Function. Domain and range of functions
Each function has two variables - an independent variable and a dependent variable, the values of which depend on the values of the independent variable. For example, in the function y = f(x) = 2x + y The independent variable is "x" and the dependent variable is "y" (in other words, "y" is a function of "x"). The valid values of the independent variable "x" are called the domain of the function, and the valid values of the dependent variable "y" are called the domain of the function.
Steps
Part 1
Finding the Domain of a Function-
Select the appropriate entry for the function's scope. The scope of definition is written in square and/or parentheses. Square bracket applies when the value is within the scope of the function; if the value is not within the scope of the definition, a parenthesis is used. If a function has several non-adjacent domains, a “U” symbol is placed between them.
- For example, the scope of [-2,10)U(10,2] includes the values -2 and 2, but does not include the value 10.
-
Plot a graph quadratic function. The graph of such a function is a parabola, the branches of which are directed either up or down. Since the parabola increases or decreases along the entire X-axis, the domain of definition of the quadratic function is all real numbers. In other words, the domain of such a function is the set R (R stands for all real numbers).
- To better understand the concept of a function, select any value of “x”, substitute it into the function and find the value of “y”. A pair of values “x” and “y” represent a point with coordinates (x,y) that lies on the graph of the function.
- Plot this point on the coordinate plane and do the same process with a different x value.
- By plotting several points on the coordinate plane, you get general idea about the form of the graph of a function.
-
If the function contains a fraction, set its denominator to zero. Remember that you cannot divide by zero. Therefore, by setting the denominator to zero, you will find values of "x" that are not within the domain of the function.
- For example, find the domain of the function f(x) = (x + 1) / (x - 1) .
- Here the denominator is: (x - 1).
- Equate the denominator to zero and find “x”: x - 1 = 0; x = 1.
- Write down the domain of definition of the function. The domain of definition does not include 1, that is, it includes all real numbers except 1. Thus, the domain of definition of the function is: (-∞,1) U (1,∞).
- The notation (-∞,1) U (1,∞) reads like this: the set of all real numbers except 1. The infinity symbol ∞ stands for all real numbers. In our example, all real numbers that are greater than 1 and less than 1 are included in the domain.
-
If a function contains a square root, then the radical expression must be greater than or equal to zero. Remember that the square root of negative numbers not extracted. Therefore, any value of “x” at which the radical expression becomes negative must be excluded from the domain of definition of the function.
- For example, find the domain of the function f(x) = √(x + 3).
- Radical expression: (x + 3).
- The radical expression must be greater than or equal to zero: (x + 3) ≥ 0.
- Find "x": x ≥ -3.
- The domain of this function includes the set of all real numbers that are greater than or equal to -3. Thus, the domain of definition is [-3,∞).
Part 2
Finding the range of a quadratic function-
Make sure you are given a quadratic function. The quadratic function has the form: ax 2 + bx + c: f(x) = 2x 2 + 3x + 4. The graph of such a function is a parabola, the branches of which are directed either up or down. Exist various methods finding the range of values of a quadratic function.
- The easiest way to find the range of a function containing a root or fraction is to graph the function using a graphing calculator.
-
Find the x coordinate of the vertex of the function graph. For a quadratic function, find the x coordinate of the vertex of the parabola. Remember that the quadratic function is: ax 2 + bx + c. To calculate the x coordinate, use the following equation: x = -b/2a. This equation is a derivative of the basic quadratic function and describes a tangent whose slope is zero (the tangent to the vertex of the parabola is parallel to the X-axis).
- For example, find the range of the function 3x 2 + 6x -2.
- Calculate the x coordinate of the vertex of the parabola: x = -b/2a = -6/(2*3) = -1
-
Find the y-coordinate of the vertex of the function graph. To do this, substitute the found “x” coordinate into the function. The desired coordinate “y” represents the limiting value of the function range.
- Calculate the y coordinate: y = 3x 2 + 6x – 2 = 3(-1) 2 + 6(-1) -2 = -5
- The coordinates of the vertex of the parabola of this function are (-1,-5).
-
Determine the direction of the parabola by substituting in the function by at least one value "x". Choose any other x value and plug it into the function to calculate the corresponding y value. If the found “y” value is greater than the “y” coordinate of the vertex of the parabola, then the parabola is directed upward. If the found “y” value is less than the “y” coordinate of the vertex of the parabola, then the parabola is directed downward.
- Substitute into the function x = -2: y = 3x 2 + 6x – 2 = y = 3(-2) 2 + 6(-2) – 2 = 12 -12 -2 = -2.
- Coordinates of a point lying on the parabola: (-2,-2).
- The found coordinates indicate that the branches of the parabola are directed upward. Thus, the range of the function includes all values of "y" that are greater than or equal to -5.
- Range of values of this function: [-5, ∞)
-
The domain of a function is written similarly to the domain of a function. The square bracket is used when the value is within the range of the function; if the value is not in the range, a parenthesis is used. If a function has several non-adjacent ranges of values, a “U” symbol is placed between them.
- For example, the range [-2,10)U(10,2] includes the values -2 and 2, but does not include the value 10.
- With the infinity symbol ∞, parentheses are always used.
Determine the type of function given to you. The range of values of the function is all valid “x” values (laid along the horizontal axis), which correspond to valid “y” values. The function can be quadratic or contain fractions or roots. To find the domain of a function, you first need to determine the type of the function.
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) . Region acceptable values variable x for the expression f(x) - this 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 from the very beginning simple 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 greatest and smallest value functions 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
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
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. Long time 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.
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