Give an example of the functional dependence of any function. The simplest functional dependencies. Functional dependence: logic and meaning

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1 G(O)B POU "Zadonsky Polytechnic College" Research work Functional dependence of real processes Performed by: Chernukhin Ivan Alekseevich, Kopenkin Pavel Vladimirovich, students of group Te-1 G(O)B POU "Zadonsky Polytechnic College" Supervisor: Storozhuk Valentina Nikolaevna, teacher of mathematics, State Educational Institution "Zadonsky Polytechnic College" Address: Lipetsk region, Zadonsk, st. Labor, d uch.g.

2 CONTENTS Introduction Chapter 1. Theoretical foundations for introducing the concept of a function From the history of the emergence of a function The concept of a function and the graph of a function Chapter 2. Practical part Study of functional dependence in everyday life Illustration of functional dependence in the work of an electrician Conclusion...22 References...23 Appendix 1 Types of functions and their properties. 1

3 Introduction The success of a person in modern society depends on how competent he is in the basics of science, including mathematics. In mathematics, all phenomena and dependencies are described using functions. Function is one of the basic mathematical and general scientific concepts that expresses the relationship between variables. The concept of function was developed by great scientists: Francois Viète, Rene Descartes, Fermat, Newton, Leibniz, Bernoulli, Euler, d'Alembert, Fourier and other scientists. Among Russian scientists we can name: Euler. Chebyshev, Sobolev, Lobachevsky, Lebedev and others. Functional dependence occurs in life “at every step,” so this topic is relevant both for each person and for the entire city, and in general for all of humanity. Years pass and we change. We also depend on our heredity, on the books we read, on the temperature of our environment and on many other factors. And therefore we formulated the topic of our research as follows: “Functional dependence of real processes.” We love to find various patterns in the world around us, we love to study numbers and build graphs. Therefore, we decided to learn more about how you can connect various moments of life with functions and graphs. The purpose of our research: to show examples of a non-standard view of functional dependence in the life around us. To do this, we set ourselves the following tasks: 1. study material on this topic; 2. get acquainted with the history of the emergence of the concept of function; 3. introduce the concept of function and function graph; 4. demonstrate various functional dependencies around us and in the work of an electrician; 5. make a presentation. 2

4 Subject of research: a set of mathematical methods and models. Object of study: functions. Research methods: study and use of scientific, journalistic and educational publications, generalization, analysis, synthesis, modeling. Hypothesis: actually occurring events in a person’s life can be represented in the form of a dependency graph. The material related to the construction of graphs of functions is not studied fully enough from the point of view of the requirements presented in the exams. Therefore, graphing tasks often cause difficulties for students. Based on this fact, this topic is necessary to be discussed in detail. The theoretical significance of our research work lies in the fact that the research results can be used for technical school students when studying the topic “Function”. The practical significance of the work lies in the fact that the results of the study can be used by college students to improve their educational level when studying the application of the function in the practical activities of an electrician. We believe that this work can help to interest students and give them the opportunity to “look inside” such a complex mathematical concept as “function”. 3

5 Chapter I. Theoretical part From the history of the origin of the function. Most mathematical concepts have come a long way in development. The concept of function has gone through a difficult path. It goes back to that distant era when people first realized that the phenomena around them are interconnected. They did not yet know how to count, but they already knew that the more deer they managed to kill during the hunt, the longer the tribe would be free from hunger; the tighter the bow string is pulled, the farther the arrow will fly; The longer the fire burns, the warmer it will be in the cave. With the development of cattle breeding, agriculture, crafts and exchange, the number of relationships between quantities known to people increased. The idea of the dependence of certain quantities goes back to ancient Greek science. But the Greeks considered only questions of a geometric nature, and did not raise the question of a general study of various dependencies. Graphic representation of dependencies was widely used by G. Galileo (), P. Fermat () and R. Descartes (), who introduced the concept of “variable quantity”. According to Descartes’ definition: “A function of a variable quantity is a quantity formed in any way from this variable quantity and constants.” The development of mechanics and technology required the introduction of a general concept of function, which was done by the German philosopher and mathematician G. Leibniz. The next step in the development of the concept of function was made by Bernoulli's student, member of the St. Petersburg Academy of Sciences, Leonard Euler (). In Differential Calculus, published in 1755, L. Euler gives a general definition of a function: 4

6 When some quantities depend on others in such a way that when the latter change, they themselves are subject to change, then the former are called functions of the latter. This name, Euler continues further, has an extremely broad character; it embraces all the ways in which one quantity is determined by means of others. In general terms, the concept of a generalized function was introduced by the Frenchman Laurent Schwartz. In 1834, in his work On the disappearance of trigonometric strings, N. I. Lobachevsky, developing the above-mentioned Euler definition of a function in 1755, wrote: The general concept requires that a function of x be called a number that is given for each x and, together with x, gradually changes. The value of a function can be given either by an analytical expression or by a condition, which provides a means of testing all the numbers and choosing one of them; or, finally, dependence can exist and remain unknown... The broad view of the theory allows for the existence of dependence only in the sense that numbers, one with another in connection, are taken as if given together. Even before Lobachevsky, a similar point of view on the concept of function was expressed by the Czech mathematician B. Bolzano. In 1837, the German mathematician P. Lejeune-Dirichlet formulated the general definition of the concept of a function as follows: y is a function of the variable x (on the interval a (x (b), if each value of x (on this interval) corresponds to a completely definite value of y, and it is indifferent , how this correspondence is established - by an analytical formula, a graph, a table, or even just words.In 1936, the 28-year-old Soviet mathematician and mechanic S. L. Sobolev was the first to consider a special case of a generalized function.5

7 Conclusion: It should be noted that starting from the 17th century. One of the most important mathematical concepts is the concept of function. It played and still plays a big role in understanding the real world. Tracing the historical path of development of the concept of function, one involuntarily comes to the conclusion that evolution is far from over and will probably never end, just as the evolution of mathematics as a whole will never end. New discoveries and inquiries from natural sciences and other sciences will lead to new extensions of the concept of function and other mathematical concepts. Mathematics is an unfinished science; it has developed over thousands of years, is developing in our era and will continue to develop in the future. 6

8 1.2. Definition of a function and graph of a function. Function is one of the basic mathematical and general scientific concepts. It played and still plays a big role in understanding the real world. A function is not only a mathematical concept, but also: a function is the work produced by an organ, an organism; role, meaning of something; function in mathematics, the law of dependence of one quantity on another; function capability, option, skill of a program or device. function responsibility, range of activities; function of a character in a literary work; function is a type of subroutine in computer science, a social function. Definition. A numerical function with a domain of definition D is a correspondence in which each number x from the set D is associated, according to some rule, with a single number y depending on x. It is customary to call x the independent variable or argument, and y the dependent variable or the value of the function. Write the indicated relationship between x and y in general form as follows: y = f (x) or y = F (x), etc. The graph of the function y = f (x) is the set of all points of the plane, coordinates (x, y ) which satisfy the relation y = f(x). There are several ways to specify functions: analytical, verbal, graphical, tabular. Analytical method. The most common analytical method for specifying a function, at 7

9 in which the function is given by a formula that establishes what computational operations must be performed on x in order to find y. Example: y = k x; V = s h ; s = a b Verbal method (proverbs, sayings) The further into the forest, the more firewood. You can't spoil porridge with oil. Talk less, work more. If you love to ride, you also love to carry sleds. Graphic method. The graphical method of specifying a function is also common. The graph of the function y=f(x), where x is from the set E, is the set of points in the plane with rectangular coordinates (x,y), where x is from E, y=f(x). The graphical method consists of drawing a line (graph), in which the abscissas represent the values of the argument, and the ordinates the corresponding values of the function. This method allows you to visualize the functional dependence. Example: distance km time t, s Tabular method. With the tabular method of specifying, a function is specified in the form of a table in which for each argument value the corresponding function value is indicated. The tabular method is well known (table of squares and table of cubes of natural numbers, etc.). This method immediately gives the numerical value of the function. This is its advantage over others

10 ways. Example. Table of squares of numbers from 1 to 10: Types of functions (Appendix 1): 1) linear: y = ax + b; 2) quadratic: y = ax 2 + bx + c; 3) inverse proportionality: y = k x ; n 4) n-root: y = x; 5) module: y = x; 6) trigonometric: y = sinx, y = cosx, y = tgx, y = ctgx; 7) exponential: y = a x; 8) logarithmic: y = log a x; 9) inverse trigonometric functions: y = arcsinx, y = arccosx, y = arctgx, y = arcctgx. Modern encyclopedias and explanatory dictionaries consider function in 5 meanings: 1. A phenomenon that depends on another and changes as this other phenomenon changes. 2. A variable quantity that changes depending on the change in another quantity. 3. Work performed by an organ or organism. 4. Responsibility, range of activities of a person, work to be performed. 5. Meaning, purpose, role. Function as a concept has one generic meaning, but in different areas of activity it manifests itself differently, remaining an unambiguous entity. The concept of a function plays an important role, being a basic concept in the study of algebra and the principles of analysis. 9

11 Conclusion: Function is one of the basic mathematical and general scientific concepts that expresses the dependence of some variables on others. It played and still plays a big role in understanding the real world. The concept of a function plays an important role, being a basic concept in the study of algebra and the principles of analysis. 10

12 Chapter 2. Practical part Study of functional dependence in everyday life. In everyday life, we constantly encounter functional dependencies. We found many examples of functions and depicted them using graphs. which Problem 1. At a temperature of 0 o C, the rail has a length l 0 = 12.5 m. with increasing temperature, thermal expansion of the rail occurs and its length, expressed in meters, changes according to the law l(t o) = l 0 (1 + t o), where = 1, coefficient of thermal expansion in degrees Celsius to the minus first power, t o temperature (in degrees Celsius). At what temperature will the rail lengthen by 6 mm. Express the answer in degrees Celsius. Solution. Let us express t from the given formula: t = l l 0 αl 0. Note, l l 0 = Δl = 6mm = m, then t = Answer: = 100 = 40 C Example 1. From the cold they brought a jar of ice into the room and began to observe the change temperature of the substance in the jar: the ice gradually melted, when it had all melted, the temperature of the water began to rise until it equaled the temperature in the room. Figure 1 shows a graph of temperature versus time. Rice. 1 11

13 Example 2. Consider dividing the birthday cake among guests. What determines the number of servings? on the number of guests. What does the weight of a serving depend on? also on the number of guests. In the first case, the more guests, the greater the number of servings we must divide the cake into (Fig. 2). Here you can clearly imagine a direct proportional relationship. Rice. 2 In the second case, the more guests, the smaller the portion weight. Here we see an inverse proportional relationship (Fig. 3). Rice. 3 Example 3. We live in the age of information technology. Every day we receive a lot of information from various sources: television, radio, newspapers, magazines, and, of course, from the Internet. It is known that the volume of information doubles every five years. Rice. 4 12

14 If we plot the dependence of the volume of information on time, we will get a certain curve, which in mathematics is called an exponential and is a graph of an exponential function (Fig. 4). Example 4. Hair grows on a person’s head and is regularly cut. The graph of the resulting dependence (provided that haircuts are done regularly) is similar to the function of the fractional part of a number, shifted upward by a units: y = x + a (Fig. 5). Example 5. During our time at school, we move to the next grade every year. This dependence is similar to the function of the integer part of the number y = (x) on a limited interval (Fig. 6). Rice. 5 Fig. 6 Example 6. Changes in temperature in our climate zone obey the laws of trigonometric functions (Fig. 7) Fig. 7 Example 7. Gardening processes can also be represented as a function and plotted. For example, an apple grew, ripened, and then it was dried (Fig. 8). We got some piecewise function. 13

15 Fig. 8 Functions are mathematical portraits of stable patterns cognizable by humans. To illustrate the characteristic properties of functions, let us turn to proverbs and sayings. After all, proverbs are also a reflection of stable patterns, verified by the centuries-old experience of the people. A graph can illustrate the meaning of any proverb. “The further into the forest, the more firewood” Fig. 9 The graph will present the amount of firewood as a function of the path. For example, the proverb “What kind of life you live, such glory will you gain” on the graph will look like this (Fig. 10): Fig.

16 From the graph it follows that if throughout your life you commit negative deeds and actions, then the fame about you will be negative, and vice versa. Or the proverb “Over-seeding is worse than under-seeding” on the graph will look like this (Fig. 11): Fig. 11 From the graph it is clear that if there are few seeds, then the harvest will be small, if there are too many seeds, then they will grow poorly, and you will lose the seeds and you will not reap the harvest, you need to plant the optimal number of seeds and the harvest will be high. Conclusion: In everyday life, we constantly encounter functional dependencies. I found many examples of functions that were depicted using graphs. 15

17 2.2. Illustration of functional dependence in the work of an electrician. Task 1. Electrical safety (the effect of electric current on the human body) The effect of current on the human body (in ma) The magnitude of the current Various magnitudes of current with a frequency of 50 Hz act as follows: 1-2) 5 10 ma pain in the muscles, their convulsive contractions, it is difficult to tear your hands away from the electrodes; 2-3) pain, hands cannot be taken away from the electrodes; 3-4) pain in the arms and chest, breathing is difficult, respiratory paralysis and loss of consciousness are possible; 4-5) clinical death is possible with prolonged action; 5-6) 100 mA or more with a duration of more than 3 s, clinical death is possible. Research task 2. Investigate: a) The dependence of current on voltage. Current, 0.5 1 1.5 amperes Voltage, volts 16

18 power 2 1.5 1 current 0, Values of Y voltage The experimental results show that voltage and current depend on each other, and this dependence is directly proportional, i.e. As the voltage increases, the current increases. b) Dependence of current on conductor resistance at the same voltage. I conducted an experiment with three different conductors at a voltage of 2 volts and received the following results: experiment Resistance of the conductor, Ohm Current in the circuit, A.5 and current 2.5 2 1.5 1 0, Column 1 resistance Experience shows that resistance and current are also dependent quantities; the greater the resistance of the conductor, the less the current. 17

19 People take into account the dependence of current, voltage, and resistance in their daily life, for example: power lines are made of metals with low resistance (copper, aluminum). Task 3. It is impossible to imagine the life of a modern person without alternating electric current, since all appliances: household, electric heating, televisions, computers, etc., operate from an alternating current network. The voltage in our sockets changes according to the following law: u= U max cos(wt), where U max = 308 V, w=314. Construct a graph of the function. Task 4. Draw a detailed three-phase current diagram. Expanded three-phase current diagram. 18

20 Task 5. Present the functional dependence of the electric current signal shape. Electric current signal shapes: 1. constant in the form of a straight line on a time graph; 2. variable sinusoidal harmonic, well described by basic trigonometric relations; 3. a meander, roughly reminiscent of a sinusoid, but with sharp, pronounced angles, which in some cases can be well smoothed out; 4. pulsating, when the direction remains the same without change, and the amplitude fluctuates periodically from zero to maximum value according to a well-defined law. Task 6. In this graph we see a comparative characteristic of the power of household appliances. Which household appliances have the most power? 19

21 Power of household appliances (W) Problem 7. How beneficial is LED lighting compared to conventional incandescent lamps from an economic point of view? Street LED lighting is very beneficial from an economic point of view. Only a small amount of electricity (for example, 100 watts) is enough to illuminate a fairly large area. For comparison: the luminous flux of an LED lamp consuming 100 watts is about Lumen, the same as from 6 conventional incandescent lamps, that is, a saving of more than 80%. 20

22 Task 8. Compile a comparative description of LED lamps, fluorescent and incandescent lamps. Characteristics LED lamp Fluorescent lamp Incandescent lamp Power consumption 5 W 15W 40 W Light output efficiency 90 Lm/W 30 Lm/W 10.5 Lm/W Luminous flux 450 Lm Lm Operating temperature 70 C 60 C 180 C Service life Up to hours Up to hours Up to hours Eco-friendly yes Contains mercury yes 21

23 I. Conclusion In the course of our work, we analyzed and studied the literature on the history of the development of function, and studied, using examples, functional dependence in the life around us. A brief overview of the development of the concept of function leads to the idea that the evolution is far from over and will probably never end, just as the evolution of mathematics in general will never end. New discoveries and inquiries from natural sciences and other sciences will lead to new extensions of the concept of function and other mathematical concepts. Function is one of the basic mathematical and general scientific concepts that expresses the dependence of some variables on others. It played and still plays a big role in understanding the real world. The concept of a function plays an important role, being a basic concept in the study of algebra and the principles of analysis. A special feature of our work is the selection of examples of functional dependencies from everyday life. We realized that there are endlessly many such examples. As a result of the work, we achieved an understanding of the importance of studying mathematics and got the opportunity to show our fellow students the beauty and significance of mathematics. By doing the work, we acquired not only the necessary knowledge, skills and abilities, but also certain personal experience. The theoretical significance of our research work lies in the fact that the research results can be used for technical school students when studying the topic “Function”. The practical significance of my work lies in the fact that the results of the study can be used by college students to improve their educational level when studying the use of functions in the practical activities of an electrician. 22

24 Literature 1. Vilenkin N. Ya. Functions in nature and technology: A book for extracurricular reading, 2nd ed., revised. M.: Education, Nagibin F.F. Mathematical box.- M., Enlightenment, Ulyanovskaya N. N. O, function, how important are you // Mathematics Encyclopedic dictionary of a young mathematician. Compiled by Savin A.P.-M., Enlightenment, Encyclopedic Dictionary of a Young Mathematician. - M.: Pedagogy Rybnikov K.A. The emergence and development of mathematical science, Moscow, Prosveshchenie, 1987. 7. Kolyagin Yu. M. “Algebra grades 10 and 11” - 3rd ed. - M.: Prosveshchenie Publishing House, Glazer G.I. History of mathematics in school: grades 9-10 - M.: Prosveshchenie Internet resources: function.ru

25 Types of functions. Appendix 1 Let's consider the main existing types of functions and their properties. 24

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MUNICIPAL AUTONOMOUS EDUCATIONAL INSTITUTION GAGINSKAYA SECONDARY SCHOOL OF GAGINSKY DISTRICT OF NIZHNY NOVGOROD REGION ADOPTED BY THE PEDAGOGICAL COUNCIL OF MAOU GAGINSKAYA SSH dated 08/29/208 APPROVED BY ORDER

INVESTIGATIVE COMMITTEE OF THE RUSSIAN FEDERATION FEDERAL STATE EDUCATIONAL INSTITUTION "ST. PETERSBURG CADET CORPS OF THE INVESTIGATIVE COMMITTEE OF THE RUSSIAN FEDERATION 199178,

Functions. Domain and set of values. Function graph. Plotting graphs of functions specified in various ways. Properties of functions: monotonicity, even and odd, periodicity, boundedness. Intervals of increasing and decreasing, highest and lowest values, extremum points (local maximum and minimum). Convexity of function. Graphic interpretation. Examples of functional dependencies in real processes and phenomena.

Complex function (composition of functions). Mutually inverse functions. Domain and range of values of the inverse function. Graph of the inverse function. Finding the inverse function of a given one.

The dependence of one variable on another is called functional dependence. The dependence of a variable on a variable is called function, if each value corresponds to a single value.

Designation: .

A variable is called an independent variable or argument, and a variable is called a dependent variable. It is said to be a function of . The value corresponding to the given value is called the function value.

All values that , takes, form the domain of the function; all values that , takes, form the set of function values.

Designations:

– domain of definition of the function;

– range of function values;

– value of the function at point .

– argument values. – function values. If a function is given by a formula, then the domain of definition is considered to consist of all values of the variable for which this formula makes sense.

The graph of a function is the set of all points on the coordinate plane whose abscissas are equal to the values of the argument, and whose ordinates are equal to the corresponding values of the function. If several values (and not just one) correspond to a certain value, then such a correspondence is not a function. In order for a set of points on a coordinate plane to be a graph of a certain function, it is necessary and sufficient that any straight line parallel to the axis intersect with the graph at no more than one point.

Methods for specifying a function

1) The function can be specified analytically as a formula. For example,

2) The function can be specified by a table of many pairs.

3) The function can be specified graphically. Pairs of values are depicted on the coordinate plane.

Definition: A function is said to be even if for any of the domain . The graph of an arbitrary even function is shown in the figure below.

Lecture 3. General concepts and definitions. Classification of functions. Function limit. Infinitely small and infinitely large functions. Basic theorems about infinitesimal functions.

Function

When solving various problems, you usually have to deal with constant and variable quantities.

Definition

A constant quantity is a quantity that retains the same value either in general or in a given process: in the latter case it is called a parameter.

A variable quantity is a quantity that can take on different numerical values.

Concept of function

When studying various phenomena, we usually deal with a set of variable quantities that are interconnected in such a way that the values of some quantities (independent variables) completely determine the values of others (dependent variables and functions).

Definition

A variable quantity y is called a (single-valued) function of a variable quantity x if they are related to each other in such a way that each value of x under consideration corresponds to a single well-defined value of the quantity y (formulated by N.I. Lobachevsky).

Designation y=f(x) (1)

x– independent variable or argument;

y– dependent variable (function);

f– characteristic of the function.

The set of all values of the independent variable for which the function is defined is called the domain of definition or the domain of existence of this function. The domain of definition of a function can be: a segment, a half-interval, an interval, or the entire numerical axis.

Each radius value corresponds to a circle area value. Area is a function of radius defined over an infinite interval

2. Function (2). Function defined at

To visualize the behavior of a function, construct a function graph.

Definition

Function graph y=f(x) is called a set of points M(x,y) plane OXY, whose coordinates are related by this functional dependence. Or the graph of a function is a line whose equation is an equality that defines the function.

For example, the graph of function (2) is a semicircle of radius 2 with its center at the origin.

The simplest functional dependencies

Let's look at a few simple functional dependencies

Direct functional dependence

Definition

Two variables are called directly proportional if when one of them changes in a certain ratio, the other changes in the same ratio.

y=kx, Where k– proportionality coefficient.

Graph of a function

Linear dependence

Definition

Two variable quantities are related by a linear relationship, if , where are some constant quantities.

Graph of a function

Inverse proportional relationship

Definition

Two variables are called inversely proportional if when one of them changes in some ratio, the other changes in the opposite ratio.

Quadratic dependence

The quadratic dependence in the simplest case has the form , where k is some constant value. The graph of a function is a parabola.

Sinusoidal dependence.

When studying periodic phenomena, the sinusoidal dependence plays an important role

- the function is called a harmonic.

A– amplitude;

Frequency;

Initial phase.

The function is periodic with period. Function values at points x And x+T, differing by period, are the same.

The function can be reduced to the form , Where . From here we get that the harmonic graph is a deformed sinusoid with amplitude A and period T, shifted along the OX axis by the amount

Methods for specifying a function

Typically, three ways of specifying a function are considered: analytical, tabular, and graphical.

Analytical method of specifying a function

If a function is expressed using a formula, then it is specified analytically.

For example

If the function y=f(x) is given by a formula, then its characteristic f denotes the set of actions that need to be performed in a certain order on the value of the argument x to get the corresponding function value.

Example . Three actions are performed on the argument value.

Tabular method of specifying a function

This method establishes correspondence between variables using a table. Knowing the analytical expression of a function, we can represent this function for the argument values that interest us using a table.

Is it possible to move from a tabular function assignment to an analytical expression?

Note that the table does not give all the values of the function, and intermediate values of the function can only be found approximately. This is the so-called interpolation functions. Therefore, in the general case, it is impossible to find an exact analytical expression for a function using tabular data. However, it is always possible to construct a formula, and more than one, which, for the values of the argument available in the table, will give the corresponding tabular values of the function. This kind of formula is called interpolation.

Graphical way to specify a function

Analytical and tabular methods do not provide a clear idea of the function.

The graphical method of specifying a function does not have this drawback. y=f(x), when the correspondence between the argument x and function y set using a schedule.

The concept of an implicit function

A function is called explicit if it is given by a formula whose right-hand side does not contain the dependent variable.

Function y from argument x is called implicit if it is given by the equation

F(x,y)=0(1) unresolved regarding the dependent variable.

Concept of inverse function

Let the function be given y=f(x)(1). By specifying the values of the argument x, we obtain the values of the function y.

It is possible, considering y argument, and X– function, set values y and get values x. In this case, equation (1) will determine x, as an implicit function of y. This last function is called reverse in relation to this function y.

Assuming that equation (1) is resolved with respect to x, we obtain an explicit expression for the inverse function

(2), where the function for all valid values y satisfies the condition

Accurate mathematical analysis of socio-economic problems is always based on the concept of function. A function is a rule by which the elements of one numerical set are compared with the elements of another numerical set. The functions are designated as:

x is the argument of the function, the explanatory or independent variable, y is the value of the function, the explained or dependent variable.

There are four ways to define a function: tabular, graphical, analytical, algorithmic. Some of the listed ways of specifying functions (sometimes called “functional dependencies”) will be discussed below using specific examples.

Example 1: The results of studying seasonal demand for some goods are given in the table.

	Period of time	Demand, thousand pieces

For each specific point in time, the table indicates the value of demand for the product at that point in time, that is, the rule is specified in a table.

Example 2: Studying the demand for a certain product depending on changes in its price made it possible to construct the following graphical dependence:

The graphical method of presentation is obviously the most convenient in terms of clarity of data presentation, but the least convenient in terms of accuracy.

Example 3. Food consumption Y depending on family income X can be described by the following relationship

Example 4. Inventory management costs U They consist of storage costs and delivery costs. In turn, each term depends on the volume of the commodity mass

Where A, b - coefficients characterizing the conditions of storage and delivery of goods.

Example 5: In microeconomics, which studies consumer behavior in the market for goods and services, the utility function is widely used U. For the case of two goods, for example, tea and coffee, it may look like:

Here U1, U2 - volumes of consumption of each type of product.

The examples given are enough to draw some conclusions.

Firstly, Analytical dependencies (formulas) are completely different, but they all consist of a finite number of simple dependencies, which are called basic elementary functions .

Figure 1 shows the six main elementary functions (in order from left to right):

1. Linear -

2. Quadratic

3. Hyperbole

4. Logarithmic

5. Demonstrative

6. Power

Functions used in practice consist of combinations of several elementary ones and are built according to the “function from function” principle. For example, let z = F(y). In turn, the variable U is also a function depending on x - that is, y = F(x). Then they say that the function z is a complex function of the form Z = F(f(x)).

It can be noted that a function can depend on one variable, then it can be depicted on a plane in the form of a graph in a coordinate system. If there are two arguments, like the utility function, then it can be depicted on the plane as a set of level lines (see Fig. 2).

Secondly , Each of the listed methods of specifying a function does not exclude any other. They just complement each other. In some problems it is more convenient to use an analytical dependence, while in others it is more convenient to use a graphical one.

In addition, it can be noted that it is possible to make a transition from one method of representing functions to another.

IN- Third, functional dependencies are useful not only because they allow one to calculate the variable being explained for given values of the explanatory variables, but also because they allow one to identify the qualitative features of the phenomenon being described.

For example, let’s study how food costs change (see example 2) if income increases by the amount Dx. The new cost value will be determined as

Thus, it is obvious that the increase in food costs does not depend on income X, and depends only on the increase in income Dx. This means that if an individual received 800 rubles. per month and his salary increases by 20 rubles, then from this amount he will allocate an additional 14 rubles for food. Another person will do the same with a salary of 1600 rubles. per month if his salary also increases by 20 rubles.

Task No. 16

Interpretation of a real dependence graph.

Examples of dependency graphs surrounding real processes; reading and interpretation.

This task tests the ability to analyze graphs of functions that describe the dependence of m / y quantities.

Theory.

Definition.

1. Function is law, by which each element value x from some set X a single element is matched y from many Y.

2. The dependence of the variable y on the variable x is called a function if each value of x corresponds to a single value of y. The variable x is called the independent variable or argument, and the variable y is called the dependent variable. The value of y corresponding to a given value of x is called the value of the function.

Write down: y = f (x). The letter f denotes this function, that is, the functional relationship between the variables x and y; f(x) is function value, corresponding to the value argument X. They also say that f(x) is the value of the function at point x. All values that the independent variable takes form domain of a function. All values that the function f (x) takes (for x belonging to its domain of definition) form function range.

Methods for specifying a function

To specify a function, you must specify a way in which, for each argument value, the corresponding function value can be found. The most common way to specify a function is using the formula y = f (x),
where f (x) is some expression with variable x. In this case, they say that the function is given by a formula or that the function is given analytically.

In practice, the tabular method of specifying a function is often used. With this method, a table is provided indicating the function values for the argument values available in the table.

Function graph.

Let the function be given analytically by the formula y = f (x). If on the coordinate plane we mark all points that have the following property: the abscissa of the point belongs to the domain of definition of the function, and the ordinate is equal to the corresponding value of the function, then the set of points (x; f (x)) is the graph of the function.

In practice, to construct a graph of a function, they compile a table of function values for certain values of the argument, plot the corresponding points on the plane and connect the resulting points with a line. In this case, it is assumed that the graph of the function is a smooth line, and the found points quite accurately show the progress of the change in the function. The advantage of a graphic image over a tabular one is its clarity and easy visibility; The disadvantage is the low degree of accuracy. A successful choice of scales is of great practical importance. Using the graph, you can find (approximately) the value of the function for those argument values that are not included in the table.