Numerical functions and their properties. Lesson “Functions and their properties

They have many properties:

1. The function is called monotonous on a certain interval A, if it increases or decreases on this interval

2. The function is called increasing on a certain interval A, if for any numbers of their set A the following condition is satisfied:.

The graph of an increasing function has a special feature: when moving along the x-axis from left to right along the interval A the ordinates of the graph points increase (Fig. 4).

3. The function is called decreasing at some interval A, if for any numbers there are many of them A the condition is met:.

The graph of a decreasing function has a special feature: when moving along the x-axis from left to right along the interval A the ordinates of the graph points decrease (Fig. 4).

4. The function is called even on some set X, if the condition is met: .

The graph of an even function is symmetrical about the ordinate axis (Fig. 2).

5. The function is called odd on some set X, if the condition is met: .

The graph of an odd function is symmetrical about the origin (Fig. 2).

6. If the function y = f(x)
f(x) f(x), then they say that the function y = f(x) accepts smallest value at=f(x) at X= x(Fig. 2, the function takes the smallest value at the point with coordinates (0;0)).

7. If the function y = f(x) is defined on the set X and there exists such that for any the inequality f(x) f(x), then they say that the function y = f(x) accepts highest value at=f(x) at X= x(Fig. 4, the function does not have the largest and smallest values) .

If for this function y = f(x) all the listed properties have been studied, then they say that study functions.

Lessons 1-2. Definition of a numerical function and methods for specifying it

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Target: discuss the definition of a function and how to define it.

I. Communicating the topic and purpose of the lessons

II. Review of 9th grade material

Various aspects of this topic have already been covered in grades 7-9. Now we need to expand and summarize the information about the functions. Let us remind you that the topic is one of the most important for the entire mathematics course. Various functions will be studied until graduation and further in higher education institutions. This topic is closely related to solving equations, inequalities, word problems, progressions, etc.

Definition 1. Let two sets of real numbers be given D and E and the law is indicated f according to which each number x∈ D matches the singular number y ∈ E (see picture). Then they say that the function y = f(x ) or y(x) with domain of definition (O.O.) D and the area of change (O.I.) E. In this case, the value x is called the independent variable (or argument of the function), the value y is called the dependent variable (or the value of the function).

Function Domain f denote D(f ). The set consisting of all numbers f(x ) (function range f), denote E(f).

Example 1

Consider the functionTo find y for each value of x, you must perform the following operations: subtract the number 2 (x - 2) from the value of x, extract the square root of this expressionand finally add the number 3The set of these operations (or the law according to which the value y is sought for each value of x) is called the function y(x). For example, for x = 6 we findThus, to calculate the function y at a given point x, it is necessary to substitute this value x into the given function y(x).

Obviously, for a given function, for any admissible number x, only one value of y can be found (that is, for each value of x there corresponds one value of y).

Let us now consider the domain of definition and the range of variation of this function. It is possible to extract the square root of the expression (x - 2) only if this value is non-negative, i.e. x - 2 ≥ 0 or x ≥ 2. FindSince by definition of an arithmetic rootthen we add the number 3 to all parts of this inequality, we get:or 3 ≤ y< +∞. Находим

Rational functions are often used in mathematics. In this case, functions of the form f(x ) = p(x) (where p(x) is a polynomial) are called entire rational functions. Functions of the form(where p(x) and q(x ) - polynomials) are called fractional-rational functions. Obviously a fractionis defined if the denominator q(x ) does not vanish. Therefore, the domain of definition of the fractional rational function- the set of all real numbers from which the roots of the polynomial are excluded q(x).

Example 2

Rational functiondefined for x - 2 ≠ 0, i.e. x ≠ 2. Therefore, the domain of definition of this function is the set of all real numbers not equal to 2, i.e., the union of the intervals (-∞; 2) and (2; ∞).

Recall that the union of sets A and B is a set consisting of all elements included in at least one of the sets A or B. The union of sets A and B is denoted by the symbol A U B. Thus, the union of segments and (3; 9) is an interval (non-intersecting intervals) are denoted by .

Returning to the example, we can write:Since for all acceptable values of x the fractiondoes not vanish, then the function f(x ) takes all values except 3. Therefore

Example 3

Let us find the domain of definition of the fractional rational function

The denominators of fractions vanish at x = 2, x = 1 and x = -3. Therefore, the domain of definition of this function

Example 4

Addiction is no longer a function. Indeed, if we want to calculate the value of y, for example, for x = 1, then using the upper formula we find: y = 2 1 - 3 = -1, and using the lower formula we get: y = 12 + 1 = 2. Thus, one value x(x = 1) correspond to two values of y (y = -1 and y = 2). Therefore, this dependence (by definition) is not a function.

Example 5

Graphs of two dependencies are shown y(x ). Let's determine which of them is a function.

In Fig. and the graph of the function is given, since at any point x 0 only one value y0 corresponds. In Fig. b is a graph of some kind of dependence (but not a function), since such points exist (for example, x 0 ), which correspond to more than one value y (for example, y1 and y2).

Let us now consider the main ways of specifying functions.

1) Analytical (using a formula or formulas).

Example 6

Let's look at the functions:

Despite its unusual form, this relationship also defines a function. For any value of x it is easy to find the value of y. For example, for x = -0.37 (since x< 0, то пользуясь верхним выражением), получаем: у(-0,37) = -0,37. Для х = 2/3 (так как х >0, then we use the lower expression) we have:From the method of finding y it is clear that any value x corresponds to only one value y.

c) 3x + y = 2y - x2. Let us express the value y from this relationship: 3x + x2 = 2y - y or x2 + 3x = y. Thus, this relation also defines the function y = x2 + 3x.

2) Tabular

Example 7

Let's write out a table of squares y for numbers x.


		2,25		6,25

The table data also defines a function - for each (given in the table) value of x, a single value of y can be found. For example, y(1.5) = 2.25, y(5) = 25, etc.

3) Graphic

In a rectangular coordinate system, to depict the functional dependence y(x), it is convenient to use a special drawing - a graph of the function.

Definition 2. Graph of a function y(x ) is the set of all points of the coordinate system, the abscissas of which are equal to the values of the independent variable x, and the ordinates are equal to the corresponding values of the dependent variable y.

By virtue of this definition, all pairs of points (x0, y0) that satisfy the functional dependence y(x) are located on the graph of the function. Any other pairs of points that do not satisfy the dependency y(x ), the functions do not lie on the graph.

Example 8

Given a function Does the point with coordinates belong to the graph of this function: a) (-2; -6); b) (-3; -10)?

1. Find the value of the function y atSince y(-2) = -6, then point A (-2; -6) belongs to the graph of this function.

2. Determine the value of the function y at Since y (-3) = -11, then point B (-3; -10) does not belong to the graph of this function.

According to this graph of the function y = f(x ) it is easy to find the domain of definition D(f ) and range E(f ) functions. To do this, the graph points are projected onto the coordinate axes. Then the abscissas of these points form the domain of definition D(f ), ordinates - range of values E(f).

Let's compare different ways to define a function. The analytical method should be considered the most complete. It allows you to create a table of function values for some argument values, build a graph of the function, and conduct the necessary research of the function. At the same time, the tabular method allows you to quickly and easily find the value of the function for some argument values. The graph of a function clearly shows its behavior. Therefore, one should not oppose different methods of specifying a function; each of them has its own advantages and disadvantages. In practice, all three ways of specifying a function are used.

Example 9

Given the function y = 2x2 - 3x +1.

Let's find: a) y (2); b) y (-3x); c) y(x + 1).

In order to find the value of a function for a certain value of the argument, it is necessary to substitute this value of the argument into the analytical form of the function. Therefore we get:

Example 10

It is known that y(3 - x) = 2x2 - 4. Let's find: a) y(x); b) y(-2).

a) Let us denote it by letter z = 3, then x = 3 - z . Let's substitute this value x into the analytical form of this function y(3 - x) = 2x2 - 4 and get: y (3 - (3 - z)) = 2 (3 - z)2 - 4, or y (z) = 2 (3 - z)2 - 4, or y (z) = 2 (9 - 6 z + z 2) - 4, or y (z) = 2x2 - 12 z + 14. Since it does not matter what letter the function argument is denoted - z, x, t or any other, we immediately get: y(x) = 2x2 - 12x + 14;

b) Now it’s easy to find y(-2) = 2 · (-2)2 - 12 · (-2) + 14 = 8 + 24 + 14 = 46.

Example 11

It is known that Let's find x(y).

Let us denote by the letter z = x - 2, then x = z + 2, and write down the condition of the problem: or To we will write the same condition for the argument (- z ): For convenience, we introduce new variables a = y (z) and b = y (- z ). For such variables we obtain a system of linear equations

We are interested in the unknown a.

To find it, we use the method of algebraic addition. Therefore, let's multiply the first equation by the number (-2), the second equation by the number 3. We get:

Let's add these equations:where Since the function argument can be denoted by any letter, we have:

In conclusion, we note that by the end of grade 9 the following properties and graphs were studied:

a) linear function y = kx + m (graph is a straight line);

b) quadratic function y = ax2 + b x + c (graph - parabola);

c) fractional linear function(graph - hyperbola), in particular functions

d) power function y = xa (in particular, the function

e) functions y = |x|.

For further study of the material, we recommend repeating the properties and graphs of these functions. The following lessons will cover the basic methods of converting graphs.

1. Define a numerical function.

2. Explain how to define a function.

3. What is called the union of sets A and B?

4. What functions are called rational integers?

5. What functions are called fractional rational? What is the domain of definition of such functions?

6. What is called the graph of a function f(x)?

7. Give the properties and graphs of the main functions.

IV. Lesson assignment

§ 1, No. 1 (a, d); 2 (c, d); 3 (a, b); 4 (c, d); 5 (a, b); 6 (c); 7 (a, b); 8 (c, d); 10 ( a ); 13 (c, d); 16 (a, b); 18.

V. Homework

§ 1, No. 1 (b, c); 2 (a, b); 3 (c, d); 4 (a, b); 5 (c, d); 6 (g); 7 (c, d); 8 (a, b); 10 (b); 13 (a, b); 16 (c, d); 19.

VI. Creative tasks

1. Find the function y = f(x), if:

Answers:

2. Find the function y = f(x) if:

Answers:

VII. Summing up the lessons

SUMMARY LESSON ON THE TOPIC “FUNCTIONS AND THEIR PROPERTIES”.

Lesson Objectives:

Methodical: increasing the active-cognitive activity of students through individual-independent work and the use of developmental type test tasks.

Educational: repeat elementary functions, their basic properties and graphs. Introduce the concept of mutually inverse functions. Systematize students’ knowledge on the topic; contribute to the consolidation of skills in calculating logarithms, in applying their properties when solving tasks of a non-standard type; repeat the construction of graphs of functions using transformations and test your skills and abilities when solving exercises on your own.

Educational: fostering accuracy, composure, responsibility, and the ability to make independent decisions.

Developmental: develop intellectual abilities, mental operations, speech, memory. Develop a love and interest in mathematics; During the lesson, ensure that students develop independent thinking in learning activities.

Lesson type: generalization and systematization.

Equipment: board, computer, projector, screen, educational literature.

Lesson epigraph:“Mathematics must then be taught, because it puts the mind in order.”

(M.V. Lomonosov).

DURING THE CLASSES

Checking homework.

Repetition of exponential and logarithmic functions with base a = 2, construction of their graphs in the same coordinate plane, analysis of their relative position. Consider the interdependence between the main properties of these functions (OOF and OFP). Give the concept of mutually inverse functions.

Consider exponential and logarithmic functions with base a = ½ c

in order to ensure that the interdependence of the listed properties is observed and for

decreasing mutually inverse functions.

Organization of independent test-type work for the development of thinking skills

systematization operations on the topic “Functions and their properties.”

FUNCTION PROPERTIES:

1). y = ‌│х│ ;

2). Increases throughout the entire definition area;

3). OOF: (- ∞; + ∞) ;

4). y = sin x;

5). Decreases at 0< а < 1 ;

6). y = x³;

7). OPF: (0; + ∞) ;

8). General function;

9). y = √ x;

10). OOF: (0; + ∞) ;

eleven). Decreases over the entire definition area;

12). y = kx + b;

13). OSF: (- ∞; + ∞) ;

14). Increases at k > 0;

15). OOF: (- ∞; 0) ; (0; + ∞) ;

16). y = cos x;

17). Has no extremum points;

18). OSF: (- ∞; 0) ; (0; + ∞) ;

19). Decreases at k< 0 ;

20). y = x²;

21). OOF: x ≠ πn;

22). y = k/x;

23). Even;

25). Decreases for k > 0;

26). OOF: [ 0; + ∞) ;

27). y = tan x;

28). Increases with k< 0;

29). OSF: [ 0; + ∞) ;

thirty). Odd;

31). y = log x ;

32). OOF: x ≠ πn/2;

33). y = ctg x ;

34). Increases when a > 1.

During this work, survey students on individual assignments:

No. 1. a) Graph the function

b) Graph the function

No. 2. a) Calculate:

b) Calculate:

No. 3. a) Simplify the expression
and find its value at

b) Simplify the expression
and find its value at
.

Homework: No. 1. Calculate: a)
;

V)
;

G)
.

No. 2. Find the domain of definition of the function: a)
;

V)
; G)
.

Sections: Mathematics

Class: 9

Lesson type: Lesson on generalization and systematization of knowledge.

Equipment:

Interactive equipment (PC, multimedia projector).
Test, material in Microsoft Word ( Annex 1).
Interactive program “Autograph”.
Individual test - handouts ( Appendix 2).

During the classes

1. Organizational moment

The purpose of the lesson is announced.

Stage I of the lesson

Checking homework

Collect leaflets with independent homework from didactic material S-19 option 1.
Solve on the board the tasks that caused difficulties for students when doing their homework.

Lesson stage II

1. Frontal survey.

2. Blitz survey: Highlight the correct answer in the test on the board (Appendix 1, pp. 2-3).

Stage III of the lesson

Doing exercises.

1. Solve No. 358 (a). Solve the equation graphically: .

2. Cards (four weak students solve in a notebook or on the board):

1) Find the meaning of the expression: a) ; b) .

2) Find the domain of definition of the functions: a) ; b) y = .

3. Solve No. 358 (a). Solve the equation graphically: .

One student solves on the board, the rest in a notebook. If necessary, the teacher helps the student.

A rectangular coordinate system was built on the interactive whiteboard using the AutoGraph program. The student draws the corresponding graphs with a marker, finds a solution, and writes down the answer. Then the task is checked: the formula is entered using the keyboard, and the graph must coincide with the one already drawn in the same coordinate system. The abscissa of the intersection of the graphs is the root of the equation.

Solution:

Answer: 8

Solve No. 360(a). Plot and read the graph of the function:

Students complete the task independently.

The construction of the graph is checked using the AutoGraph program, the properties are written on the board by one student (domain of definition, domain of value, parity, monotonicity, continuity, zeros and constancy of sign, greatest and least values of a function).

Solution:

Properties:

1) D( f) = (-); E( f) = , increases by )