Module or absolute value a real number is called the number itself if X non-negative, and the opposite number, i.e. -x if X negative:

Obviously, but by definition, |x| > 0. The following properties of absolute values ​​are known:

• 1) xy| = |dg| |g/1;
• 2>- -H;

Uat

• 3) |x+r/|
• 4) |dt-g/|

Modulus of the difference of two numbers X - A| is the distance between points X And A on the number line (for any X And A).

It follows from this, in particular, that the solutions to the inequality X - A 0) are all points X interval (A- g, a + c), i.e. numbers satisfying the inequality a-d + G.

This interval (A- 8, A+ d) is called the 8-neighborhood of a point A.

Basic properties of functions

As we have already stated, all quantities in mathematics are divided into constants and variables. Constant value A quantity that retains the same value is called.

Variable value is a quantity that can take on different numerical values.

Definition 10.8. Variable value at called function from a variable value x, if, according to some rule, each value x e X assigned a specific value at e U; the independent variable x is usually called an argument, and the domain X its changes are called the domain of definition of the function.

The fact that at there is a function otx, most often expressed symbolically: at= /(x).

There are several ways to specify functions. The main ones are considered to be three: analytical, tabular and graphical.

Analytical way. This method consists of specifying the relationship between an argument (independent variable) and a function in the form of a formula (or formulas). Usually f(x) is some analytical expression containing x. In this case, the function is said to be defined by the formula, for example, at= 2x + 1, at= tgx, etc.

Tabular The way to specify a function is that the function is specified by a table containing the values ​​of the argument x and the corresponding values ​​of the function /(.r). Examples include tables of the number of crimes for a certain period, tables of experimental measurements, and a table of logarithms.

Graphic way. Let a system of Cartesian systems be given on the plane rectangular coordinates xOy. The geometric interpretation of the function is based on the following.

Definition 10.9. Schedule function is called the geometric locus of points of the plane, coordinates (x, y) which satisfy the condition: U-Ah).

A function is said to be given graphically if its graph is drawn. The graphical method is widely used in experimental measurements using recording devices.

Having a visual graph of a function before your eyes, it is not difficult to imagine many of its properties, which makes the graph an indispensable tool for studying a function. Therefore, plotting a graph is the most important (usually the final) part of the study of a function.

Each method has both its advantages and disadvantages. Thus, the advantages of the graphic method include its clarity, and the disadvantages include its inaccuracy and limited presentation.

Let us now move on to consider the basic properties of functions.

Even and odd. Function y = f(x) called even, if for anyone X condition is met f(-x) = f(x). If for X from the domain of definition the condition /(-x) = -/(x) is satisfied, then the function is called odd. A function that is neither even nor odd is called a function general view.

• 1) y = x 2 is an even function, since f(-x) = (-x) 2 = x 2, i.e./(-x) =/(.r);
• 2) y = x 3 - an odd function, since (-x) 3 = -x 3, t.s. /(-x) = -/(x);
• 3) y = x 2 + x is a function of general form. Here /(x) = x 2 + x, /(-x) = (-x) 2 +
• (-x) = x 2 - x,/(-x) */(x);/(-x) -/"/(-x).

The graph of an even function is symmetrical about the axis Oh, and the graph of an odd function is symmetrical about the origin.

Monotone. Function at=/(x) is called increasing in between X, if for any x, x 2 e X from the inequality x 2 > x, it follows /(x 2) > /(x,). Function at=/(x) is called decreasing, if x 2 > x, it follows /(x 2) (x,).

The function is called monotonous in between X, if it either increases over this entire interval or decreases over it.

For example, the function y = x 2 decreases by (-°°; 0) and increases by (0; +°°).

Note that we have given the definition of a function that is monotonic in the strict sense. Generally to monotonic functions include non-decreasing functions, i.e. such for which from x 2 > x, it follows/(x 2) >/(x,), and non-increasing functions, i.e. such for which from x 2 > x, it follows/(x 2)

Limitation. Function at=/(x) is called limited in between X, if such a number exists M > 0, which |/(x)| M for any x e X.

For example, the function at =-

is bounded on the entire number line, so

Periodicity. Function at = f(x) called periodic, if such a number exists T^ Oh what f(x + T = f(x) for all X from the domain of the function.

In this case T is called the period of the function. Obviously, if T - period of the function y = f(x), then the periods of this function are also 2Г, 3 T etc. Therefore, the period of a function is usually called the smallest positive period (if it exists). For example, the function / = cos.g has a period T= 2P, and the function y = tg Zx - period p/3.

Back forward

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Goals:

Equipment: projector, screen, personal computer, multimedia presentation

During the classes

1. Organizational moment.

2. Updating students' knowledge.

2.1. Answer students' questions about homework.

2.2. Solve the crossword puzzle (repetition of theoretical material) (Slide 2):

– After solving the crossword puzzle, read the name of the topic of today’s lesson in the highlighted vertical column. (Slides 3, 4)

3. Explanation of a new topic.

3.1. – Guys, you have already met the concept of a module, you have used the notation | a| . Previously, we were talking only about rational numbers. Now we need to introduce the concept of modulus for any real number.

Each real number corresponds to a single point on the number line, and, conversely, each point on the number line corresponds to a single real number. All the basic properties of operations on rational numbers are preserved for real numbers.

The concept of the modulus of a real number is introduced. (Slide 5).

Definition. Modulus of a non-negative real number x call this number itself: | x| = x; modulus of a negative real number X call the opposite number: | x| = – x .

– Write down the topic of the lesson and the definition of the module in your notebooks:

In practice, various module properties, For example. (Slide 6) :

Complete orally No. 16.3 (a, b) – 16.5 (a, b) to apply the definition, properties of the module. (Slide 7) .

3.4. For any real number X can be calculated | x| , i.e. we can talk about function y = |x| .

Task 1. Construct a graph and list the properties of the function y = |x| (Slides 8, 9).

One student on the board is graphing a function

Fig 1.

Properties are listed by students. (Slide 10)

1) Domain of definition – (– ∞; + ∞) .

2) y = 0 at x = 0; y > 0 at x< 0 и x > 0.

3) The function is continuous.

4) y naim = 0 for x = 0, y naib does not exist.

5) The function is limited from below, not limited from above.

6) The function decreases on the ray (– ∞; 0) and increases on the ray )