Which of the intervals is shown on the line? Function.Function graph

B) Number line

Consider the number line (Fig. 6):

Consider the set of rational numbers

Each rational number is represented by some point on number axis. So, the numbers are marked in the figure.

Let's prove that .

Proof. Let there be a fraction: . We have the right to consider this fraction irreducible. Since , then - the number is even: - odd. Substituting its expression, we find: , from which it follows that - even number. We have obtained a contradiction that proves the statement.

So, not all points on the number axis represent rational numbers. Those points that do not represent rational numbers represent numbers called irrational.

Any number of the form , , is either an integer or an irrational number.

Numeric intervals

Numerical segments, intervals, half-intervals and rays are called numerical intervals.

Let us represent the numbers on the coordinate line a And b, as well as the number x between them.

The set of all numbers that meet the condition a ≤ x ≤ b, called numerical segment or just a segment. It is designated as follows: [ a; b] - It reads like this: a segment from a to b.

The set of numbers that meet the condition a< x < b , called interval. It is designated as follows: ( a; b)

It reads like this: interval from a to b.

Sets of numbers satisfying the conditions a ≤ x< b или a<x ≤ b, are called half-intervals. Designations:

Set a ≤ x< b обозначается так:[a; b), reads like this: half-interval from a to b, including a.

Many a<x ≤ b is indicated as follows:( a; b], reads like this: half-interval from a to b, including b.

Now let's imagine beam with a dot a, to the right and left of which there is a set of numbers.

a, meeting the condition x ≥ a, called numerical beam.

It is designated as follows: [ a; +∞)-Reads like this: a numerical ray from a to plus infinity.

Set of numbers to the right of a point a, corresponding to the inequality x>a, called open number beam.

It is designated as follows: ( a; +∞)-Reads like this: an open numerical ray from a to plus infinity.

a, meeting the condition x ≤ a, called numerical ray from minus infinity toa .

It is designated as follows:( - ∞; a]-Reads like this: a numerical ray from minus infinity to a.

Set of numbers to the left of the point a, corresponding to the inequality x< a , called open number ray from minus infinity toa .

It is designated as follows: ( - ∞; a)-Reads like this: an open number ray from minus infinity to a.

The set of real numbers is represented by the entire coordinate line. They call him number line. It is designated as follows: ( - ∞; + ∞ )

3) Linear equations and inequalities with one variable, their solutions:

An equation containing a variable is called an equation with one variable, or an equation with one unknown. For example, an equation with one variable is 3(2x+7)=4x-1.

The root or solution of an equation is the value of a variable at which the equation becomes a true numerical equality. For example, the number 1 is a solution to the equation 2x+5=8x-1. The equation x2+1=0 has no solution, because the left side of the equation is always greater than zero. The equation (x+3)(x-4) =0 has two roots: x1= -3, x2=4.

Solving an equation means finding all its roots or proving that there are no roots.

Equations are called equivalent if all the roots of the first equation are roots of the second equation and vice versa, all the roots of the second equation are roots of the first equation or if both equations have no roots. For example, the equations x-8=2 and x+10=20 are equivalent, because the root of the first equation x=10 is also the root of the second equation, and both equations have the same root.

When solving equations, the following properties are used:

If you move a term in an equation from one part to another, changing its sign, you will get an equation equivalent to the given one.

If both sides of an equation are multiplied or divided by the same non-zero number, you get an equation equivalent to the given one.

The equation ax=b, where x is a variable and a and b are some numbers, is called a linear equation with one variable.

If a¹0, then the equation has a unique solution.

If a=0, b=0, then any value of x satisfies the equation.

If a=0, b¹0, then the equation has no solutions, because 0x=b is not executed for any value of the variable.
Example 1. Solve the equation: -8(11-2x)+40=3(5x-4)

Let's open the parentheses in both sides of the equation, move all terms with x to the left side of the equation, and terms that do not contain x to the right side, we get:

16x-15x=88-40-12

Example 2. Solve the equations:

x3-2x2-98x+18=0;

These equations are not linear, but we will show how such equations can be solved.

3x2-5x=0; x(3x-5)=0. The product is equal to zero, if one of the factors is equal to zero, we get x1=0; x2= .

Answer: 0; .

Factor the left side of the equation:

x2(x-2)-9(x-2)=(x-2)(x2-9)=(x-2)(x-3)(x-3), i.e. (x-2)(x-3)(x+3)=0. This shows that the solutions to this equation are the numbers x1=2, x2=3, x3=-3.

c) Imagine 7x as 3x+4x, then we have: x2+3x+4x+12=0, x(x+3)+4(x+3)=0, (x+3)(x+4)= 0, hence x1=-3, x2=- 4.

Answer: -3; - 4.
Example 3. Solve the equation: ½x+1ç+½x-1ç=3.

Let us recall the definition of the modulus of a number:

For example: ½3½=3, ½0½=0, ½- 4½= 4.

In this equation, under the modulus sign are the numbers x-1 and x+1. If x is less than –1, then the number x+1 is negative, then ½x+1½=-x-1. And if x>-1, then ½x+1½=x+1. At x=-1 ½x+1½=0.

Thus,

Likewise

a) Consider given equation½x+1½+½x-1½=3 for x£-1, it is equivalent to the equation -x-1-x+1=3, -2x=3, x=, this number belongs to the set x£-1.

b) Let -1< х £ 1, тогда данное уравнение равносильно уравнению х+1-х+1=3, 2¹3 уравнение не имеет решения на данном множестве.

c) Consider the case x>1.

x+1+x-1=3, 2x=3, x= . This number belongs to the set x>1.

Answer: x1=-1.5; x2=1.5.
Example 4. Solve the equation:½x+2½+3½x½=2½x-1½.

We'll show you short note solving the equation, revealing the sign of the modulus “over intervals”.

x £-2, -(x+2)-3x=-2(x-1), - 4x=4, x=-2О(-¥; -2]

–2<х£0, х+2-3х=-2(х-1), 0=0, хÎ(-2; 0]

0<х£1, х+2+3х=-2(х-1), 6х=0, х=0Ï(0; 1]

x>1, x+2+3x=2(x-1), 2x=- 4, x=-2П(1; +¥)

Answer: [-2; 0]
Example 5. Solve the equation: (a-1)(a+1)x=(a-1)(a+2), for all values ​​of parameter a.

There are actually two variables in this equation, but consider x to be the unknown and a to be the parameter. It is required to solve the equation for the variable x for any value of the parameter a.

If a=1, then the equation has the form 0×x=0; any number satisfies this equation.

If a=-1, then the equation looks like 0×x=-2; not a single number satisfies this equation.

If a¹1, a¹-1, then the equation has a unique solution.

Answer: if a=1, then x is any number;

if a=-1, then there are no solutions;

if a¹±1, then .

B) Linear inequalities with one variable.

If the variable x is given any numerical value, then we get numerical inequality, expressing either a true or false statement. Let, for example, the inequality 5x-1>3x+2 be given. For x=2 we get 5·2-1>3·2+2 – true statement(correct number statement); at x=0 we get 5·0-1>3·0+2 – a false statement. Any value of a variable at which this inequality with a variable turns into a true numerical inequality is called a solution to the inequality. Solving an inequality with a variable means finding the set of all its solutions.

Two inequalities with the same variable x are said to be equivalent if the sets of solutions to these inequalities coincide.

The main idea of ​​solving the inequality is as follows: we replace the given inequality with another, simpler, but equivalent to the given one; we again replace the resulting inequality with a simpler inequality equivalent to it, etc.

Such replacements are made on the basis of the following statements.

Theorem 1. If any term of an inequality with one variable is transferred from one part of the inequality to another with opposite sign, while leaving the inequality sign unchanged, we obtain an inequality equivalent to the given one.

Theorem 2. If both sides of an inequality with one variable are multiplied or divided by the same positive number, leaving the sign of the inequality unchanged, then an inequality equivalent to the given one will be obtained.

Theorem 3. If both sides of an inequality with one variable are multiplied or divided by the same negative number, changing the sign of the inequality to the opposite one, we get an inequality equivalent to the given one.

An inequality of the form ax+b>0 is called linear (respectively, ax+b<0, ax+b³0, ax+b£0), где а и b – действительные числа, причем а¹0. Решение этих неравенств основано на трех теоремах равносильности изложенных выше.

Example 1. Solve the inequality: 2(x-3)+5(1-x)³3(2x-5).

Opening the brackets, we get 2x-6+5-5x³6x-15,

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

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

A set X is bounded from above (from below) if there is a number c such that for any x ∈ X the inequality x≤с (x≥c) holds. The number c in this case is called the upper (lower) bound of the set X. A set that is bounded both above and below is called bounded. The smallest (largest) of the upper (lower) bounds 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 between them also belong to this set. There are several somewhat different types of non-empty numerical intervals: Straight, open beam, closed beam, 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 a 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 is called an open number ray. They write or accordingly: .

Closed beam

The set of numbers such that is called a closed number line. They write or accordingly:.

A set of numbers is called a number segment.

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

Interval

A set of numbers such that 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 whose ends is removed to infinity, and a number line - as an interval, both ends of which are removed to infinity.

Half-interval

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

They write or, respectively,

3.Function.Graph of the function. Methods for specifying a function.

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

The notation F = y(x) means that a function is being considered that allows for any value of the independent variable x (from among those that the argument x can generally take) to find the corresponding value of the dependent variable y.

Methods for specifying a function.

The function can be specified by a formula, for example:

y = 3x2 – 2.

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

4.Main characteristics of the function: monotonicity, parity, periodicity.

Answer - Periodicity Definition. A function f is called periodic if there is such a number
, that f(x+
)=f(x), for all x D(f). Naturally, there are countless numbers of such numbers. The smallest positive number ^ T is called the period of the function. Examples. A. y = cos x, T = 2 . V. y = tg x, T = . S. y = (x), T = 1. D. y = , this function is not periodic. Parity Definition. A function f is called even if the property f(-x) = f(x) holds for all x in D(f). If f(-x) = -f(x), then the function is called odd. If none of the indicated relations are satisfied, then the function is called a general function. Examples. A. y = cos (x) - even; V. y = tg (x) - odd; S. y = (x); y=sin(x+1) – functions of general form. Monotony Definition. A function f: X -> R is called increasing (decreasing) if for any
the condition is met:
Definition. A function X -> R is called 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 = cos x - piecewise monotonic function.

Numerical intervals. Context. Definition

An equality (equation) has one point on the number line (although this point depends on the transformations done and the root chosen). The solution to the equation itself will be a numerical set (sometimes consisting of a single number). However, all this is on the number line (visualization of the set real numbers) will only be displayed pointwise, but there are also more generic types relationships between two numbers - inequalities. In them, the number line is divided by a certain number and cut off from it certain part- the values ​​of an expression or a numerical interval.

It is logical to discuss the topic of numerical intervals together with inequalities, but this does not mean that it is connected only with them. Numerical intervals (intervals, segments, rays) are a set of variable values ​​that satisfy a certain inequality. That is, in essence, this is the set of all points on the number line, limited by some kind of framework. Therefore, the topic of numerical intervals is most closely related to the concept variable. Where there is a variable, or an arbitrary point x on the number line, and it is used, there are also numerical intervals, intervals - x values. Often the value can be anything, but this is also a numerical interval covering the entire number line.

Let's introduce the concept numerical interval. Among number sets, that is, sets whose objects are numbers, distinguish so-called numerical intervals. Their value is that it is very easy to imagine a set corresponding to a specified numerical interval, and vice versa. Therefore, with their help it is convenient to write down many solutions to an inequality. Whereas the set of solutions to the equation will not be a numerical interval, but simply several numbers on the number line, with inequalities, in other words, any restrictions on the value of a variable, numerical intervals appear.

A number interval is the set of all points on the number line, limited by a given number or numbers (points on the number line).

A numerical interval of any kind (a set of x values ​​enclosed between certain numbers) can always be represented in three ways mathematical notation: special notations for intervals, chains of inequalities (single inequality or double inequality) or geometrically on the number line. Essentially, all these designations have the same meaning. They provide a constraint(s) on the values ​​of some mathematical object, variable size(some variable, any expression with a variable, function, etc.).

From the above it can be understood that since it is possible to limit the area of ​​the number line in different ways (there are different types inequalities), then there are different types of numerical intervals.

Types of numerical intervals

Each type of number interval has proper name, special designation. To indicate numerical intervals, round and square brackets are used. A parenthesis means that the final, boundary-defining point on the number line (end) of this parenthesis is not included in the set of points of this interval. Square bracket means the end enters the gap. With infinity (on this side the interval is not limited) use a parenthesis. Sometimes instead parentheses you can write square, rotated in reverse side: (a;b) ⇔]a;b[

There may be confusion with the names of the intervals: there are huge amount options. Therefore, it is always better to indicate them accurately. In English literature only the term is used interval ("interval") - open, closed, half-open (half-closed). There are many variations.

Using intervals in mathematics denotes very large number things: there are intervals of isolation when solving equations, intervals of integration, intervals of convergence of series. When studying a function, intervals are always used to denote its range of values ​​and domain of definition. Gaps are very important, for example, there are Bolzano-Cauchy theorem(you can find out more on Wikipedia).

Systems and sets of inequalities

System of inequalities

So, a variable x or the value of some expression can be compared with some constant value- this is an inequality, but you can compare this expression with several quantities - a double inequality, a chain of inequalities, etc. This is exactly what was shown above - as an interval and a segment. Both is system of inequalities.

So, if the task is to find the set general solutions two or more inequalities, then we can talk about solving a system of inequalities (just like with equations - although we can say that equations are a special case).

Then it is obvious that the value of the variable used in the inequalities, at which each of them becomes true, is called the solution to the system of inequalities.

All inequalities included in the system are united curly brace- "(". Sometimes they are written in the form double inequality(as shown above) or even chain of inequalities. Example of a typical entry: f x ≤ 30 g x 5 .

Systems solution linear inequalities with one variable in general case comes down to these 4 types: x > a x > b (1) x > a x< b (2) x < a x >b(3)x< a x < b (4) . Здесь предполагается, что b > a.

Any system can be solved graphically using the number line. Where the solutions of the inequalities that make up the system intersect, there will be a solution to the system itself.

Let us present a graphical solution for each case.