Valid function values. Function range (set of function values)

Many problems lead us to search for a set of function values ​​on a certain segment or throughout the entire domain of definition. Such tasks include various evaluations of expressions and solving inequalities.

In this article, we will define the range of values ​​of a function, consider methods for finding it, and analyze in detail the solution of examples from simple to more complex. All material will be provided with graphic illustrations for clarity. So this article is a detailed answer to the question of how to find the range of a function.

Definition.

The set of values ​​of the function y = f(x) on the interval X is the set of all values ​​of a function that it takes when iterating over all .

Definition.

Function range y = f(x) is the set of all values ​​of a function that it takes when iterating over all x from the domain of definition.

The range of the function is denoted as E(f) .

The range of a function and the set of values ​​of a function are not the same thing. We will consider these concepts equivalent if the interval X when finding the set of values ​​of the function y = f(x) coincides with the domain of definition of the function.

Also, do not confuse the range of the function with the variable x for the expression on the right side of the equality y=f(x) . The range of permissible values ​​of the variable x for the expression f(x) is the domain of definition of the function y=f(x) .

The figure shows several examples.

Graphs of functions are shown with thick blue lines, thin red lines are asymptotes, red dots and lines on the Oy axis show the range of values ​​of the corresponding function.

As you can see, the range of values ​​of a function is obtained by projecting the graph of the function onto the y-axis. It can be one single number (first case), a set of numbers (second case), a segment (third case), an interval (fourth case), an open ray (fifth case), a union (sixth case), etc.

So what do you need to do to find the range of values ​​of a function?

Let's start with the simplest case: we will show how to determine the set of values ​​of a continuous function y = f(x) on the segment.

It is known that a function continuous on an interval reaches its maximum and minimum values ​​on it. Thus, the set of values ​​of the original function on the segment will be the segment . Consequently, our task comes down to finding the largest and smallest values ​​of the function on the segment.

For example, let's find the range of values ​​of the arcsine function.

Example.

Specify the range of the function y = arcsinx .

Solution.

The area of ​​definition of the arcsine is the segment [-1; 1] . Let's find the largest and smallest value of the function on this segment.

The derivative is positive for all x from the interval (-1; 1), that is, the arcsine function increases over the entire domain of definition. Consequently, it takes the smallest value at x = -1, and the largest at x = 1.

We have obtained the range of arcsine function .

Example.

Find the set of function values on the segment.

Solution.

Let's find the largest and smallest value of the function on a given segment.

Let us determine the extremum points belonging to the segment:

We calculate the values ​​of the original function at the ends of the segment and at points :

Therefore, the set of values ​​of a function on an interval is the interval .

Now we will show how to find the set of values ​​of a continuous function y = f(x) in the intervals (a; b) , .

First, we determine the extremum points, extrema of the function, intervals of increase and decrease of the function on a given interval. Next, we calculate at the ends of the interval and (or) the limits at infinity (that is, we study the behavior of the function at the boundaries of the interval or at infinity). This information is enough to find the set of function values ​​on such intervals.

Example.

Define the set of function values ​​on the interval (-2; 2) .

Solution.

Let's find the extremum points of the function falling on the interval (-2; 2):

Dot x = 0 is a maximum point, since the derivative changes sign from plus to minus when passing through it, and the graph of the function goes from increasing to decreasing.

there is a corresponding maximum of the function.

Let's find out the behavior of the function as x tends to -2 on the right and as x tends to 2 on the left, that is, we find one-sided limits:

What we got: when the argument changes from -2 to zero, the function values ​​increase from minus infinity to minus one-fourth (the maximum of the function at x = 0), when the argument changes from zero to 2, the function values ​​decrease to minus infinity. Thus, the set of function values ​​on the interval (-2; 2) is .

Example.

Specify the set of values ​​of the tangent function y = tgx on the interval.

Solution.

The derivative of the tangent function on the interval is positive , which indicates an increase in function. Let's study the behavior of the function at the boundaries of the interval:

Thus, when the argument changes from to, the function values ​​increase from minus infinity to plus infinity, that is, the set of tangent values ​​on this interval is the set of all real numbers.

Example.

Find the range of the natural logarithm function y = lnx.

Solution.

The natural logarithm function is defined for positive values ​​of the argument . On this interval the derivative is positive , this indicates an increase in the function on it. Let's find the one-sided limit of the function as the argument tends to zero on the right, and the limit as x tends to plus infinity:

We see that as x changes from zero to plus infinity, the values ​​of the function increase from minus infinity to plus infinity. Therefore, the range of the natural logarithm function is the entire set of real numbers.

Example.

Solution.

This function is defined for all real values ​​of x. Let us determine the extremum points, as well as the intervals of increase and decrease of the function.

Consequently, the function decreases at , increases at , x = 0 is the maximum point, the corresponding maximum of the function.

Let's look at the behavior of the function at infinity:

Thus, at infinity the values ​​of the function asymptotically approach zero.

We found out that when the argument changes from minus infinity to zero (the maximum point), the function values ​​increase from zero to nine (to the maximum of the function), and when x changes from zero to plus infinity, the function values ​​decrease from nine to zero.

Look at the schematic drawing.

Now it is clearly visible that the range of values ​​of the function is .

Finding the set of values ​​of the function y = f(x) on intervals requires similar research. We will not dwell on these cases in detail now. We will meet them again in the examples below.

Let the domain of definition of the function y = f(x) be the union of several intervals. When finding the range of values ​​of such a function, the sets of values ​​on each interval are determined and their union is taken.

Example.

Find the range of the function.

Solution.

The denominator of our function should not go to zero, that is, .

First, let's find the set of function values ​​on the open ray.

Derivative of a function is negative on this interval, that is, the function decreases on it.

We found that as the argument tends to minus infinity, the function values ​​asymptotically approach unity. When x changes from minus infinity to two, the values ​​of the function decrease from one to minus infinity, that is, on the interval under consideration, the function takes on a set of values. We do not include unity, since the values ​​of the function do not reach it, but only asymptotically tend to it at minus infinity.

We proceed similarly for the open beam.

On this interval the function also decreases.

The set of function values ​​on this interval is the set .

Thus, the desired range of values ​​of the function is the union of the sets and .

Graphic illustration.

Special attention should be paid to periodic functions. The range of values ​​of periodic functions coincides with the set of values ​​on the interval corresponding to the period of this function.

Example.

Find the range of the sine function y = sinx.

Solution.

This function is periodic with a period of two pi. Let's take a segment and define the set of values ​​​​on it.

The segment contains two extremum points and .

We calculate the values ​​of the function at these points and on the boundaries of the segment, select the smallest and highest value:

Hence, .

Example.

Find the range of a function .

Solution.

We know that the arc cosine range is the segment from zero to pi, that is, or in another post. Function can be obtained from arccosx by shifting and stretching along the abscissa axis. Such transformations do not affect the range of values, therefore, . Function obtained from stretching three times along the Oy axis, that is, . And the last stage of transformation is a shift of four units down along the ordinate. This leads us to double inequality

Thus, the required range of values ​​is .

Let us give the solution to another example, but without explanations (they are not required, since they are completely similar).

Example.

Define Function Range .

Solution.

Let us write the original function in the form . The range of values ​​of the power function is the interval. That is, . Then

Hence, .

To complete the picture, we should talk about finding the range of values ​​of a function that is not continuous on the domain of definition. In this case, we divide the domain of definition into intervals by break points, and find sets of values ​​on each of them. By combining the resulting sets of values, we obtain the range of values ​​of the original function. We recommend you remember

Definition
Function y = f (x) is called a law (rule, mapping), according to which, each element x of the set X is associated with one and only one element y of the set Y.

The set X is called domain of the function.
Set of elements y ∈ Y, which have preimages in the set X, is called set of function values(or range of values).

Domain functions are sometimes called definition set or many tasks functions.

Element x ∈ X called function argument or independent variable.
Element y ∈ Y called function value or dependent variable.

The mapping f itself is called characteristic of the function.

The characteristic f has the property that if two elements and from the definition set have equal values: , then .

The symbol denoting the characteristic may be the same as the symbol of the function value element. That is, you can write it like this: . It is worth remembering that y is an element from the set of function values, and is the rule by which the element x is associated with the element y.

The process of calculating a function itself consists of three steps. In the first step, we select an element x from the set X. Next, using the rule, the element x is associated with an element of the set Y. In the third step, this element is assigned to the variable y.

Private value of the function call the value of a function given a selected (particular) value of its argument.

Graph of function f called a set of pairs.

Complex functions

Definition
Let the functions and be given. Moreover, the domain of definition of the function f contains a set of values ​​of the function g. Then each element t from the domain of definition of the function g corresponds to an element x, and this x corresponds to y. This correspondence is called complex function: .

A complex function is also called composition or superposition of functions and sometimes denoted as follows: .

In mathematical analysis, it is generally accepted that if a characteristic of a function is denoted by one letter or symbol, then it specifies the same correspondence. However, in other disciplines, there is another way of notation, according to which mappings with the same characteristic, but different arguments, are considered different. That is, the mappings are considered different. Let's give an example from physics. Let's say we consider the dependence of momentum on coordinates. And let us have a dependence of coordinates on time. Then the dependence of the impulse on time is a complex function. But, for brevity, it is designated as follows: . With this approach, and - this various functions. At identical values arguments they can give different meanings. This notation is not accepted in mathematics. If a reduction is required, you should enter new characteristic. For example . Then it is clearly visible that and is different functions.

Valid functions

The domain of a function and the set of its values ​​can be any set.
For example, number sequences are functions whose domain is the set of natural numbers, and the set of values ​​is real or complex numbers.
The cross product is also a function, since for two vectors and there is only one value of the vector. Here the domain of definition is the set of all possible pairs of vectors. The set of values ​​is the set of all vectors.
A Boolean expression is a function. Its domain of definition is the set of real numbers (or any set in which the comparison operation with the element “0” is defined). The set of values ​​consists of two elements - “true” and “false”.

Numerical functions play an important role in mathematical analysis.

Numeric function is a function whose values ​​are real or complex numbers.

Real or real function is a function whose values ​​are real numbers.

Maximum and minimum

Real numbers have a comparison operation. Therefore, the set of values ​​of a real function can be limited and have the largest and smallest values.

The actual function is called limited from above (from below), if there is a number M such that the inequality holds for all:
.

The number function is called limited, if there is a number M such that for all:
.

Maximum M (minimum m) function f, on some set X, the value of the function is called for a certain value of its argument, for which for all,
.

Top edge or exact upper bound A real function bounded above is the smallest number that bounds its range of values ​​from above. That is, this is a number s for which, for everyone and for any, there is an argument whose function value exceeds s′: .
The upper bound of a function can be denoted as follows:
.

The upper bound of an upper bounded function

Bottom edge or exact lower limit A real function bounded from below is the largest number that bounds its range of values ​​from below. That is, this is a number i for which, for everyone and for any, there is an argument whose function value is less than i′: .
The infimum of a function can be denoted as follows:
.

The infimum of a lower bounded function is the point at infinity.

Thus, any real function, on a non-empty set X, has an upper and lower bound. But not every function has a maximum and a minimum.

As an example, consider a function defined on an open interval.
It is limited, on this interval, from above by the value 1 and below - the value 0 :
for all .
This function has an upper and lower bound:
.
But it has no maximum and minimum.

If we consider the same function on the segment, then on this set it is bounded above and below, has an upper and lower bound and has a maximum and a minimum:
for all ;
;
.

Monotonic functions

Definitions of increasing and decreasing functions
Let the function be defined on some set of real numbers X. The function is called strictly increasing (strictly decreasing)
.
The function is called non-decreasing (non-increasing), if for all such that the following inequality holds:
.

Definition of a monotonic function
The function is called monotonous, if it is non-decreasing or non-increasing.

Multivalued functions

An example of a multivalued function. Its branches are indicated by different colors. Each branch is a function.

As follows from the definition of the function, each element x from the domain of definition is associated with only one element from the set of values. But there are mappings in which the element x has several or infinite number images

As an example, consider the function arcsine: . It is the inverse of the function sinus and is determined from the equation:
(1) .
For a given value of the independent variable x, belonging to the interval, this equation is satisfied by infinitely many values ​​of y (see figure).

Let us impose a restriction on the solutions of equation (1). Let
(2) .
Under this condition, a given value corresponds to only one solution to equation (1). That is, the correspondence defined by equation (1) under condition (2) is a function.

Instead of condition (2), you can impose any other condition of the form:
(2.n) ,
where n is an integer. As a result, for each value of n, we will get our own function, different from others. Many similar functions are multivalued function. And the function determined from (1) under condition (2.n) is branch of a multivalued function.

This is a set of functions defined on a certain set.

Multivalued function branch is one of the functions included in the multi-valued function.

Single-valued function is a function.

References:
O.I. Besov. Lectures on mathematical analysis. Part 1. Moscow, 2004.
L.D. Kudryavtsev. Course of mathematical analysis. Volume 1. Moscow, 2003.
CM. Nikolsky. Course of mathematical analysis. Volume 1. Moscow, 1983.

Meaning is one of the most controversial and controversial issues in language theory. The issue of determining the meaning of a word (meaning lexical meaning) is widely covered in the works of domestic and foreign linguists. However, despite its centuries-old history, it has still not received not only a generally accepted, but even a sufficiently clear answer.

In modern linguistics, two approaches to the problem of determining meaning can be distinguished: referential(referential) and functional(functional). Scientists who adhere to the referential approach strive to describe meaning as a component of a word with the help of which a concept is conveyed, and which thus gives the word the ability to objectively reflect existing reality, designate objects, qualities, actions and abstract concepts. Proponents of the functional approach study the functions of words in speech and pay less attention to the question “what is meaning?” than to “what functions does meaning?” Ginzburg R.Z., Khidekel S.S., Kyazeva G.Yu. Lexicology in English. - M., 1979 (in Engl.) - P. 13..

All the major works on the theory of semantics have so far been based on the referential approach. The central idea of ​​this approach is to identify three factors that characterize the meaning of a word: “the word (the symbol)” (sound form of the word), “the mental content” (concept) and “the referent” (the term “referent” - that object (action) , quality), which means the word). In accordance with this approach, meaning is understood as a complex whole, consisting of a designated object and a concept about this object. This relationship is presented by scientists in the form of a schematic image, namely triangles that differ slightly from each other. The most famous is the Ogden-Richards triangle. Stern G. Meaning and change of meaning with special reference to the English language. - Goeteborg, 1931, - P.45., given in the book of the German linguist Gustav Stern “Meaning and change of meaning with special reference to the English language”.

Thought or reference

(the mental content)

Symbol Referent

The term “symbol” here refers to the word; “thought” or “reference” is a concept. The dotted line means that there is no direct connection between the referent and the word: it is established only with the help of the concept. German linguist Gustav Stern argues that the meaning of a word is completely determined by its relationship with its three factors: word, referent and concept. In accordance with the above, G. Stern offers the following definition of the meaning of a word: “The meaning of a word in real speech is identical with those elements of the subjective understanding of the object denoted by the word by the speaker or listener, which, in their opinion, are expressed by this word” Stern G. Ibid. - P. 37..

S. Ulmann Ulmann S. Words and their use. - L., 1951. - P. 32-33., defining the meaning, proposes to simplify the terminology and replaces “symbol” with “name” (name), and “thought or reference” with “sense” (meaning). He also proposes to exclude the term “referent” from the definition, explaining this by the lack of a direct connection between the word and the referent and trying to explain in more detail the connection between the two key terms - name and meaning. The scientist emphasizes the two-way connection between a word and the concept that this word denotes. Not only does a word spoken or written bring to mind the corresponding concept, but the very concept that comes to mind forces us to find the appropriate word. When I think of a table, I will definitely name the word “table”, and also, when I hear the word “table”, I will definitely imagine it. Thus, Ullman arrives at this definition of meaning: Meaning is a reciprocal relationship between the name and the sense, which enables the one to call up the other.(Meaning is a two-way relationship between name and meaning (word and concept), which, when mentioned, allows the first to instantly recall the second and vice versa).

A.I. Smirnitsky Smirnitsky A.I. Lexicology of the English language. - M., 1956. - P. 149-152. states that the meaning of a word cannot be identified with either its referent, i.e. the object that it denotes, nor with the sound of this word. Taking into account the above, he offers the following definition of the meaning of a word: the meaning of a word is this is a known representation of an object, phenomenon or relationship in consciousness(or a mental formation similar in nature, constructed from reflections individual elements reality - mermaid, goblin, witch, etc.), included in the structure of a word as its so-called internal side in relation to which the sound of the word acts as a material shell, necessary not only for expressing meaning and for communicating it to other people, but also for its very emergence, formation, existence and development.

In contrast to proper names, pronouns do not name anything, but only point to someone or something, mainly revealing its relationship to the speaker: you, my, that, hers. The meaning of pronouns is extremely general.

Interjections do not name or indicate anything. Their meaning lies in the fact that they express, not concepts, but the feelings and will of the speaker. An interjection can express a feeling in general: Oh! Ah! Dear me! oh my! oh dear! Or some specific feeling, for example: despondency (alas!), annoyance (damn!), approval (hear! hear!), disdain (pooh!), surprise (gosh!), etc. Imperative, i.e. expressing will, interjections can be a call to calm down or shut up: come, come! easy! there, there! hush! etc.

Although they have meaning, proper names, pronouns and interjections do not express concepts.

The meaning is determined not only by the connection of the word with objects of reality, but also by the place of the word in the system of this language. (Comparing lexical systems different languages, we see that the nature and essence of the dependence of meaning on the structure of language becomes especially obvious). Based on this, the meaning of a word can be defined as the mental content assigned to a given sound form, conditioned by the system of a given language, and common to a given language community. Arbekova T.I. Lexicology of the English language: Practical. well. - M., 1977. - P. 52-53.

The meaning of words is determined by the entire lexical-semantic system of the language and is the result of a reflection of socially conscious objective reality. Lexical meaning is formed under the conditions of specific connections and relationships between words of a given language. Unlike concepts that are common to different languages, the lexical meaning of a word is always nationally specific, like all vocabulary in general. In addition to the concept it expresses, the meaning of a word may also include other components: emotional connotation, stylistic characteristics, correlation with other words of the same language. It is layered with additional ideas and various kinds of semantic associations. Depending on which part of speech a word belongs to, its lexical meaning is associated with a certain range of grammatical meanings and may be influenced by them, so that each part of speech has its own semantic features. The non-identity of meaning and concept is also manifested in the fact that one concept can be expressed by the meaning of two or more words, and, conversely, one polysemantic word can, in its meanings, unite a whole group of interconnected concepts. The lexical meaning of a word may, finally, not coincide with the concept in scope or content. Arnold I.V. Decree. op. - P. 55.

1. Suggestibility is associated with general personal and intellectual immaturity, has a certain functional role in ontogenesis as a factor in primary, not yet interiorized, interpsychic relationships between people (V.N. Kulikov).

2. Suggestibility is a trait of a hysterical personality, which is characterized by imitative forms of hysterical behavior (A. Yakubik).

3. Suggestibility is a personality trait associated with intellectual deficiency, a negative attitude of the subject towards himself, lack of self-confidence, low self-esteem - which determine the orientation in behavior towards the opinions and assessments of other people.

4. Suggestibility is a relative trait that manifests itself in a significant situation; what is personally significant is more often taken for granted (S. V. Kravkov, V. A. Bakeev).

Task No. 8. Can the examples below be classified as cases of pathology of volitional behavior? Why?

1. “The complete opposite of Porfiry Vladimirovich was represented by his brother, Pavel Vladimirovich. It was the complete personification of a person devoid of any actions. Even as a boy, he did not show the slightest inclination to study, or to play, or to be sociable, but he loved to live alone, alienated from people. He used to hide in a corner, pout and start fantasizing. It seems to him that he has eaten too much oatmeal, that this has made his legs thin, and he is not studying. Or - that he is not Pavel - a noble son, but Davydka-pas-tukh... that he clicks an arapnik and does not study. ...The years passed, and Pavel Vladimirovich gradually developed into that apathetic and mysteriously gloomy personality from whom, in end result, it turns out a person devoid of actions. Perhaps he was kind, but he did no good to anyone; Perhaps he was not stupid, but he never committed a single smart act in his entire life. He was hospitable, but no one flattered himself on his hospitality; he willingly spent money, but no useful or pleasant result from this spending ever occurred for anyone; He never offended anyone, but no one imputed this to his dignity...” (M. E. Saltykov-Shchedrin).

Onegin locked himself at home,

Yawning, he took up the pen,

I wanted to write, but the work is hard

He felt sick; Nothing

It didn't come from his pen...

And again, betrayed by idleness,

Languishing with spiritual emptiness,

He sat down - with a laudable purpose

Appropriating someone else's mind for yourself;

He lined the shelf with a group of books,

I read and read, but to no avail:

There is boredom, there is deception or delirium;

There is no conscience in that, there is no meaning in that;

Everyone is wearing different chains;

And the old thing is outdated,

And the old are delirious of the newness.

How women he left book

And a shelf with their dusty family,

Covered it with mourning taffeta.

Jumping Dragonfly

The red summer sang;

I didn’t have time to look back,

How winter rolls into your eyes.

The pure field has died;

There are no more bright days,

Like under every leaf

Both the table and the house were ready.

Everything has passed: with the cold winter

Need, hunger comes;

The dragonfly no longer sings:

And who cares?

Sing on a hungry stomach!

Angry melancholy,

She crawls towards the Ant:

“Don’t leave me, dear godfather!

Let me gather my strength

And only until spring days

Feed and warm!

- “Gossip, this is strange to me:

Did you work during the summer?” -

Ant tells her.

“Was it before that, my dear?

In our soft ants

Songs, playfulness every hour,

So much so that my head was turned.”

- “Oh, so you...”

- “I am without a soul

I sang all summer.”

- - “Did you sing everything? this business:

So come and dance!”

(I. A. Krylov)

4. Often those who are tired of being at home leave huge mansions and suddenly return, because they find that they are no better at home. He rushes quickly, chasing the trotters, to the estate, as if he needs to rush to a fire; again, again begins to yawn as soon as he touches the threshold of the estate; either the dejected one goes to sleep and seeks oblivion, or he hastily rushes to the city, and here he is there again ( Lucretius).

Task No. 9. Analyze the given example from criminal practice and explain what personality traits contribute to suggestion. Can we consider that these properties and traits form a pathopsychological syndrome that deforms volitional behavior, and why?

B., 29 years old, was accused of theft Money. Since childhood, she has been distinguished by perseverance, diligence, and diligence. She graduated from 8th grade and medical school with honors. She got married at 23 and has 2 children from her marriage. For a long time she lived with her husband’s parents, relations with whom were conflicting. She was very tired, her mood was depressed, she cried often, she was irritable, slept poorly, and lost weight. She got a job as a cashier in a hairdressing salon and intended to subsequently work in her specialty.

On the way from work, a woman approached B. on the street, who told her that she looked bad, asked where and with whom she lived, where she worked, and promised to help her with “fortune telling.” She scheduled the next meeting on the day B. received a large sum of money from the bank. At the same time, there were accomplices present, two other women, “assisting” the leader, confirming her “capabilities.”

After 10 days, B., having received money from the bank and delivered it to work, went to meet that woman. Having learned that B. came without money, the accomplices began to demand the money needed for “fortune telling” and threatened her with a deterioration in her health and relationship with her husband. B. returned to the accounting department and took money from the safe.

On the street, during the process of “fortune telling,” she gave money to one of the women, after which all three disappeared. She was prosecuted for theft.

During the investigation, she blamed herself for what happened, saying that the “fortune teller” influenced her with her appearance. When the “fortune teller” approached her on the street and with a sympathetic face inquired about her well-being, promising to help, she had no doubt about the sincerity of the “fortune teller’s” words. At that moment, a woman appeared nearby who intended to bring the “fortune teller” a significant amount of money for the service allegedly rendered earlier. She was so fascinated by the words of the “fortune teller” that she was ready to carry out any of her orders. The first two days after the meeting, her health improved; in the following days, she was anxiously waiting for something, often remembered what happened to her, and willingly went to a repeat meeting.

At the meeting, she felt some anxiety and excitement, she told the fortune teller that she could not bring money, but she began to threaten her that misfortunes awaited her because of this. The “assistants” said the same thing. B. got scared, went for the money and gave it to the fortune teller. Then, on her orders, she closed her eyes and stood there for three minutes. She opened her eyes and, not seeing the “fortune teller”, for some time she believed that this was how it should be, then she realized that she had been deceived, “everything broke inside”, she began to rush around the street, looking for the “fortune teller” and her companions, but they were nowhere to be found. When she returned to work, she reported the incident to the police.

Topics for essays

1. General state modern theoretical studies of will.

2. Children's games and their significance in the development of will.

3. Formation of volitional regulation of behavior in children.

4. The main directions and ways of developing the will.

Literature

1. Vygotsky L. S. The problem of will and its development in childhood // Collection. op. - T. 3. - M., 1982.

2. Zimin P.P. Will and its education in adolescents. - Tashkent, 1985.

3. Ivannikov V. A. Psychological mechanisms of volitional regulation. - M., 1998.

4. Ilyin E. P. Psychology of will. - St. Petersburg: Peter, 2000.

5. Maklakov A. G. General psychology. - St. Petersburg: Peter, 2000.

6. Rubinstein S. L. Fundamentals of general psychology. - St. Petersburg: Peter, 1999.

7. Selivanov V.I. Psychology of volitional activity. - Ryazan: Ryazan State Pedagogical Institute, 1974.

8. Selivanov V.I. Will and its education. - M.: Knowledge, 1976.

9. Chkhartishvili Sh. N. The problem of will in psychology // Questions of psychology. 1967.

Topic 1.8. Emotional-volitional organization of the subject (will). Practical lesson.

Will is the ability (function) of a person, manifested in self-determination and self-regulation of his activities and various mental processes. It is carried out through a voluntary and conscious form of motivation. The psychological mechanism for voluntary change of impulse is a change in the meaning of the action. Therefore, behind volitional efforts there is a special activity that occurs in the internal plane of consciousness, to mobilize all human capabilities.

The will is realized in the form of incentive and inhibitory activity of the psyche. Thanks to volitional regulation, cognitive mental processes are transferred to the category of voluntary and efforts become possible that allow a person to carry out purposeful activities.

Actions controlled and regulated by the will can be simple or complex. Depending on the extent to which an individual understands the meaning of his volitional activity and whether he attributes responsibility to external circumstances or, on the contrary, to his own efforts and abilities, his locus of control is determined.

When assessing a person according to the “strong-willed-weak-willed” criterion, one should take into account his ability to create an additional incentive to action through a change in his semantic side. The initiation of action, as well as strength, pace, speed, duration of work, and overcoming external and internal (psychological) obstacles depend on this. Since volitional regulation is determined by semantic changes in consciousness, it depends on such personality components as worldview, the nature of the semantic sphere, and conviction.

According to the criteria of activity, volitional properties are distinguished, which include perseverance, determination, energy, perseverance, etc.

From the diversity of volitional properties, the workshop included research on determining subjective control, persistence and impulsiveness.

Task 26

Subjective control research

Purpose of the study: determine the locus of subjective control.

Material and equipment: test questionnaire developed by E.F. Bazhin et al. based on J. Rotter’s locus of control scale, answer sheet, pen.

Function y=f(x) is such a dependence of the variable y on the variable x, when each valid value of the variable x corresponds to a single value of the variable y.

Function definition domain D(f) is the set of all possible values ​​of the variable x.

Function Range E(f) is the set of all admissible values ​​of the variable y.

Graph of a function y=f(x) is a set of points on the plane whose coordinates satisfy a given functional dependence, that is, points of the form M (x; f(x)). The graph of a function is a certain line on a plane.

If b=0 , then the function will take the form y=kx and will be called direct proportionality.

D(f) : x \in R;\enspace E(f) : y \in R

Schedule linear function- straight.

The slope k of the straight line y=kx+b is calculated using the following formula:

k= tan \alpha, where \alpha is the angle of inclination of the straight line to the positive direction of the Ox axis.

1) The function increases monotonically for k > 0.

For example: y=x+1

2) The function decreases monotonically as k< 0 .

For example: y=-x+1

3) If k=0, then giving b arbitrary values, we obtain a family of straight lines parallel to the Ox axis.

For example: y=-1

Inverse proportionality

Inverse proportionality called a function of the form y=\frac (k)(x), where k is a non-zero real number

D(f) : x \in \left \( R/x \neq 0 \right \); \: E(f) : y \in \left \(R/y \neq 0 \right \).

Function graph y=\frac (k)(x) is a hyperbole.

1) If k > 0, then the graph of the function will be located in the first and third quarters of the coordinate plane.

For example: y=\frac(1)(x)

2) If k< 0 , то график функции будет располагаться во второй и четвертой координатной плоскости.

For example: y=-\frac(1)(x)

Power function

Power function is a function of the form y=x^n, where n is a non-zero real number

1) If n=2, then y=x^2. D(f) : x \in R; \: E(f) : y \in; main period of the function T=2 \pi