Properties of numerical functions. Numerical functions and their properties This material is compiled according to the Federal State Standards
Numeric function This correspondence between a number set is called X and many R real numbers, in which each number from the set X matches a single number from a set R. A bunch of X called domain of the function
. Functions are indicated by letters f, g, h etc. If f– function defined on the set X, then real number y, corresponding to the number X there are many of them X, often denoted f(x) and write
y = f(x). Variable X this is called an argument. Set of numbers of the form f(x) called function range
The function is specified using a formula. For example , y = 2X - 2. If, when specifying a function using a formula, its domain of definition is not indicated, then it is assumed that the domain of definition of the function is the domain of definition of the expression f(x).
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.
Limits.
A number A is called the limit of a function as x tends to ∞ if for any E>0, there exists δ (E)>0 such that for all x satisfies the inequality |x|>δ the inequality |F(x)-A| A number A is called the limit of a function as X tends to X 0 if for any E>0, there exists δ (E)>0 such that for all X≠X 0 satisfies the inequality |X-X 0 |<δ выполняется неравенство |F(x)-A| UNILATERAL LIMITS. When defining the limit, X tends to X0 in an arbitrary manner, that is, from any direction. When X tends to X0, so that it is always less than X0, then the limit is called the limit at X0 on the left. Or a left-handed limit. The right-hand limit is defined similarly. 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 b) Simplify the expression Homework: No. 1. Calculate: a) V) G) No. 2. Find the domain of definition of the function: a) V) Lessons 1-2. Definition of a numerical function and methods for specifying it 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 function 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. Find 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 function 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 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: 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 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 us denote by the letter z = x - 2, then x = z + 2, and write down the condition of the problem: or 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: 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 OGOI SPO Ryazan Pedagogical College ABSTRACT Topic: “Numerical functions and their properties. Direct and inverse proportional relationships" Titova Elena Vladimirovna Specialty: 050709 “Teaching in primary school with additional training in the field of preschool education” Course: 1 Group: 2 Department: school Head: Pristuplyuk Olga Nikolaevna Introduction…………………………………………………………………………………3 1.2 Methods of specifying functions…………………………………………….6 2.1 The concept of direct proportionality………………..9 3.1
Functional propaedeutics in the initial course of mathematics....11 3.2 Solving problems involving proportionally dependent quantities……18 List of references……………………………..22 Introduction In mathematics, the idea of a function appeared along with the concept of quantity. It was closely related to geometric and mechanical concepts. The term function (from Latin – execution) was first introduced by Leibniz in 1694. By function he understood abscissas, ordinates and other segments associated with a point describing a certain line. This analytical expression, composed of a variable and a constant. In other words, the function is expressed by various types of formulas: y=ax+b, y= =axІ+bx+c, etc. The main trends in the development of modern school education are expressed in the ideas of humanization, humanitarization, activity-based and personality-oriented approach to organizing the educational process. In the basis of teaching mathematics in secondary schools, the principle of priority of the developmental function of teaching comes to the fore. Consequently, studying the concept of a numerical function in elementary school is a fairly significant component in the formation of schoolchildren’s mathematical concepts. For a primary school teacher, it is necessary to focus on studying this concept, since there is a direct relationship between the function and many areas of human activity, which will later help children enter the world of science. Besides ,
Students, as a rule, formally grasp the definition of the concept of a function and do not have a holistic understanding of functional dependence, i.e. cannot apply their knowledge to solving mathematical and practical problems; associate a function exclusively with an analytical expression in which the variable at expressed through a variable X; cannot interpret representations of function in different models; find it difficult to construct graphs of functions based on their properties, etc. The reasons for these difficulties are related not only and not so much to the methodology of studying functional material in an algebra course, but to the unpreparedness of students’ thinking to perceive and assimilate the concept of “function”. 1. Numeric functions 1.1 Development of the concept of functional dependence in mathematics Let us analyze the progress of development of pedagogical ideas in the field of teaching the most important component of mathematics - functional dependence. The functional line of the school mathematics course is one of the leading courses in algebra, algebra and the beginnings of analysis. The main feature of the educational material of this line is that with its help you can establish various connections in teaching mathematics. Over the course of several centuries, the concept of function has changed and improved. The need to study functional dependence in a school mathematics course has been the focus of pedagogical press since the second half of the 19th century. Such well-known methodologists as M.V. Ostrogradsky, V.N. Shklarevich, S.I. Shokhor-Trotsky, V.E. Serdobinsky, V.P. Sheremetevsky paid much attention to this issue in their works. First stage- the stage of introducing the concept of function (mainly through an analytical expression) into the school mathematics course. Second phase The introduction of the concept of function into a high school algebra course is characterized mainly by a transition to a graphical representation of functional dependence and an expansion of the range of functions studied. Third stage The development of the Russian school began in the 20s. twentieth century. An analysis of the methodological literature of the Soviet period showed that the introduction of the concept of function into the school mathematics course was accompanied by heated discussions, and allowed us to identify four main problems around which there were differences of opinion among methodologists, namely: 1) the purpose and significance of studying the concept of function by students; 2) approaches to defining a function; 3) the issue of functional propaedeutics; 4) the place and volume of functional material in the school mathematics course. Fourth stage due to the transfer of the economy of the RSFSR to a planned basis In 1934, the school received the first stable textbook by A.P. Kiselev, “Algebra,” revised under the editorship of A.P. Barsukov in two parts. Its second part included sections “Functions and their graphs”, “Quadratic function”. In addition, in the section “Generalization of the concept of degree” the exponential function and its graph were considered, and in the section “Logarithms” the logarithmic function and its graph were considered. It was in it that the function was defined through the concept of a variable quantity: “That variable quantity, the numerical values of which change depending on the numerical values of another, is called a dependent variable, or a function of another variable quantity.” However, it does not reflect the idea of correspondence and there is no mention of an analytical expression, which allows us to conclude that this definition has a significant flaw. The scientist regarded the formation of an idea of a function as a manifestation of formalism in teaching. He believed that in secondary school the concept of function should be taught on the basis of the concept of correspondence. This period is characterized by insufficient time to study functions, ill-conceived exercise systems, students’ lack of understanding of the true essence of the concept of function, and a low level of functional and graphic skills of school graduates. Thus, the need to reform the teaching of mathematics in secondary schools has again arisen. The restructuring of all school mathematics on the basis of a set-theoretic approach marked the fifth stage in the development of the idea of functional dependence. The idea of the set-theoretic approach was undertaken by a group of French scientists united under the pseudonym Nicolas Bourbaki. An international meeting was held in Roymont (France, 1959), at which the overthrow of all conventional courses was proclaimed. The focus was on the structures and unifications of all school mathematics based on set theory. An important role in the development of reform ideas was played by the articles of V.L. Goncharov, in which the author pointed out the importance of early and long-term functional propaedeutics, and proposed the use of exercises consisting of performing a number of pre-specified numerical substitutions in the same given letter expression. The stabilization of programs and textbooks created the ground for positive changes in the quality of students' functional knowledge. In the late sixties and early seventies, along with negative reviews, those began to appear in the press, which noted a certain improvement in the knowledge of school graduates about functions and graphs. However, the overall level of students' mathematical development remained generally insufficient. School mathematics courses continued to spend an inordinate amount of time on formal preparation and did not pay enough attention to developing students' ability to learn independently. So, numeric function is a correspondence between a numerical set R of real numbers, in which each number from the set X corresponds to a single number from the set R. Accordingly, X represents the domain of definition of the function (DOF). The function itself is denoted by lowercase letters of the Latin alphabet (f, d, e, k). If a function f is given on a set X, then the real number y corresponding to the number x from the set X is denoted as f(x) (y=f(x)). The variable x is called argument. The set of numbers of the form f(x) for all x is called function rangef.
Most often, functions are specified by various types of formulas: y=2x+3, y=xІ, y=3xі, y=?3xІ, where x is a real number, y is the corresponding singular number. However, with one formula you can set a bunch of functions, the difference of which is determined only by the domain of definition: Y= 2x-3, where x belongs to the set of real numbers and y=2x-3, X - belonging to the set of natural numbers. Often, when specifying a function using a formula, the OOF is not specified (the OOF is the domain of definition of the expression f(x)). It is also quite convenient to represent numerical functions visually, i.e. using a coordinate plane. Like many others, numeric functions have the following properties: Increasing, decreasing, monotonicity, domain of definition and domain of value of a function, boundedness and unboundedness, even and odd, periodicity. Domain and range of functions. In elementary mathematics, functions are studied only on the set of real numbers R. This means that the argument of a function can only take those real values for which the function is defined, i.e. it also accepts only real values. The set X of all admissible real values of the argument x for which the function y = f(x) is defined is called the domain of the function. The set Y of all real values of y that a function takes is called the range of the function. Now we can give a more precise definition of a function: the rule (law) of correspondence between the sets X and Y, according to which for each element from the set X one and only one element from the set Y can be found, is called a function. Even and odd functions. If for any x from the domain of definition of the function the following holds: f (- x) = f (x), then the function is called even; if it occurs: f (- x) = - f (x), then the function is called odd. The graph of an even function is symmetrical with respect to the Y axis (Fig. 5), and the graph of an odd function is symmetrical with respect to the origin (Fig. 6). Periodic function. A function f (x) is periodic if there is a non-zero number T such that for any x from the domain of definition of the function the following holds: f (x + T) = f (x). This smallest number is called the period of the function. All trigonometric functions are periodic. But the most important property for learning function in primary grades is monotone. Monotonic function. If for any two values of the argument x1 and x2 the condition x2 > x1 implies f (x2) > f (x1), then the function | f(x) | called increasing; if for any x1 and x2 the condition x2 > x1 implies f (x2) In elementary school, the function manifests itself in the form of direct and inverse proportional relationships. Direct proportionality- this is, first of all,
function, which can be given using the formula y=kx, where k is a non-zero real number. The name of the function y = kx is associated with the variables x and y contained in this formula. If attitude two quantities are equal to some number different from zero, then they are called directly proportional. K is the proportionality coefficient. In general, the function y=kx is a mathematical model of many real situations considered in the initial mathematics course. For example, let’s say that there are 2 kg of flour in one package, and x such packages were purchased, then the entire mass of flour purchased is y. This can be written as a formula like this: y=2x, where 2=k. Direct proportionality has a number of properties: xseveral times the corresponding positive value y increases (decreases) by the same amount. 2.3 The concept of inverse proportionality. Properties of inverse proportionality: If the values of the variablesxAndywill be positive real numbers, then with increasing (decreasing) variablexseveral times the corresponding value of y decreases (increases) by the same amount. Practical part The concept of functional dependence is one of the leading ones in mathematical science, therefore, the formation of this concept among students is an important task in the teacher’s purposeful activities to develop mathematical thinking and creative activity of children. The development of functional thinking presupposes, first of all, the development of the ability to discover new connections and master general educational techniques and skills. In the initial course of mathematics, a significant role should be given to functional propaedeutics, which provides for preparing students to study systematic courses in algebra and geometry, and also instills in them the dialectical nature of thinking, an understanding of the causal relationships between the phenomena of the surrounding reality. In this regard, we will outline the main directions of propaedeutic work at the initial stage of teaching the subject according to the program of L.G. Peterson: The concept of sets, the correspondence of elements of two sets and functions. Dependence of the results of arithmetic operations on changes in components. Tabular, verbal, analytical, graphical methods of specifying a function. Linear dependence. Coordinate system, first and second coordinate, ordered pair. Solving the simplest combinatorial problems: compiling and counting the number of possible permutations, subsets of elements of a finite set.. Using systematic enumeration of natural values of one and two variables when solving plot problems. Filling out tables with arithmetic calculations, data from the conditions of applied problems. Selecting data from a table by condition. Relationship between proportional quantities; applied study of their graphs. The content of the initial mathematics course allows students to form an understanding of one of the most important ideas in mathematics - idea of conformity.When completing tasks to find the meanings of expressions and filling out tables, students establish that each pair of numbers corresponds to no more than one number obtained as a result. However, to understand this, the contents of the tables must be analyzed. Make up all possible examples of adding two single-digit numbers with the answer being 12. When completing this task, students establish a relationship between two sets of values of terms. The established correspondence is a function, since each value of the first term corresponds to a single value of the second term with a constant sum. There are 10 apples in a vase. How many apples will be left if you take 2 apples? 3 apples? 5 apples? Write the solution in the table. What does the result depend on? By how many units does it change? Why? This problem actually presents the function at = 10 - X, where the variable X takes the values 2, 3, 5. As a result of completing this task, students must conclude: the greater the subtrahend, the smaller the difference. The idea of functional correspondence is also present in exercises like: Connect with an arrow the mathematical expressions and the corresponding numerical values: 15 + 6 27 35 Introduction letter symbols allows you to introduce students to the most important concepts of modern mathematics - variable, equation, inequality, which contributes to the development of functional thinking, since the idea of functional dependence is closely related to them. When working with a variable, students realize that the letters included in an expression can take on different numerical values, and the letter expression itself is a generalized notation of numerical expressions. The experience of students communicating with exercises on establishing patterns in number sequences and their continuation: 1, 2, 3, 4… (at = X + 1) 1, 3, 5, 7… (at= 2 · X + 1) Concept quantities, along with the concept of number, is the main concept of the initial mathematics course. The material in this section is a rich source for the implementation of indirect functional propaedeutics. Firstly, this is the dependence (inversely proportional) between the selected unit of quantity (measure) and its numerical value (measure) - the larger the measure, the smaller the number obtained as a result of measuring the quantity with this measure. Therefore, it is important that when working with each quantity, students gain experience in measuring quantities with different standards in order to consciously choose, first a convenient one, and then a single measurement. Secondly, when studying the quantities characterizing the processes of movement, work, purchase and sale, ideas are formed about the relationship between speed, time and distance, price, quantity and cost in the process of solving word problems of the following types - reduction to unity (finding the fourth proportional) , finding the unknown by two differences, proportional division. It is especially difficult for students to understand the relationship between these quantities, since the concept of “proportional dependence” is not the subject of special study and assimilation. In the program L.G. Peterson methodically solves this problem by using the following techniques: - Solving problems with missing data (“open” condition): Vasya’s home to school is 540 m, and Pasha’s is 480 m. Who lives closer? Who will get there faster? Sasha bought notebooks for 30 rubles and pencils for 45 rubles. Which items did he spend the most money on? What items did he buy more? By analyzing the texts of these problems, students discover that they lack data and that the answers to the questions depend on price and speed. - Fixing the conditions of the tasks not only in a table (as proposed in the classical method), but also in the form of a diagram. This allows you to “visualize” the dependencies considered in the problem. So, if moving objects cover the same distance of 12 km in different times (2 hours, 3 hours, 4 hours, 6 hours), then using the diagram the inverse relationship is clearly interpreted - the more parts (time), the smaller each part (speed). - Change one of the task data and compare the results of solving problems. 48 kg of apples were brought to the school canteen. How many boxes could they bring if all the boxes contained the same amount of apples? Students complete the conditions of the problem and record the relationship between quantities using various means of structuring theoretical knowledge - in a table, diagram and verbally. Here it is useful to pay attention to the multiple ratio of the quantities under consideration - how many times greater is one of the quantities, the same number of times greater (smaller) is the other, with the third being constant. In elementary school, students are implicitly introduced to tabular, analytical, verbal, graphical methods of specifying functions. For example, the relationship between speed, time and distance can be expressed: A) verbally: “to find the distance, you need to multiply the speed by the time”; B) analytically: s= v t; B) tabular: v =5 km/h Graphical way to specify the dependency between v, t, s allows us to form an idea of speed as a change in the location of a moving object per unit of time (along with the generally accepted one - as a distance traveled per unit of time) And a comparison of the graphs of the motion of two bodies (moving independently of each other) clarifies the idea of speed as a quantity characterizing the speed of movement. Compound Numeric Expressions(with and without parentheses), calculating their values according to the rules of the order of actions allows students to realize that the result depends on the order in which actions are performed. Arrange the parentheses to form correct equalities. 20 + 30: 5=10, 20 + 30: 5 = 26 In the course of L.G. Peterson, students are implicitly introduced to linear dependence, as a special case of a function. This function can be specified by a formula of the form at= kh + b, Where X- independent variable, k And b- numbers. Its domain is the set of all real numbers. Having traveled 350 kilometers, the train began to travel for t hours at a speed of 60 km/h. How many kilometers did the train travel in total?(350 + 60 · t) By completing tasks with named numbers, students realize the dependence numerical values of quantities from the use of different units of measurement. The same segment was measured first in centimeters, then in decimeters. In the first case, the number we got was 135 more than in the second. What is the length of the segment in centimeters? (Dependency= 10 · X) In the process of studying the initial course of mathematics, students form the concept of a natural series of numbers, a segment of a natural series, assimilate the properties of a natural series of numbers - infinity, orderliness, etc., form the idea of the possibility of an unlimited increase in a natural number or a decrease in its share. In the mathematics course for grades 3-4, significant attention is paid to teaching students to use formulas, their independent conclusion. Here it is important to teach students to present the same information in different forms - graphically and analytically, giving students the right to choose the form in accordance with their cognitive styles. Students are particularly interested in tasks related to analyzing tables of variable values, “discovering” dependencies between them and writing them down as formulas. To develop students’ ability to derive formulas, you need to teach them to write various statements in mathematical language (in the form of equalities): A pen is three times more expensive than a pencil ( R = To + 3); Number A When divided by 5, the remainder is 2 ( A= 5 · b + 2); The length of the rectangle is 12 cm greater than the width ( A = b + 12). A prerequisite is to discuss possible options for the values of these quantities and fill out the corresponding tables. A special place in the course of L.G. Peterson takes on assignments related to mathematical research: Represent the number 16 as a product of two factors in different ways. For each method, find the sum of the factors. In which case was the smaller amount obtained? Do the same with numbers 36 and 48. What is your guess? When completing similar tasks (to study the relationship between the number of angles of a polygon and the total value of degree measures of angles, between the value of the perimeter of figures of different shapes with the same area, etc.), students improve their skills in working with a table, since it is convenient to record the solution in a table. In addition, the tabular method of fixing the solution is used when solving non-standard mathematical problems using the method of ordered search or rational selection. There are 13 children in the class. Boys have as many teeth as girls have fingers and toes. How many boys and how many girls are there in the class? (Each boy has exactly 32 teeth). Exercises on number composition; Particular methods of calculations (36 + 19 = 35 + 20; 36 - 19 = 37 - 20; 12 · 5 = 12 · 10: 2); Estimation of sum, difference, product, quotient. When performing tasks like these, it is important to present information in a multisensory manner. How will the sum change if one term is increased by 10 and the second is decreased by 5? How will the area of a rectangle (or the product of two numbers) change if one of the sides (one of the numbers) is increased by 3? A significant portion of students complete such tasks by substituting specific numerical values. Methodologically competent in this situation would be to interpret the condition graphically and analytically. (A+ 3) · b = A· b+ 3 ·b
The concept of function in high school is associated with coordinate system. In the course of L.G. Peterson contains material for propaedeutic work in this direction: Numerical segment, numerical ray, coordinate ray; Pythagorean table, coordinates on the plane (coordinate angle); Traffic schedules; Pie, bar, and line charts that visually represent relationships between discrete quantities. So, the study of arithmetic operations, increasing and decreasing a number by several units or several times, the relationship between components and results of arithmetic operations, solving problems on finding the fourth proportional, on the relationship between speed, time and distance; price, quantity and value; the mass of an individual item, their quantity and total mass; productivity, time and work; etc., on the one hand, underlie the formation of the concept of function, and on the other hand, they are studied on the basis of functional concepts. It should be noted that graphic modeling has quite a large propaedeutic significance: graphic interpretation of the problem conditions, drawing, drawing, etc. Information presented in graphic form is easier to perceive, capacious and rather conditional, designed to convey information only about the essential features of an object and to develop students’ graphic skills. In addition, the result of propaedeutics of functional dependence should be high mental activity of younger schoolchildren, the development of intellectual, general subject and specific mathematical skills. All this creates a solid basis not only for solving methodological problems of primary mathematics - the formation of computational skills, the ability to solve word problems, etc., but also for the implementation of the developmental possibilities of mathematical content and, no less important, for the successful study of functions in secondary school. 3.2 Solving problems involving proportionally dependent quantities Solving a problem means using a logically correct sequence of actions and operations with numbers, quantities, explicitly or implicitly available in the problem, relationships to fulfill the requirement of the task (answer its question). The main ones in mathematics are: arithmetic And algebraic ways to solve problems. At arithmetic way the answer to the question of the problem is found as a result of performing arithmetic actions on numbers. Different arithmetic methods for solving the same problem are different relationships between data, data and unknowns, data and what is sought, underlying the choice of arithmetic operations, or the sequence using these relationships when choosing actions. Solving a word problem using arithmetic is a complex activity. decisive. However, there are several stages in it: 1. Perception and analysis of the content of the task. 2. Search and drawing up a plan for solving the problem. 3. Execution of the solution plan. Formulation of the conclusion about the fulfillment of the requirement tasks (answering the task question). 4. Checking the solution and eliminating errors, if any. Proportional division problems are introduced in different ways: you can offer to solve a ready-made problem, or you can first compose it by transforming the problem to find the fourth proportional. In both cases, the success of the solution proportional division problems will be determined by a solid ability to solve problems of finding the fourth proportional, therefore, as preparation must include solving problems of the appropriate type to find fourth proportional. That is why the second one is preferable the mentioned options for introducing proportional division problems. Moving on to solving ready-made problems from the textbook, as well as problems compiled teacher, including various groups of quantities, you first need to establish which quantities discussed in the problem, then write the problem briefly in the table, having previously divided the question of the problem into two questions, if it contains the word every. As a rule, students complete the solution independently, analyze conducted only with individual students. Instead of a short note, you can make drawing. For example, if the problem involves pieces of cloth, coils of wire, and etc., then they can be represented by segments by writing the corresponding numerical the values of these quantities. Note that you should not perform a short run every time. recording or drawing, if the student, after reading the problem, knows how to solve it, then let him decide, and those who find it difficult to use a short note or drawing To solve the task. Gradually the tasks should become more complex by introducing additional data (for example: “The first piece contained 16 m of matter, and the second 2 times less.”) or asking a question (for example: “How many meters Was there more matter in the first piece than in the second?). When familiarizing yourself with the solution to the disproportionate division problem, you can go another way: first solve ready-made problems, and later execute transforming the problem of finding the fourth proportional into a problem of proportional division and, after solving them, compare both the problems themselves and their decisions. Exercises help generalize the ability to solve problems of the type considered. creative nature. Let's name some of them. Before solving it, it is useful to ask which of the questions in the problem will be answered a larger number and why, and after deciding to check whether it corresponds to this type the resulting numbers, which will be one of the ways to check the solution. You can further find out whether the answer could have produced the same numbers and under what conditions. Useful exercises for students to compose problems and then solve them, and task transformation exercises. This is, first of all, compilation problems similar to the one solved. So, after solving the problem with quantities: price, quantity and cost - offer to compose and solve a similar problem with the same quantities or with others, such as speed, time and distance. This is the compilation of problems for their solution, written down as separate actions, and in the form of expression, is the compilation and solution of problems according to their short schematic notation 1 way: X = 15*30 / 8 = 56 rubles 25 kopecks Method 2: the amount of cloth has increased 15/8 times, which means they will pay 15/8 times more money X =30*15/8 = 56 rubles 25 kopecks 2.
A certain gentleman called a carpenter and ordered him to build a courtyard. He gave him 20 workers and asked how many days they would build his yard. The carpenter answered: in 30 days. But the master needs to build it in 5 days, and for this he asked the carpenter: how many people do you need to have so that you can build a courtyard with them in 5 days; and the carpenter, perplexed, asks you, arithmetician: how many people does he need to hire to build a yard in 5 days? An unfinished short condition is written on the board: Option I: proportion Option II: without proportions I. II. X = 20*6 = 120 workers 3.
They took 560 soldiers with food for 7 months, but they were ordered to serve for 10 months, and they wanted to remove people from themselves so that there would be enough food for 10 months. The question is, how many people should be reduced? An ancient task. Solve this problem without proportion: (The number of months increases by a factor, which means the number of soldiers decreases by a factor. 560 – 392 = 168 (soldiers must be reduced) In ancient times, to solve many types of problems, there were special rules for solving them. The familiar problems of direct and inverse proportionality, in which we need to find the fourth from three values of two quantities, were called “triple rule” problems. If, for three quantities, five values were given, and it was necessary to find the sixth, then the rule was called “quintuple.” Similarly, for four quantities there was a “septenary rule”. Problems involving the application of these rules were also called “complex triple rule” problems. 4.
Three chickens laid 3 eggs in 3 days. How many eggs will 12 hens lay in 12 days? How many times has the number of chickens increased? (4 times) How did the number of eggs change if the number of days did not change? (increased 4 times) How many times has the number of days increased? (4 times) How did the number of eggs change? (increased 4 times) X = 3*4*4 =48(eggs) 5
. If a scribe can write 15 leaves in 8 days, how many scribes will it take to write 405 leaves in 9 days? From increasing work days Let's consider a more complex problem with four quantities. 6.
To illuminate 18 rooms, 120 tons of kerosene were used in 48 days, with 4 lamps burning in each room. How many days will 125 pounds of kerosene last if 20 rooms are illuminated and 3 lamps are lit in each room? The number of days of using kerosene decreases with the increase in rooms in 20
times. X = 48 * * : = 60 (days) The final value is X = 60. This means that 125 pounds of kerosene lasts for 60 days. Conclusion The methodological system for studying functional dependence in elementary school, developed in the context of modular education, represents an integrity made up of the interrelation of the main components (target, content, organizational, technological, diagnostic) and principles (modularity, conscious perspective, openness, focus of learning on the development of the student’s personality , versatility of methodological consulting). The modular approach is a means of improving the process of studying functional dependence in primary school students, which allows: students to master a system of functional knowledge and methods of action, practical (operational) skills; the teacher - to develop their mathematical thinking based on functional material, to cultivate independence in learning. Methodological support for the process of studying functions in primary school is built on the basis of modular programs, which are the basis for identifying fundamental patterns that are mandatory for understanding the topic, successful and complete assimilation of the content of educational material, and the acquisition by students of solid knowledge, skills and abilities. Bibliography. 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) 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.
and find its value at
and find its value at 
.
;
;
.
;
; G) 
.
To 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).
Since by definition of an arithmetic root
then we add the number 3 to all parts of this inequality, we get:
or 3 ≤ y< +∞. Находим
defined 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; ∞).
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.
From the method of finding y it is clear that any value x corresponds to only one value y.
Does the point with coordinates belong to the graph of this function: a) (-2; -6); b) (-3; -10)?
Let's find x(y).
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
where 
Since the function argument can be denoted by any letter, we have:
n1.doc
Ryazan
Theoretical part
1.1 Development of the concept of functional dependence in mathematics…………………………….……………………………4
Numeric functions
1.3 Function properties………………………………………………………7
2. Direct and inverse proportional relationships
2.2 Properties of direct proportional dependence…………………………………………….10
2.3 The concept of inverse proportionality and its properties………………………………………………………………-
Practical part
Conclusion……………………………………………………………......21
In the first half of the 18th century. there was a transition from a visual representation of the concept of a function to an analytical definition. The Swiss mathematician Johann Bernoulli, and then the academician Leonhard Euler, believed that the function
Today we know that a function can be expressed not only in mathematical language, but also graphically. The discoverer of this method was Descartes. This discovery played a huge role in the further development of mathematics: the transition from points to numbers, from lines to equations, from geometry to algebra took place. Thus, it became possible to find common techniques for solving problems.
On the other hand, thanks to the coordinate method, it became possible to depict geometrically different dependencies.
Thus, graphs provide a visual representation of the nature of the relationship between quantities; they are often used in various fields of science and technology.
This means that before the concept of “function” is introduced, it is necessary to carry out work on the formation of functional thinking skills, so that “at the moment when the general idea of functional dependence should enter the consciousness of students, this consciousness will be sufficiently prepared for the substantive and effective, and not just the formal perception of a new concept and associated ideas and skills” (A.Ya. Khinchin)
The development of the idea of functional dependence proceeded in several stages:
I. Ya. Khinchin paid much attention to this problem in his works.
The modern concept of function differs significantly from previous ones. It more fully reflects all the properties and dependencies that it possesses.
1.2 Methods for specifying functions
1.3 Function properties.
A function is considered defined if: the domain of definition of the function X is specified; 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.
Limited and unlimited functions. A function is called bounded if there is a positive number M such that | f(x) | M for all values of x. If such a number does not exist, then the function is unlimited.
2. Direct and inverse proportional relationships.
2.1 The concept of direct proportionality.
2.2 Properties of direct proportionality.
If the values of the variablesxAndy
The domain of definition of the function y=kx is the set of real numbers R;
A direct proportionality graph is a straight line passing through the origin;
For k>0, the function y=kx increases over the entire domain of definition (for k
If the function f is direct proportionality, then (x1,y1),(x2,y2) are pairs of corresponding variables x and y, where x is not equal to zero, which means x1/x2=y1/y2.
Inverse proportionality- This function, which can be given using the formula y=k/x, where k is a non-zero real number. The name of the function y = k/x is associated with the variables x and y, the product of which is equal to some real number that is not equal to zero.
The domain of definition and range of values of the function y=k/x is the set of real numbers R;
Direct proportionality graph – hyperbola;
When k 0, respectively, decreases throughout the entire domain of definition, branches - down)
If the function f is inverse proportionality, then (x1,y1),(x2,y2) are pairs of corresponding variables x and y, where x is not equal to zero, which means x1/x2=y2/y1.
3.1 Functional propaedeutics in the initial course of mathematics
d) graphically (using a coordinate ray or angle).
When analyzing the numbers presented in the table, students easily notice that the numbers in the first line increase by one, the numbers in the second line increase by four. The teacher’s task is to pay attention to the relationship between the values of variables A And b. In order to strengthen the applied orientation of mathematical education, this situation should be “revitalized” and transferred to plot status.
Teaching mathematics according to the program L.G. Peterson ensures that students understand the relationship between the results and components of arithmetic operations, and an idea of “speed” of change in the result of arithmetic operations depending on changes in components:
Chickens
days
eggs
3
3
3
12
12
X
You need to find out:
(the number of scribes increases with the increase in sheets by times and decreases
(scribes)).
The number of days of using kerosene increases with the increase in the amount of kerosene in
times and from reducing the lamps by a factor.
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.
.
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)).