следующим образом изображается на координатной прямой


We pass to the numerical intervals of the following form - number rays. Geometrically, they differ from open beams in that the beginning of the beam is not discarded. In other words, the geometric image of numerical intervals of this type is a full-fledged ray.

As for specifying numerical rays using inequalities, they correspond to non-strict inequalities x≤a or x≥a . They are denoted by (−∞, a] and . And the geometric image of a numerical segment is a segment together with its ends:


For example, a numerical segment, which is given by a double inequality, can be denoted as , on the coordinate line it corresponds to a segment with ends at points having coordinates root of two and root of three.

It remains only to say about the numerical intervals called half-intervals. They represent, so to speak, an intermediate option between an interval and a segment, since they include one of the boundary points. The half-intervals are given by double inequalities a

Table of numerical intervals

So, in the previous paragraph, we defined and described the following numerical intervals:

• open number beam;
• number beam;
• interval;
• half-interval.

For convenience, we summarize all the data on numerical intervals in a table. Let's put in it the name of the numerical interval, the inequality corresponding to it, the notation and the image on the coordinate line. We get the following range table:

Bibliography.

• Algebra: textbook for 8 cells. general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; ed. S. A. Telyakovsky. - 16th ed. - M. : Education, 2008. - 271 p. : ill. - ISBN 978-5-09-019243-9.
• Mordkovich A. G. Algebra. Grade 9 At 2 pm Part 1. Textbook for students of educational institutions / A. G. Mordkovich, P. V. Semenov. - 13th ed., Sr. - M.: Mnemosyne, 2011. - 222 p.: ill. ISBN 978-5-346-01752-3.

Answer - The set (-∞;+∞) is called a number line, and any number is called a point of this line. Let a be an arbitrary point on the real line and δ

Positive number. The interval (a-δ; a+δ) is called the δ-neighborhood of the point a.

The set X is bounded from above (from below) if there is such a number c that for any x ∈ X the inequality x≤с (x≥c) is satisfied. The number c in this case is called the upper (lower) bound of the set X. A set bounded both above and below is called bounded. The smallest (largest) of the upper (lower) faces of a set is called the exact upper (lower) bound of this set.

A numerical interval is a connected set of real numbers, that is, such that if 2 numbers belong to this set, then all the numbers enclosed between them also belong to this set. There are several, in a sense, different types of non-empty numerical intervals: Line, open ray, closed ray, line segment, half-interval, interval

Number line

The set of all real numbers is also called the number line. They write.

In practice, there is no need to distinguish between the concept of a coordinate or number line in the geometric sense and the concept of a number line introduced by this definition. Therefore, these different concepts are denoted by the same term.

open beam

The set of numbers such that or is called an open number ray. Write or respectively: .

closed beam

The set of numbers such that or is called a closed number ray. Write or respectively:

The set of numbers such that is called a number segment.

Comment. The definition does not state that . It is assumed that the case is possible. Then the numerical interval turns into a point.

Interval

A set of numbers such as is called a numerical interval.

Comment. The coincidence of the designations of an open beam, a straight line and an interval is not accidental. An open ray can be understood as an interval, one of the ends of which is removed to infinity, and a number line - as an interval, both ends of which are removed to infinity.

Half interval

The set of numbers such that or is called a numerical half-interval.

Write or, respectively,

3.Function.Function graph. Ways to set a function.

Answer - If two variables x and y are given, then they say that the variable y is a function of the variable x, if such a relationship between these variables is given that allows for each value to uniquely determine the value of y.

The notation F = y(x) means that we are considering a function that allows for any value of the independent variable x (out of those that the argument x can take at all) to find the corresponding value of the dependent variable y.

Ways to set a function.

A function can be defined by a formula, for example:

y \u003d 3x2 - 2.

The function can be given by a graph. Using the graph, you can determine which value of the function corresponds to the specified value of the argument. Usually this is an approximate value of the function.

4. The main characteristics of the function: monotonicity, parity, periodicity.

Answer - Periodicity Definition. A function f is called periodic if there exists such a number
, that f(x+
)=f(x), for all x D(f). Naturally, there are an infinite number of such numbers. The smallest positive number ^ T is called the period of the function. Examples. A. y \u003d cos x, T \u003d 2 . B. y \u003d tg x, T \u003d . S. y = (x), T = 1. D. y = , this function is not periodic. Parity Definition. A function f is called even if for all x from D(f) the property f(-x) = f(x) is satisfied. If f (-x) = -f (x), then the function is called odd. If none of these relations is satisfied, then the function is called a function of general form. Examples. A. y \u003d cos (x) - even; B. y \u003d tg (x) - odd; S. y \u003d (x); y=sin(x+1) – general functions. Monotony Definition. A function f: X -> R is called increasing (decreasing) if for any
condition is met:
Definition. A function X -> R is said to be monotonic on X if it is increasing or decreasing on X. If f is monotone on some subsets of X, then it is called piecewise monotone. Example. y \u003d cos x is a piecewise monotone function.