Abstract: This article provides a general and detailed overview of the area of definition and values that many readers find difficult. The ability to find answers to simple and complex functions using convenient methods is introduced.
Key words: Domain of definition, domain of values, quadratic function, trigonometric functions, linear functions
Another important issue that always comes to our mind is related to the manners, behavior and, in a word, worldview of our youth. Today, times are changing rapidly. Those who feel these changes the most are young people. Let the youth be in harmony with the demands of their time. But at the same time, he should not forget his identity. Let the call of who we are, the descendants of great people always echo in the heart of the file and encourage us to stay true to ourselves. What can we achieve? Education, education and only education. (I.A. Karimov.)
Every person wants the science he studies to be more perfect and looks for its favorable aspects, works tirelessly on himself, as it can be seen that every science and field has its own difficulties and aspects to consider. For example, let’s look at mathematics, from afar it seems difficult and sometimes impossible, but if we look closely at mathematics, it becomes much easier to understand its beauty and meaning. As a proof of this, we will provide information about the simplest and most convenient methods of defining functions and examples in the field of values in mathematics. First, “What is a domain of definition and a domain of values?” Let’s form general ideas about this concept. That is, we do not abstract certain concepts in our brain with the rule. Actually the rule confuses the point, this is my personal opinion.
The field of definition is the values that the function can accept, and the expression formed by these values is called the field of values of the function. As a clear example of this, we will get acquainted with the following functions. First, let’s get acquainted with examples related to the field of detection.
1 The domain of linear quadratic and cubic functions like …. is always . This is simply because there are no exceptions to the values can accept.
2. What is the domain of the function? We find a solution to this as follows. It cannot accept only the number 0 in the denominator. That is, it satisfies all values other than 0. We set the condition . This results in the following inequality. We remove the number 0 from the numbers up to and get the following inequality.
Answer: D=(-∞; 0) (0; ∞)
3. Let’s find the field of definition for the function. In this case, we work as and get . That is, when the denominator takes the number 2, it remains 0, so the number 2 should be removed from the inequality.
Answer: .
4. Find the domain of the function zzzz
1) we do not pay attention to in the picture. We only find the definition area for the denominator. The domain of definition in the denominator is valid for an entire function.
2) is formed
3) We remove the numbers 2 and 1 from the numbers (-∞; ∞). And the following inequality is formed.
Answer:
5. Let’s pay attention to the definition area of the function . For functions under even roots, the only condition is enough. And the following answer is formed.
Answer:
6. Find the area where is defined
1) is formed
Answer:
7. Find the domain of the function f(x).
1) we apply the expression under the root according to the above rule.
2) We give the condition . Why is the inequality sign > instead of The reason is very simple, if the sign is non-deterministic, a value will be generated that will make the denominator 0, so we specify a non-deterministic sign.
3)
Answer:
8. Find the domain of the function
1) An even number always appears under the module. We can say that this is an invariant axiom. That is, it is an opinion that does not require any proof.
2) So the expression under the module can accept any number.
Answer: (-∞; ∞)
9. Etc., the field of definition of functions of the form takes values (-∞;0) (0; ∞).
Answer:
10. The definition area of . functions is accepts numbers up to That is, is always valid in all values.
Answer:
But there are some exceptional cases. Let’s take a look at them. For example, given a function , a very simple solution is the same as the function with domain of definition. That is, is formed. Don’t get distracted by one thing, it’s not good to rush to assume that the function is always . Regardless of what function is given, we should always pay attention to the values that can take.
Or consider the function .
1) It is enough to find the domain of the function
2) is formed.
The domain of the function is (-∞; ∞), the numbers 3 and 1 are removed and the following answer is obtained.
Answer
Let’s look at examples from the field of values.
1. Find the domain of the function .
1) The example is solved by replacing the expressions and with numbers.
2) this expression has a solution for all real values of It can go on like this.
Answer:
2. Find the domain of the function .
1) A, b, c expression is replaced by a numerical value
2)
3) is derived from the expression.
4) The expression is equal to 0.
5) The expression is substituted for the expression
6)
7) means, after satisfying the condition , the answer is . If , it would be the opposite, i.e.
Answer:
3. Find the domain of the function
1) Let’s think a little through this example condition. When the denominator is 0, the expression has an unacceptable range of values.
2) This expression produces . And this number 0 is removed from the number line. And the following response is generated.
Answer:
4. We try to find the range of values of the expression . Based on the reasoning above, the denominator should not be 0, and we should exclude that number from the answer.
1) is formed
Answer:
5. Find the domain of the function
1) Such examples are actually very simple. It is necessary to understand the way of work, not to memorize it. These simple examples are the basis for working on difficult examples.
Let’s look at the range of values of functions .
1) in these functions, which we have seen above, is an unknown number of arbitrary infinite values.
2) but always has values even when it is any infinite number
Answer:
7. Find the domain of the functions. Among the functions, the most simple way to find the values and the field of determination is precisely these functions.
Answer:
Conclusion: The methods and recommendations given in this article about the examples and problems given in the field of values and the field of definition are very useful. In the society we live in, many things seem complicated, but everything is very easy. It is only necessary to know how to place these complexities in the child’s mind and to have the right psychological approach. Taking into account his nature and thinking, a certain topic should be explained in a childish way in the language he is interested in.
References:
1. Karimov I.A. “A perfect generation is the foundation of the development of Uzbekistan” Tashkent: Spirituality 1997.
2. Jumayev M.E., Tajiyeva Z.G. “Methodology of teaching mathematics in primary grades” Tashkent: science and technology, 2005.
3. Toshboyeva, S. R., & Turg’unova, N. M. (2021). THE ROLE OF MATHEMATICAL OLYMPIADS IN THE DEVELOPMENT OF INDIVIDUAL CONSCIOUSNESS. Theoretical & Applied Science, (4), 247-251.
4. Toshboyeva, S. R., & Shavkatjonqizi, S. M. (2021). Specific ways to improve mathematical literacy in the process of sending students to hinger education. Academicia: An international multidisciplinary research journal, 11(10), 234-240.
5. Toshboyeva, S. R. (2020). Competent approach in teaching probability theory and mathematical statistics. EPRA International Journal of Research and Development (IJRD).
6. Rahmonberdiyevna, T. S., & Soxibovna, A. M. (2021). Techniques for Teaching Elementary Students Rational Numbers and Convenient ways to Perform Operations on Them. International journal of culture and modernity, 11, 283-287.
7. Raxmonberdiyevna, T. S., & Shavkatjonqizi, S. M. (2021). Methods for the development of stochastic competence in mathematics lessons at school. ACADEMICIA: An International Multidisciplinary Research Journal, 11(5), 863-866.
8. Rakhmonberdiyevna, T. S. (2022). CREATIVITY AS A PEDAGOGICAL PROBLEM. Conferencea, 138-141.
9. Rakhmonberdiyevna, T. S. (2022). RESEARCH OF CREATIVE ACTIVITY OF THE FUTURE PRIMARY CLASS TEACHER. Conferencea, 155-157.
10. Rahmonberdiyevna, T. S. (2022). MULOHAZALAR VA PREDIKATLAR USTIDA AMALLAR MAVZUSI BO ‘YICHA BA’ZI MASALALARNI YECHISHNING SODDA YO ‘LLARI. RESEARCH AND EDUCATION, 1(2), 234-237.
11. Тошбоева, С. Р. (2022). БЎЛАЖАК БОШЛАНҒИЧ СИНФ ЎҚИТУВЧИЛАРИНИНГ ИЖОДИЙ ҚОБИЛИЯТЛАРИНИ РИВОЖЛАНТИРИШНИНГ НАЗАРИЙ АСОСЛАРИ. Journal of new century innovations, 4(1), 294-297.
12. Тошбоева, С. Р. (2022). БЎЛАЖАК БОШЛАНҒИЧ СИНФ ЎҚИТУВЧИЛАРНИНГ ИЖОДИЙ ҚОБИЛИЯТЛАРИНИ МАЛАКАВИЙ АМАЛИЁТ ЖАРАЁНИДА РИВОЖЛАНТИРИШНИНГ ПЕДАГОГИК ХУСУСИЯТЛАРИ. Journal of new century innovations, 4(1), 289-293.
13. Toshboeva, S. R., Mallaboeva, B. M., & Yusupova, L. A. K. (2022). LAWS AND THEIR APPLICATION BASED ON SOME INTERESTING COMBINATORIAL PROBLEMS. Oriental renaissance: Innovative, educational, natural and social sciences, 2(10), 1341-1349.
14. Rahmonberdievna, T. S., & Mukhtorovna, M. B. (2022). LAWS AND THEIR APPLICATION BASED ON SOME INTERESTING COMBINATORIAL PROBLEMS.
15. www.pdffactory.com.