Acceptable Range: Theory and Practice
Shamshurin A.V. 1
Gagarina N.A. 1
1 Municipal budgetary educational institution "Secondary comprehensive school No. 31"
The text of the work is placed without images and formulas.
Full version work is available in the "Files of work" tab in PDF format
Introduction
I started by looking at a lot of math topics on the Internet and chose this topic because I am sure that the importance of finding the DPV plays a huge role in solving equations and problems. In his research work I considered equations in which it is enough only to find the ODZ, danger, optionality, limitedness of the ODZ, some prohibitions in mathematics. The most important thing for me is to pass the exam well in mathematics, and for this you need to know: when, why and how to find the ODZ. This prompted me to study the topic, the purpose of which was to show that mastering this topic will help students to correctly complete the assignments for the exam. To achieve this goal, I have researched additional literature and other sources. It became interesting to me, but the students of our school know: when, why and how to find ODZ. Therefore, I conducted a test on the topic “When, why and how to find ODZ?” (10 equations were given). Number of students - 28. Managed - 14%, the danger of ODZ (taken into account) - 68%, optional (taken into account) - 36%.
Target: identification: when, why and how to find ODZ.
Problem: the equations and inequalities in which you need to find the ODZ have not found a place in the course of systematic presentation of algebra, which is probably why my peers and I often make mistakes when solving such examples, devoting a lot of time to solving them, while forgetting about the ODZ.
Tasks:
- Show the significance of ODZ in solving equations and inequalities.
- Conduct practical work on this topic and summarize its results.
I think the knowledge and skills I have gained will help me decide whether to look for ODZ or not? I will stop making mistakes by learning how to do ODZ correctly. Whether I succeed, time will tell, or rather the exam.
Chapter 1
What is ODZ?
ODZ is tolerance range, that is, these are all values of the variable for which the expression makes sense.
Important. To find the ODZ, we do not solve the example! We solve the pieces of the example for finding forbidden places.
Some taboos in mathematics. There are very few such forbidden actions in mathematics. But not everyone remembers them...
- Expressions under the sign of even multiplicity or must be> 0 or equal to zero, ODZ: f (x)
- The expression in the denominator of a fraction cannot be equal to zero, ODZ: f (x)
- |f(x)|=g(x), ODZ: g(x) 0
How to write ODZ? Very simple. Always write ODZ next to the example. Under these known letters, looking at the original equation, we write down the x values that are allowed for the original example. Transforming an example can change the DPV and, accordingly, the answer.
Algorithm for finding ODZ:
- Determine the type of ban.
- Find values for which the expression does not make sense.
- Exclude these values from the set of real numbers R.
Solve the equation: =
|
Without ODZ |
WITH ODZ |
|
Answer: x=5 |
ODZ: => => Answer: no roots |
The range of valid values protects us from such serious errors. To be honest, it is because of the ODZ that many “drummers” turn into “triples”. Considering that the search and accounting for ODZ is an insignificant step in the solution, they skip it, and then they are surprised: “why did the teacher give 2?”. Yes, that's why I put it because the answer is wrong! These are not "nitpicks" of the teacher, but a very specific mistake, the same as an incorrect calculation or a lost sign.
Additional equations:
a) = ; b) -42=14x+; c) =0; d) |x-5|=2x-2
Chapter 2
ODZ. For what? When? How?
Acceptable range - there is a solution
- ODZ is an empty set, which means that the original example has no solutions
- = ODZ:
Answer: no roots.
- = ODZ:
Answer: no roots.
0, the equation has no roots
Answer: no roots.
Additional examples:
a) + =5; b) + = 23x-18; c) =0.
- There is one or more numbers in the ODZ, and a simple substitution quickly determines the roots.
ODZ: x=2, x=3
Check: x=2, + , 0<1, верно
Check: x=3, + , 0<1, верно.
Answer: x=2, x=3.
- > ODZ: x=1, x=0
Check: x=0, > , 0>0, wrong
Check: x=1, > , 1>0, true
Answer: x=1.
- + \u003d x ODZ: x \u003d 3
Check: +=3, 0=3, wrong.
Answer: no roots.
Additional examples:
a) = ; b) + =0; c) + \u003d x -1
Danger of ODZ
Note that identical transformations can:
- do not affect the ODZ;
- lead to an extended ODZ;
- lead to a narrowing of the ODZ.
It is also known that as a result of some transformations that change the original ODZ, it can lead to incorrect decisions.
Let's explain each case with an example.
1) Consider the expression x + 4x + 7x, the ODZ of the variable x for this is the set R. We present similar terms. As a result, it will take the form x 2 +11x. Obviously, the ODZ of the variable x of this expression is also the set R. Thus, the performed transformation did not change the ODZ.
2) Take the equation x+ - =0. In this case, ODZ: x≠0. This expression also contains similar terms, after reduction of which, we come to the expression x, for which the ODZ is R. What we see: as a result of the transformation, the ODZ has expanded (zero has been added to the ODZ of the x variable for the original expression).
3) Let's take an expression. The ODV of the variable x is determined by the inequality (x−5) (x−2)≥0, ODV: (−∞, 2]∪∪/ Access mode: Materials of the websites www.fipi.ru, www.eg
Annex 1
Practical work "ODZ: when, why and how?"
|
Option 1 |
Option 2 |
|
│х+14│= 2 - 2х |
|
|
│3-х│=1 - 3х |
Annex 2
Answers to tasks practical work"ODZ: when, why and how?"
|
Option 1 |
Option 2 |
|
Answer: no roots |
Answer: x is any number except x=5 |
|
9x+ = +27 ODZ: x≠3 Answer: no roots |
ODZ: x=-3, x=5. Answer: -3;5. |
|
y= -decreases, y= -increases So the equation has at most one root. Answer: x=6. |
ODZ: → →х≥5 Answer: x≥5, x≤-6. |
|
│х+14│=2-2х ODZ:2-2х≥0, х≤1 х=-4, х=16, 16 does not belong to ODZ |
Decreases - increases The equation has at most one root. Answer: no roots. |
|
0, ODZ: x≥3, x≤2 Answer: x≥3, x≤2 |
8x+ = -32, ODZ: x≠-4. Answer: no roots. |
|
x=7, x=1. Answer: no solution |
Increasing - decreasing Answer: x=2. |
|
0 ODZ: x≠15 Answer: x is any number except x=15. |
│3-х│=1-3х, ODZ: 1-3х≥0, х≤ x=-1, x=1 does not belong to the ODZ. Answer: x=-1. |
How ?
Solution examples
If something is missing somewhere, then there is something somewhere
We continue to study the "Functions and Graphics" section, and the next station of our journey is. An active discussion of this concept began in the article on sets and continued in the first lesson on function graphs, where I looked at elementary functions, and, in particular, their scope. Therefore, I recommend that dummies start with the basics of the topic, since I will not dwell on some of the basic points again.
It is assumed that the reader knows the domain of definition of the following functions: linear, quadratic, cubic function, polynomials, exponent, sine, cosine. They are defined on (set of all real numbers). For tangents, arcsines, so be it, I forgive you =) - rarer graphs are not remembered immediately.
The domain of definition seems to be a simple thing, and a natural question arises, what will the article be about? In this lesson, I will consider common tasks for finding the domain of a function. In addition, we will repeat inequalities with one variable, the skills to solve which will be required in other tasks higher mathematics. The material, by the way, is all school, so it will be useful not only to students, but also to students. The information, of course, does not pretend to be encyclopedic, but on the other hand, there are not far-fetched “dead” examples here, but roasted chestnuts, which are taken from real practical works.
Let's start with an express cut into the topic. Briefly about the main thing: we are talking about a function of one variable. Its domain of definition is set of "x" values, for which exist the meaning of "games". Consider a hypothetical example: 
The domain of this function is the union of intervals:
(for those who forgot: - the union icon). In other words, if we take any value of "x" from the interval , or from , or from , then for each such "x" there will be a value of "y".
Roughly speaking, where the domain of definition is, there is a graph of the function. But the half-interval and the “ce” point are not included in the definition area and there is no graph there.
How to find the scope of a function? Many remember the children's rhyme: "stone, scissors, paper", and in this case it can be safely rephrased: "root, fraction and logarithm." Thus, if you are life path there is a fraction, root or logarithm, then you should immediately be very, very alert! Tangent, cotangent, arcsine, arccosine are much less common, and we will also talk about them. But first, sketches from the life of ants:
The scope of a function that contains a fraction
Suppose given a function containing some fraction . As you know, you cannot divide by zero: , so those x values that turn the denominator to zero are not included in the scope of this function.
I will not dwell on the simplest functions like 
and so on, because everyone can see points that are not included in their domain of definition. Consider more meaningful fractions:
Example 1
Find the scope of a function
Solution: there is nothing special in the numerator, but the denominator must be non-zero. Let's equate it to zero and try to find the "bad" points:
The resulting equation has two roots: 
. Value Data not included in the scope of the function. Indeed, substitute or into the function and you will see that the denominator goes to zero.
Answer: domain: 
The entry reads as follows: “the domain of definition is all real numbers with the exception of the set consisting of values 
". I remind you that the backslash icon in mathematics denotes logical subtraction, and curly braces denote a set. The answer can be equivalently written as a union of three intervals:
Whoever likes it.
At points 
function endures endless breaks, and the straight lines given by equations 
are vertical asymptotes for the graph of this function. However, this is a slightly different topic, and further I will not particularly focus on this.
Example 2
Find the scope of a function
The task is essentially oral and many of you will find the definition area almost immediately. Answer at the end of the lesson.
Will a fraction always be "bad"? No. For example, a function is defined on the entire number axis. Whatever value of "x" we take, the denominator will not turn to zero, moreover, it will always be positive:. Thus, the scope of this function is: .
All functions like 
defined and continuous on .
A little more complicated is the situation when the denominator occupied square trinomial:
Example 3
Find the scope of a function 
Solution: Let's try to find the points where the denominator goes to zero. For this we will decide quadratic equation:
The discriminant is negative, which means real roots no, and our function is defined on the whole number line.
Answer: domain:
Example 4
Find the scope of a function 
This is an example for independent solution. Solution and answer at the end of the lesson. I advise you not to be lazy with simple problems, because misunderstandings will accumulate for further examples.
Function scope with root
Function with square root is defined only for those values of "x" when radical expression is non-negative: . If the root is located in the denominator, then the condition is obviously tightened: . Similar calculations are valid for any root of a positive even degree: 
, however, the root is already the 4th degree in function studies I don't remember.
Example 5
Find the scope of a function 
Solution: radical expression must be non-negative:
Before continuing the solution, let me remind you the basic rules for working with inequalities, known since school.
I draw Special attention! We are now considering the inequalities with one variable- that is, for us there is only one dimension along the axis. Please do not confuse with inequalities of two variables, where the entire coordinate plane is geometrically involved. However, there are also pleasant coincidences! So, for inequality, the following transformations are equivalent:
1) Terms can be transferred from part to part by changing their (terms) signs.
2) Both sides of the inequality can be multiplied by a positive number.
3) If both parts of the inequality are multiplied by negative number, you need to change the sign of the inequality itself. For example, if there was “more”, then it will become “less”; if it was “less than or equal to”, then it will become “greater than or equal to”.
In the inequality, we move the “three” to the right side with a change of sign (rule No. 1):
Multiply both sides of the inequality by –1 (rule #3):
Multiply both sides of the inequality by (rule number 2):
Answer: domain: 
The answer can also be written in the equivalent phrase: "the function is defined at".
Geometrically, the domain of definition is depicted by shading the corresponding intervals on the x-axis. In this case:
Once again, I recall the geometric meaning of the domain of definition - the graph of the function 
exists only in the shaded area and is absent at .
In most cases, a purely analytical finding of the domain of definition is suitable, but when the function is very confused, you should draw an axis and make notes.
Example 6
Find the scope of a function
This is a do-it-yourself example.
When there is a square binomial or trinomial under the square root, the situation becomes a little more complicated, and now we will analyze the solution technique in detail:
Example 7
Find the scope of a function 
Solution: the radical expression must be strictly positive, that is, we need to solve the inequality . At the first step, we try to factorize the square trinomial: 
The discriminant is positive, we are looking for the roots: 
So the parabola 
intersects the x-axis at two points, which means that part of the parabola is located below the axis (inequality), and part of the parabola is above the axis (the inequality we need).
Since the coefficient , then the branches of the parabola look up. It follows from the above that the inequality is satisfied on the intervals (the branches of the parabola go up to infinity), and the vertex of the parabola is located on the interval below the abscissa axis, which corresponds to the inequality:
! Note:
if you do not fully understand the explanations, please draw the second axis and the whole parabola! It is advisable to return to the article and the manual Hot School Mathematics Formulas.
Please note that the points themselves are punctured (not included in the solution), since our inequality is strict.
Answer: domain:
In general, many inequalities (including the considered one) are solved by the universal interval method, known again from the school curriculum. But in the cases of square two- and three-terms, in my opinion, it is much more convenient and faster to analyze the location of the parabola relative to the axis. And the main method - the method of intervals, we will analyze in detail in the article. Function nulls. Constancy intervals.
Example 8
Find the scope of a function
This is a do-it-yourself example. The sample commented in detail on the logic of reasoning + the second way of solving and another important transformation of the inequality, without knowing which the student will limp on one leg ..., ... hmm ... at the expense of the foot, perhaps he got excited, rather - on one finger. Thumb.
Can a square root function be defined on the entire number line? Certainly. All familiar faces: . Or a similar sum with an exponent: . Indeed, for any values \u200b\u200bof "x" and "ka": , therefore, even more so.
Here's a less obvious example: 
. Here the discriminant is negative (the parabola does not cross the x-axis), while the branches of the parabola are directed upwards, hence the domain of definition: .
The question is the opposite: can the scope of a function be empty? Yes, and a primitive example immediately suggests itself 
, where the radical expression is negative for any value of "x", and the domain of definition is: (an empty set icon). Such a function is not defined at all (of course, the graph is also illusory).
with odd roots 
etc. things are much better - here root expression can also be negative. For example, a function is defined on the entire number line. However, the function has a single point still not included in the domain of definition, since the denominator is turned to zero. For the same reason for the function 
points are excluded.
Function domain with logarithm
The third common function is the logarithm. As an example, I will draw natural logarithm, which comes across in about 99 examples out of 100. If some function contains a logarithm, then its domain of definition should include only those x values that satisfy the inequality . If the logarithm is in the denominator: then additionally condition is imposed (because ).
Example 9
Find the scope of a function
Solution: in accordance with the above, we compose and solve the system:
Graphic solution for Dummies:
Answer: domain:
I’ll dwell on one more technical point - after all, I don’t have a scale and no divisions along the axis. The question arises: how to make such drawings in a notebook on checkered paper? Is it possible to measure the distance between points in cells strictly according to scale? It is more canonical and stricter, of course, to scale, but a schematic drawing that fundamentally reflects the situation is also quite acceptable.
Example 10
Find the scope of a function 
To solve the problem, you can use the method of the previous paragraph - to analyze how the parabola is located relative to the x-axis. Answer at the end of the lesson.
As you can see, in the realm of logarithms, everything is very similar to the situation with a square root: the function 
(square trinomial from Example No. 7) is defined on intervals , and the function 
(square binomial from Example No. 6) on the interval . It's embarrassing to even say that type functions are defined on the entire number line.
Helpful information
: the type function is interesting, it is defined on the entire number line except for the point. According to the property of the logarithm, "two" can be taken out by a factor outside the logarithm, but in order for the function not to change, "x" must be enclosed under the module sign: 
. Here's another one for you practical use» module =). This is what you need to do in most cases when you demolish even degree, for example: 
. If the base of the degree is obviously positive, for example, then there is no need for the module sign and it is enough to get by with parentheses: .
In order not to repeat ourselves, let's complicate the task:
Example 11
Find the scope of a function 
Solution: in this function we have both the root and the logarithm.
The root expression must be non-negative: , and the expression under the logarithm sign must be strictly positive: . Thus, it is necessary to solve the system:
Many of you know very well or intuitively guess that the solution of the system must satisfy to each condition.
Examining the location of the parabola relative to the axis, we come to the conclusion that the interval satisfies the inequality (blue shading):
Inequality , obviously, corresponds to the "red" half-interval .
Since both conditions must be met simultaneously, then the solution of the system is the intersection of these intervals. "Common interests" are observed on the half-interval.
Answer: domain:
Typical inequality, as demonstrated in Example No. 8, is not difficult to resolve analytically.
The found domain of definition will not change for "similar functions", for example, for 
or 
. You can also add some continuous functions, for example: , or like this: 
, or even like this: . As they say, the root and the logarithm are stubborn things. The only thing is that if one of the functions is "reset" to the denominator, then the domain of definition will change (although in the general case this is not always true). Well, in the theory of matan about this verbal ... oh ... there are theorems.
Example 12
Find the scope of a function 
This is a do-it-yourself example. Using a blueprint is quite appropriate, since the function is not the easiest.
A couple more examples to reinforce the material:
Example 13
Find the scope of a function
Solution: compose and solve the system: 
All actions have already been sorted out in the course of the article. Draw on a numerical line the interval corresponding to the inequality and, according to the second condition, exclude two points:
The value turned out to be completely irrelevant.
Answer: domain
A small mathematical pun on a variation of the 13th example:
Example 14
Find the scope of a function 
This is a do-it-yourself example. Who missed, he is in flight ;-)
The final section of the lesson is devoted to more rare, but also "working" functions:
Function scopes
with tangents, cotangents, arcsines, arccosines
If some function includes , then from its domain of definition excluded points 
, Where Z is the set of integers. In particular, as noted in the article Graphs and properties of elementary functions, the function has the following values:
That is, the domain of definition of the tangent: 
.
We will not kill much:
Example 15
Find the scope of a function
Solution: in this case, the following points will not be included in the domain of definition:
Let's drop the "two" of the left side into the denominator of the right side:
As a result 
:
Answer: domain: 
.
In principle, the answer can also be written as a union of an infinite number of intervals, but the construction will turn out to be very cumbersome:
The analytical solution is in complete agreement with geometric transformation graphics: if the function argument is multiplied by 2, then its graph will shrink to the axis twice. Notice how the period of the function has halved, and break points increased twice. Tachycardia.
Similar story with cotangent. If some function includes , then points are excluded from its domain of definition. In particular, for the function, we shoot the following values with an automaton burst:
In other words:
Scientific adviser:
1. Introduction 3
2. Historical background 4
3. "Place" of ODZ when solving equations and inequalities 5-6
4. Features and danger of ODZ 7
5. ODZ - there is a decision 8-9
6. Finding ODZ is extra work. Equivalence of transitions 10-14
7. ODZ in the exam 15-16
8. Conclusion 17
9. Literature 18
1. Introduction
Problem: the equations and inequalities in which you need to find the ODZ have not found a place in the course of systematic presentation of algebra, which is probably why my peers and I often make mistakes when solving such examples, devoting a lot of time to solving them, while forgetting about the ODZ.
Target: be able to analyze the situation and draw logically correct conclusions in examples where it is necessary to take into account the ODD.
Tasks:
1. Study theoretical material;
2. Solve a set of equations, inequalities: a) fractionally rational; b) irrational; c) logarithmic; d) containing inverse trigonometric functions;
3. Apply the learned materials in a situation that differs from the standard;
4. Create a paper on the topic "Region of acceptable values: theory and practice"
Project work: I started working on the project by repeating the functions known to me. The scope of many of them is limited.
ODZ occurs:
1. When deciding fractional rational equations and inequalities
2. When deciding irrational equations and inequalities
3. When deciding logarithmic equations and inequalities
4. When solving equations and inequalities containing inverse trigonometric functions
After solving many examples from various sources(benefits for the exam, textbooks, reference books), I systematized the solution of examples for following principles:
you can solve the example and take into account the ODZ (the most common way)
It is possible to solve the example without taking into account the ODZ
It is possible only taking into account the ODZ to come to the right decision.
Methods used in the work: 1) analysis; 2) statistical analysis; 3) deduction; 4) classification; 5) forecasting.
Studied analysis USE results over the past years. Many mistakes have been made in the examples in which DHS must be taken into account. This again emphasizes relevance my theme.
2. Historical outline
Like other concepts of mathematics, the concept of a function did not develop immediately, but went a long way of development. In the work of P. Fermat "Introduction and study of flat and solid places" (1636, publ. 1679) says: "Whenever in final equation there are two unknown quantities, there is a place. In essence, we are talking about functional dependence and its graphic image("place" in Fermat means a line). The study of lines according to their equations in R. Descartes' "Geometry" (1637) also indicates a clear understanding of the mutual dependence of two variables. I. Barrow ("Lectures on Geometry", 1670) in geometric shape the mutual inverseness of the actions of differentiation and integration is established (of course, without using these terms themselves). This already testifies to a completely clear mastery of the concept of function. In a geometric and mechanical form, we also find this concept in I. Newton. However, the term "function" first appears only in 1692 by G. Leibniz and, moreover, not quite in its modern sense. G. Leibniz calls various segments associated with a curve (for example, the abscissas of its points) a function. In the first printed course "Analysis of Infinitely Small for the Knowledge of Curved Lines" by Lopital (1696), the term "function" is not used.
The first definition of a function in a sense close to the modern one is found in I. Bernoulli (1718): "A function is a quantity composed of a variable and a constant." This not quite distinct definition is based on the idea of specifying a function by an analytic formula. The same idea appears in the definition of L. Euler, given by him in "Introduction to the analysis of infinite" (1748): "A function of a variable quantity is an analytical expression, composed in some way from this variable quantity and numbers or constant quantities." However, L. Euler is no stranger to modern understanding function, which does not associate the concept of a function with any of its analytical expressions. In his " Differential calculus"(1755) says:" When some quantities depend on others in such a way that when the latter change, they themselves undergo a change, then the former are called functions of the latter.
WITH early XIX centuries, more and more often define the concept of a function without mentioning its analytical representation. In the "Treatise on differential and integral calculus" (1797-1802) S. Lacroix says: "Any quantity whose value depends on one or many other quantities is called a function of these latter." In the "Analytical Theory of Heat" by J. Fourier (1822) there is a phrase: "The function f(x) denotes a completely arbitrary function, that is, a sequence of given values, subject or not to a general law and corresponding to all values x contained between 0 and some value x". Close to modern and the definition of N. I. Lobachevsky: “... General concept function requires that the function from x name the number that is given for each x and together with x gradually changes. The value of a function may be given either by an analytic expression, or by a condition that provides a means of testing all numbers and choosing one of them, or, finally, the dependence may exist and remain unknown. In the same place a little lower it is said: "The broad view of the theory admits the existence of dependence only in the sense that the numbers one with the other in connection are understood as if given together." Thus, the modern definition of a function, free from references to the analytical task, usually attributed to P. Dirichlet (1837), was repeatedly proposed before him.
The domain of definition (permissible values) of the function y is the set of values of the independent variable x for which this function is defined, i.e., the domain of change of the independent variable (argument).
3. "Place" of the region of admissible values when solving equations and inequalities
1. When solving fractional rational equations and inequalities the denominator must not be zero.
2. Solution of irrational equations and inequalities.
2.1..gif" width="212" height="51"> .
In this case, there is no need to find the ODZ: it follows from the first equation that the obtained x values satisfy the following inequality: https://pandia.ru/text/78/083/images/image004_33.gif" width="107" height="27 src="> is the system:
Since the equation and enter equally, then instead of inequality, you can include the inequality https://pandia.ru/text/78/083/images/image009_18.gif" width="220" height="49"> 
https://pandia.ru/text/78/083/images/image014_11.gif" width="239" height="51"> 
3. Solution of logarithmic equations and inequalities.
3.1. Scheme for solving the logarithmic equation
But it suffices to check only one condition of the ODZ.
3.2..gif" width="115" height="48 src=">.gif" width="115" height="48 src=">
4. Trigonometric equations of the form are equivalent to the system (instead of inequality, the system can include the inequality https://pandia.ru/text/78/083/images/image024_5.gif" width="377" height="23"> are equivalent to the equation
4. Features and danger of the range of permissible values
In mathematics lessons, we are required to find the ODZ in each example. At the same time, according to the mathematical essence of the matter, finding the ODZ is not at all obligatory, often unnecessary, and sometimes impossible - and all this without any damage to the solution of the example. On the other hand, it often happens that after solving an example, students forget to take into account the ODZ, write it down as the final answer, take into account only some conditions. This circumstance is well known, but the "war" continues every year and, it seems, will go on for a long time.
Consider, for example, the following inequality:
Here, the ODZ is sought, and the inequality is solved. However, when solving this inequality, schoolchildren sometimes believe that it is quite possible to do without searching for ODZ, more precisely, they can do without the condition
Indeed, to obtain the correct answer, it is necessary to take into account both the inequality and .
And here, for example, is the solution to the equation: https://pandia.ru/text/78/083/images/image032_4.gif" width="79 height=75" height="75">
which is equivalent to working with ODZ. However, in this example, such work is redundant - it is enough to check the fulfillment of only two of these inequalities, and any two.
Let me remind you that any equation (inequality) can be reduced to the form . The DPV is simply the scope of the function on the left side. The fact that this area must be monitored already follows from the definition of the root as a number from the area of the given function, thereby from the ODZ. Here's a funny example on this topic..gif" width="20" height="21 src="> has a domain of definition of a set of positive numbers (this is, of course, an agreement - to consider the function at, , but reasonable), and then -1 is not is the root.
5. Range of acceptable values - there is a solution
And finally, in the mass of examples, finding the ODZ allows you to get the answer without cumbersome layouts, and even orally.
1. OD3 is an empty set, which means that the original example has no solutions.
1)
2) 3) 
2. In ODZ one or more numbers are found, and a simple substitution quickly determines the roots.
1)
, x=3
2)
Here in the ODZ there is only the number 1, and after substitution it is clear that it is not a root.
3) There are two numbers in the ODZ: 2 and 3, and both are suitable.
4) > There are two numbers 0 and 1 in the ODZ, and only 1 is suitable.
DPV can be used effectively in combination with the analysis of the expression itself.
5) 
< ОДЗ: Но в правой части неравенства могут быть только положительные числа, поэтому оставляем х=2. Тогда в неравенство подставим 2.
6) 
From the ODZ it follows that, whence we have ..gif" width="143" height="24"> From the ODZ we have: . But then and . Since, then there are no solutions.
From the ODZ we have: https://pandia.ru/text/78/083/images/image060_0.gif" width="48" height="24">>, which means . Solving the last inequality, we get x<- 4, что не входит в ОДЗ. Поэтому решения нет.
3) ODZ: . Since then
On the other hand, https://pandia.ru/text/78/083/images/image068_0.gif" width="160" height="24">
ODZ:. Consider the equation on the interval [-1; 0).
It fulfills such inequalities https://pandia.ru/text/78/083/images/image071_0.gif" width="68" height="24 src=">.gif" width="123" height="24 src="> and there are no solutions. With the function and https://pandia.ru/text/78/083/images/image076_0.gif" width="179" height="25">.ODZ: x>2..gif" width="233" height ="45 src="> Let's find the ODZ:
An integer solution is possible only for x=3 and x=5. By checking, we find that the root x \u003d 3 does not fit, which means the answer is: x \u003d 5.
6. Finding the range of acceptable values is extra work. Equivalence of transitions.
Examples can be given where the situation is clear even without finding the ODZ.
1. 
Equality is impossible, because when subtracting a larger expression from a smaller one, a negative number should be obtained.
2. 
.
The sum of two non-negative functions cannot be negative.
I will also give examples where finding the ODZ is difficult, and sometimes simply impossible.
And, finally, the search for ODZ is very often just unnecessary work, without which one can perfectly do, thus proving an understanding of what is happening. There are a huge number of examples here, so I will choose only the most typical ones. In this case, the main decision technique is equivalent transformations in the transition from one equation (inequality, system) to another.
1.
. ODZ is not needed, because, having found those values of x for which x2=1, we cannot get x=0.
2. . ODZ is not needed, because we find out when the radical expression is equal to a positive number.
3. . ODZ is not needed for the same reasons as in the previous example.
4. 
ODZ is not needed, because the root expression is equal to the square of some function, and therefore cannot be negative.
5. 
6. ..gif" width="271" height="51"> Only one constraint for the radical expression is sufficient for the solution. Indeed, it follows from the written mixed system that the other radical expression is also non-negative.
8. ODZ is not needed for the same reasons as in the previous example.
9. 
The DPV is not needed, since it is sufficient that two of the three expressions under the logarithm signs be positive to ensure that the third one is positive.
10. 
.gif" width="357" height="51"> ODZ is not needed for the same reasons as in the previous example.
It is worth noting, however, that when solving by the method of equivalent transformations, knowledge of the ODZ (and the properties of functions) helps.
Here are some examples.
1. . OD3, from which follows the positiveness of the expression on the right side, and it is possible to write an equation equivalent to the given one in this form https://pandia.ru/text/78/083/images/image101_0.gif" width="112" height="27 "> ODZ:. But then, and when solving this inequality, it is not necessary to consider the case when the right side is less than 0.
3. . It follows from the ODZ that , and therefore the case when https://pandia.ru/text/78/083/images/image106_0.gif" width="303" height="48"> Transition to general view looks like that:
https://pandia.ru/text/78/083/images/image108_0.gif" width="303" height="24">
Two cases are possible: 0
Hence, the original inequality is equivalent to the following set of systems of inequalities:
The first system has no solutions, and from the second we get: x<-1 – решение неравенства.
Understanding the conditions of equivalence requires knowing some subtleties. For example, why are the following equations equivalent:
Or 
And finally, perhaps the most important. The fact is that equivalence guarantees the correctness of the answer if some transformations of the equation itself are performed, but is not used for transformations in only one of the parts. Reduction, the use of different formulas in one of the parts do not fall under the equivalence theorems. I have already given some examples of this kind. Let's look at some more examples.
1. Such a decision is natural. On the left side, by the property of the logarithmic function, let's move on to the expression ..gif" width="111" height="48">
Solving this system, we get the result (-2 and 2), which, however, is not the answer, since the number -2 is not included in the ODZ. So what do we need to install ODZ? Of course not. But since we used a certain property of the logarithmic function in the solution, we must ensure the conditions under which it is fulfilled. Such a condition is the positiveness of expressions under the sign of the logarithm..gif" width="65" height="48">.
2. 
..gif" width="143" height="27 src="> numbers are subject to substitution in this way 
. Who wants to make such tedious calculations?.gif" width="12" height="23 src="> add a condition, and it is immediately clear that only the number meets this condition https://pandia.ru/text/78/083/ images/image128_0.gif" width="117" height="27 src=">) was demonstrated by 52% of the dealers. One of the reasons for such low performance is the fact that many graduates did not select the roots obtained from the equation after squaring it.
3) Consider, for example, the solution of one of the tasks C1: "Find all x values for which the points of the graph of the function 
lie above the corresponding points of the graph of the function ". The task is reduced to solving fractional inequality containing logarithmic expression. We know how to solve such inequalities. The most common of these is the interval method. However, when using it, dealers make various mistakes. Consider the most common mistakes using the example of inequality:
X< 10. Они отмечают, что в первом случае решений нет, а во втором – корнями являются числа –1 и . При этом выпускники не учитывают условие x < 10.
8. Conclusion
Summing up, we can say that there is no universal method for solving equations and inequalities. Every time, if you want to understand what you are doing, and not act mechanically, a dilemma arises: what method of decision to choose, in particular, to look for ODZ or not? I think that my experience will help me solve this dilemma. I will stop making mistakes once I learn how to use the ODZ correctly. Whether I succeed, time will tell, or rather the exam.
9. Literature
And others. "Algebra and the beginning of analysis 10-11" problem book and textbook, M .: "Enlightenment", 2002. "Handbook of elementary mathematics." M .: "Nauka", 1966. Newspaper "Mathematics" No. 46, Newspaper "Mathematics" No. Newspaper "Mathematics" No. "History of mathematics at school grades VII-VIII". M .: "Enlightenment", 1982. and others. "The most complete edition of options for real tasks of the USE: 2009 / FIPI" - M .: "Astrel", 2009. and others. "USE. Mathematics. Universal materials for the preparation of students / FIPI "- M .: "Intellect-center", 2009. and others. "Algebra and the beginning of analysis 10-11". M .: "Prosveshchenie", 2007. , "Workshop on solving problems of school mathematics (workshop on algebra)". M .: Education, 1976. "25000 lessons of mathematics." M .: "Prosveshchenie", 1993. "Preparing for Olympiads in Mathematics." M.: "Exam", 2006. "Encyclopedia for children "MATHEMATICS"" volume 11, M.: Avanta +; 2002. Materials of the sites www. ***** www. *****.
Fractional equations. ODZ.
Attention!
There are additional
material in Special Section 555.
For those who strongly "not very..."
And for those who "very much...")
We continue to master the equations. We already know how to work with linear and quadratic equations. The last view remains fractional equations . Or they are also called much more solid - fractional rational equations . It is the same.
Fractional equations.
As the name implies, these equations necessarily contain fractions. But not just fractions, but fractions that have unknown in the denominator. At least in one. For example:
Let me remind you, if in the denominators only numbers, these are linear equations.
How to decide fractional equations? First of all, get rid of the fractions! After that, the equation, most often, turns into a linear or quadratic one. And then we know what to do... In some cases, it can turn into an identity, like 5=5 or an incorrect expression, like 7=2. But this rarely happens. Below I will mention it.
But how to get rid of fractions!? Very simple. Applying all the same identical transformations.
We need to multiply the whole equation by the same expression. So that all denominators decrease! Everything will immediately become easier. I explain with an example. Let's say we need to solve the equation:
How were they taught in elementary school? We transfer everything in one direction, reduce it to a common denominator, etc. Forget how horrible dream! This is how you do it when you add or subtract fractional expressions. Or work with inequalities. And in equations, we immediately multiply both parts by an expression that will give us the opportunity to reduce all denominators (i.e., in essence, by common denominator). And what is this expression?
On the left side, to reduce the denominator, you need to multiply by x+2. And on the right, multiplication by 2 is required. So, the equation must be multiplied by 2(x+2). We multiply:
This is the usual multiplication of fractions, but I will write in detail:
Please note that I am not opening the parenthesis yet. (x + 2)! So, in its entirety, I write it:
On the left side, it is reduced entirely (x+2), and in the right 2. As required! After reduction we get linear the equation:
Anyone can solve this equation! x = 2.
Let's solve another example, a little more complicated:
If we remember that 3 = 3/1, and 2x = 2x/ 1 can be written:
And again we get rid of what we don’t really like - from fractions.
We see that to reduce the denominator with x, it is necessary to multiply the fraction by (x - 2). And units are not a hindrance to us. Well, let's multiply. All left side And all right side:
Brackets again (x - 2) I don't reveal. I work with the bracket as a whole, as if it were one number! This must always be done, otherwise nothing will be reduced.
With a feeling of deep satisfaction, we cut (x - 2) and we get the equation without any fractions, in a ruler!
And now we open the brackets:
We give similar ones, transfer everything to the left side and get:
But before that, we will learn to solve other problems. For interest. Those rakes, by the way!
If you like this site...
By the way, I have a couple more interesting sites for you.)
You can practice solving examples and find out your level. Testing with instant verification. Learning - with interest!)
you can get acquainted with functions and derivatives.
The function is the model. Let's define X as a set of values of an independent variable // independent means any.
A function is a rule by which, for each value of the independent variable from the set X, one can find the only value of the dependent variable. // i.e. for every x there is one y.
It follows from the definition that there are two concepts - an independent variable (which we denote by x and it can take any value) and a dependent variable (which we denote by y or f (x) and it is calculated from the function when we substitute x).
FOR EXAMPLE y=5+x
1. Independent is x, so we take any value, let x = 3
2. and now we calculate y, so y \u003d 5 + x \u003d 5 + 3 \u003d 8. (y is dependent on x, because what x we substitute, we get such y)
We say that the variable y is functionally dependent on the variable x and this is denoted as follows: y = f (x).
FOR EXAMPLE.
1.y=1/x. (called hyperbole)
2. y=x^2. (called parabola)
3.y=3x+7. (called straight line)
4. y \u003d √ x. (called the branch of the parabola)
The independent variable (which we denote by x) is called the argument of the function.
Function scope
The set of all values that a function argument takes is called the function's domain and is denoted by D(f) or D(y).
Consider D(y) for 1.,2.,3.,4.
1. D (y)= (∞; 0) and (0;+∞) //the whole set of real numbers except zero.
2. D (y) \u003d (∞; +∞) / / all the many real numbers
3. D (y) \u003d (∞; +∞) / / all the many real numbers
4. D (y) \u003d)
