Definition of a function. What is a Mathematical Function Functions Related to Division

Excel has a function for finding random numbers =RAND(). The opportunity to find random number in Excel, an important component of planning or analysis, because you can predict the results of your model on a large amount of data, or simply find one random number to test your formula or experience.

We continue the series of articles about mathematical formulas in Excel. Today we will analyze the formula for writing a “module in Excel”. The modulus of a number is used to determine the absolute value of a number, for example the length of a segment. Below we provide several ways to calculate the modulus of a number in Excel, the main function is ABS, and additional calculations using the IF and SQRT functions.

We touched a little on the topic of exponents in the article about rounding large numbers. In this article we will discuss what an exponent is in Excel and, most importantly, why it can be useful in everyday life or in business.

Do you need to assign a number to each number in Excel so that you can sort them by that number? You can come up with complex constructions for text data, but for numeric data there is a special RANK function in Excel. It is classified as a static function and can be quite useful. In the article we also talk about new functions from Excel 2010 RANK.CP() […]

We continue our review of mathematical functions and possibilities. Today we're looking at the simplest formula - a degree in Excel. Raising to a power (root) by a function or simple notation, negative power. How to write down a degree beautifully will also be here. Everything is simple in principle, but this does not mean that you do not need to write an article about it. Moreover, one large article covering everything [...]

I realized that there are very few descriptions of mathematical functions on our website. Although there are a great many of them in Excel. There is a description of VAT, all sorts of printed documents and forms. But there is almost no description of the basics of the spreadsheet editor - mathematical functions. “We need to address this gap,” I thought. That's what I'm doing. First of all, factorial. Why? Just the other day, I was doing [...]

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Management is an important part of the modern socio-economic system. It is characterized by the influence of the subject in management on the object of management. In simple terms, management is management.

Processes that are inextricably linked with management in one way or another usually occur in an enterprise on the basis of the so-called functional distribution. The essence of management activities and provide management functions

Main functions

Today, the most important functions of management are planning, organizing, motivating, coordinating, and controlling.

Previously, in Russia, management functions were somewhat different and included such concepts as control, regulation, stimulation, coordination, organization and planning.

It is also worth highlighting the version presented by American scientists Michael Meskon, Michael Albert and Franklin Khedouri.

They even identified only four functions of management: planning, organization, motivation, control.

The listed management functions are somehow related to decision-making processes and communication, that is, communication.

Today, the option of having an even wider list of management functions is most often considered.

1. The first thing you need to do is set a goal. (To do this, you need to answer the question “What do I want?”).
2. The next stage is planning. Planning consists of a step-by-step description of the steps that are necessary to achieve a particular goal.
3. We should also not forget about marketing. To do this, it is necessary to answer questions such as “What do I have and what of this can help or hinder me on the path to achieving my goal?”
4. The issue should also be resolved with the organization. To do this, you need to answer the questions about “Where and what is located and how best to connect it all?”
5. New information. (“What achievements can you use to achieve your goal as soon as possible?”)
6. The issue of incentives in some cases plays a decisive role. In order to answer it, the question should be asked: “What needs to be done to ensure that the performers exactly fulfill all the requirements I prescribed?” However, you should remember that stimulation is not motivation, since motivation is a whole set of different internal motives for an individual person.
7. We must not forget about the issue of coordination. Coordination represents the results of individual performers, who must give one or another overall result. It is also desirable that there are no additional modifications.
8. We should not forget about the issue of control. “Is everything going exactly as planned?”).
9. Analysis and accounting. (Questions: “What happened in the end?” + “Was the goal achieved?” + “What hindered, and what, on the contrary, helped?” and many others).

The most important function in management is the planning function.

What is it and what is it needed for? By implementing this function, the entrepreneur, based on the analysis obtained, can formulate certain plans or programs. The planning process itself can allow you to formulate the goal much more clearly.

After this, you can try to use the results obtained to ensure better coordination of the efforts of all structural divisions of your company. This means that planning is one of the continuous processes of exploring new opportunities and methods for improving the company’s activities due to the fact that the manager is able to identify a number of new opportunities and factors in its activities.

It follows from this that the organization's plans will not be prescriptive in nature. Moreover, they will change only in accordance with a particular situation.

Function of the organization necessary to form the structure of the company. In addition, it is needed in order to provide it with everything necessary, for example, financial resources. In the plan that the organization draws up, there is the creation of conditions in order to achieve the planned goal.

Motivation function allows you to activate company employees so that they work better and more efficiently. This will improve the productivity of the entire company. The simplest method for motivating employees is to provide special cash bonuses for achieving certain goals.

Control function necessary to achieve the company's goals. It is important to understand that control must be comprehensive, otherwise there will be practically no benefit from it.

Coordination function is to establish interaction between various structures of the organization to improve the efficiency of the entire company.

Function- one of the most important concepts of mathematics, the initial concept of its leading field - mathematical analysis. School mathematics courses focus on numerical functions. The reason for this is the close connection of mathematics with the natural sciences, in particular with physics, for which numerical functions serve as a means of quantitatively describing various dependencies between quantities.

In the initial course of mathematics, the concept of function and everything connected with it is not explicitly studied, but the idea of ​​functional dependence literally permeates it, and a correct understanding of such properties of real phenomena as interdependence and changeability is the basis of the scientific worldview. Of course, all this requires from the primary school teacher certain knowledge about the function and its properties, and, above all, such knowledge that will help him carry out propaedeutics of the concept of function in primary school.

44. Concept of function. Methods for specifying functions

Let's complete two tasks for younger schoolchildren.

1) Increase every odd single-digit number by 2 times.

2) Fill out the table.

Minuend
Subtrahend
Difference

What mathematical concepts are we dealing with in these assignments?

First of all, in each task there are two numerical sets, between the elements of which a correspondence is established. In the first, these are the sets (1, 3, 5, 7) and (2, 6, 10, 14), and in the second, this is the set of values ​​of the subtrahend (0,1,2, 3,4, 5) and the set of difference values (5, 4, 3, 2, 1, 0). What is the similarity between the correspondences established between these sets? In both the first and second tasks, each number from the first set is associated with a single number from the second. In mathematics, such correspondences are called functions. In general terms, the concept of a numerical function is defined as follows:

Definition. A numerical function is a correspondence between a numerical set X and a set R of real numbers, in which each number from the set X is associated with a single number from the set R.

The set X is called domain of definition of the function.

Functions are usually denoted by the letters f, g, h, etc. If f is a function defined on a set X, then the real number y corresponding to the number x from the set X is often denoted by f(x) and written y= f(x). The variable x is called argument (or independent variable) of the function f. The set of numbers of the form f(x) for all x from the set X is called function range f.

In the first example discussed above, the function is defined on the set X = (1, 3, 5, 7) - this is its domain of definition. And the range of values ​​of this function is the set (2,6,10,14).

From the definition of a function it follows that to specify a function it is necessary to indicate, firstly, a numerical set X, i.e. the domain of definition of the function, and, secondly, the rule according to which each number from the set X corresponds to a unique real number.

Often functions are specified using formulas that indicate how to find the corresponding function value from a given argument value. For example, the formulas y = 2x-3, y = x 2, y = 3x, where x is a real number, define functions, since each real value of x can be associated with a single value y by performing the actions specified in the formula.

Note that using the same formula you can specify as many functions as you like, which will differ from each other in their domain of definition. For example, the function y = 2x-3, where x R, is different from the function y = 2x-3, where x N. Indeed, at x = -5 the value of the first function is -13, and the value of the second at x = -5 is undefined .

Often, when specifying a function using a formula, its domain is not specified. In such cases, it is considered that the domain of definition of the function f(x) is the domain of definition of the expression f(x). For example, if a function is given by the formula y = 2x-3, then its domain of definition is considered to be the set R of real numbers. If a function is given by the formula y =, then its domain of definition is the set R of real numbers, excluding the number 2 (if x = 2, then the denominator of this fraction becomes zero).

Numerical functions can be represented visually on the coordinate plane. Let y = f(x) be a function with domain X. Then its schedule is the set of points in the coordinate plane that have an abscissa x and an ordinate f(x) for all x from the set X.

Thus, the graph of the function y = 2x-3, defined on the set R, is a straight line (Fig. 1), and the graph of the function y = x 2, also defined on the set R, is a parabola (Fig. 2).

Fig.1 Fig.2

Functions can be specified using a graph. For example, the graphs shown in Figure 3 define functions, one of which has the interval [-2, 3] as its domain of definition, and the second has a finite set (-2, -1,0, 1, 2, 3).

Not every set of points on a coordinate plane represents a graph of some function. Since for each value of the argument from the domain of definition the function must have only one value, then any straight line parallel to the ordinate axis either does not intersect the graph of the function at all, or intersects it only at one point. If this condition is not met, then the set of points on the coordinate plane does not define the graph of the function. For example, the curve in Figure 4 is not a graph of a function - straight line AB, parallel to the ordinate axis, intersects it at two points. Functions can be specified using a table.

For example, the table below describes the dependence of air temperature on the time of day. This dependence is a function, since each time value t corresponds to a single value of air temperature p?;

Numeric functions have many properties. We will consider one of them - the property of monotonicity, since the teacher’s understanding of this property is important when teaching mathematics to primary schoolchildren.

Definition. A function f is called monotonic on a certain interval A if it increases or decreases on this interval.

Definition. A function f is said to be increasing on a certain interval A if for any numbers x 1, x 2 from the set A the following condition is satisfied:

x 1<х 2 f(x 1)

The graph of a function increasing over interval A has a peculiarity: when moving along the abscissa axis from left to right along interval A, the ordinates of the graph points increase (Fig. 5).

Rice. 5 Fig.6

Definition. A function f is said to be decreasing on a certain interval A if for any numbers x1, x2 from the set A the following condition is satisfied:

x 1<х 2 f(x 1)>f(x 2).

The graph of a function decreasing on interval A has a peculiarity: when moving along the abscissa axis from left to right along interval A, the ordinates of the graph points decrease (Fig. 6).

Concept functions- one of the main ones in mathematics.

You often hear this word in math lessons. You build graphs of functions, study the function, find the largest or smallest value of the function. But to understand all these actions, let's define what a function is.

A function can be defined in several ways. They will all complement each other.

1. Function is dependence of one variable on another. In other words, relationship between quantities.

Any physical law, any formula reflects such a relationship between quantities. For example, the formula is the dependence of fluid pressure on depth.

The greater the depth, the greater the fluid pressure. We can say that the pressure of a fluid is a function of the depth at which it is measured.

The designation you are familiar with precisely expresses the idea of ​​such a dependence of one quantity on another. The value of y depends on the value according to a certain law, or rule, denoted by .

In other words: we change (the independent variable, or argument) – and according to a certain rule it changes.

It is not necessary to designate the variables and . For example, the dependence of length on temperature, that is, the law of thermal expansion. The notation itself means that the value depends on .

2. Another definition can be given.

A function is a specific action over the variable.

This means that we take a value, do a certain action with it (for example, square it or calculate its logarithm) - and get the value.

In the technical literature there is a definition of a function as a device whose input is supplied and the output is obtained.

So the function is action over the variable. In this meaning, the word “function” is also used in areas far from mathematics. For example, you can talk about the functions of a mobile phone, the functions of the brain, or the functions of a deputy. In all these cases, we are talking about the actions being performed.

3. Let's give another definition of a function - the one that is most often found in textbooks.

A function is a correspondence between two sets, with each element of the first set corresponding to one and only one element of the second set.

For example, the function assigns to each real number a number twice as large as .

Let us repeat once again: for each element of the set, according to a certain rule, we associate an element of the set. The set is called domain of the function. A bunch of - range of values.

But why is there such a long clarification here: “each element of the first set corresponds to one and only one element of the second”? It turns out that the correspondences between sets are also different.

Let us consider, as an example, the correspondence between two sets—Russian citizens who have passports and their passport numbers. It is clear that this correspondence is one-to-one - each citizen has only one Russian passport. And vice versa - you can find a person by passport number.

In mathematics there are also such one-to-one functions. For example, a linear function. Each value corresponds to one and only one value. And vice versa - knowing , you can definitely find .

There may be other types of correspondences between sets. Let's take for example a group of friends and the months in which they were born:

Every person was born in a specific month. But this correspondence is not one-to-one. For example, Sergei and Oleg were born in June.

An example of such a correspondence in mathematics is the function. One and the same element of the second set corresponds to two different elements of the first set: and .

What should be the correspondence between two sets so that it is not a function? Very simple! Let's take the same group of friends and their hobbies:

We see that in the first set there are elements that correspond to two or three elements from the second set.

It would be very difficult to describe such a correspondence mathematically, wouldn’t it?

Here's another example. The pictures show curves. Which one do you think is a graph of a function and which one is not?

The answer is obvious. The first curve is a graph of some function, and the second is not. After all, there are points on it where each value corresponds to not one, but three values.

Let's list ways to specify a function.

1 . Using formula. This is a convenient and familiar way for us. For example:

These are examples of functions given by formulas.

2. Graphic method. It is the most visual. The graph shows everything at once - the increase and decrease of the function, the highest and lowest values, the maximum and minimum points. The next article will talk about studying a function using a graph.

In addition, it is not always easy to derive the exact formula of a function. For example, the dollar exchange rate (that is, the dependence of the value of the dollar on time) can only be shown on a chart.

3. Using a table. You once started studying the topic “Function” with this method - you built a table and only after that - a graph. And in the experimental study of any new pattern, when neither the formula nor the graph are yet known, this method will be the only possible one.

4 . Using a description. It happens that in different areas a function is given by different formulas. A function you know is given by a description.