ISSN:
2181-3906
2023
International scientific journal
«MODERN SCIENCE АND RESEARCH»
VOLUME 2 / ISSUE 5 / UIF:8.2 / MODERNSCIENCE.UZ
730
THE SIMPLEST AND MOST CONVENIENT METHODS OF DEFINING FUNCTIONS
IN MATHEMATICS AND EXAMPLES OF GP IN THE FIELD OF VALUES
Toshboyeva Saidaxon Rahmonberdiyevna
Fargʻona davlat universiteti oʻqituvchisi
Saidakhon.toshboyeva@gmail.com
Hojimamatova Hilola Umidjon qizi
Fargʻona davlat universiteti oʻquvchisi
hojimamatovahilola@gmail.com
https://doi.org/10.5281/zenodo.7954073
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.
ПРОСТЕЙШИЕ И УДОБНЫЕ СПОСОБЫ ОПРЕДЕЛЕНИЯ ФУНКЦИЙ В
МАТЕМАТИКЕ И ПРИМЕРЫ ГП В ОБЛАСТИ ЗНАЧЕНИЙ
Аннотация.
В этой статье представлен общий и подробный обзор области
определения и значений, которые многим читателям кажутся трудными. Введена
возможность находить ответы на простые и сложные функции удобными методами.
Ключевые слова:
область определения, область значений, квадратичная функция,
тригонометрические функции, линейные функции.
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.
ISSN:
2181-3906
2023
International scientific journal
«MODERN SCIENCE АND RESEARCH»
VOLUME 2 / ISSUE 5 / UIF:8.2 / MODERNSCIENCE.UZ
731
1
The
domain
of
linear
quadratic
and
cubic
functions
like
,
²
,
³
²
f x
kx b f x
ax
bx c f x
ax
bx
cx t
....
is always
;
. This is
simply because there are no exceptions to the values
x
can accept.
2.
1
f x
x
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
0
x
. 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
1
2
f x
x
function. In this case, we work as
2 0
x
and get
2
x
. That is, when the denominator takes the number 2, it remains 0, so the number 2
should be removed from the inequality.
Answer:
; 2 2;
D
.
4. Find the domain of the function zzzz
1) we do not pay attention to
3
x
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)
² 3
2 0;
2,
1
x
x
x
x
is formed
3) We remove the numbers 2 and 1 from the numbers (-∞; ∞). And the following inequality is
formed.
Answer:
; 1 1; 2 2;
D
5. Let's pay attention to the definition area of the function
f x
x
. For functions under
even roots, the only condition
0
x
is enough. And the following answer is formed.
Answer:
0;
D
6. Find the area where
3
f x
x
is defined
1)
3 0.
3
x
x
is formed
Answer:
3;
D
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
3
6
0
x
. 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)
3
6;
6
x
x
Answer:
6;
D
ISSN:
2181-3906
2023
International scientific journal
«MODERN SCIENCE АND RESEARCH»
VOLUME 2 / ISSUE 5 / UIF:8.2 / MODERNSCIENCE.UZ
732
8. Find the domain of the
f x
x
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.
1
1
1
,
,
....
²
³
f x
f x
f x
x
x
x
⁴
Etc., the field of definition of functions of the form
1
f x
x
takes values (-∞;0) (0; ∞).
Answer:
;0 0;
D
10. The definition area of
.
,
,
,
f x
sinx f x
cosx f x
tgx f x
ctgx
functions is
;
accepts numbers up to That is,
x
is always valid in all values.
Answer:
;
D
But there are some exceptional cases. Let's take a look at them. For example, given a function
1
f x
sin
x
, a very simple solution is the same as the function
1
f x
x
with domain of
definition. That is,
; 0 0;
D
is formed. Don't get distracted by one thing, it's not
good to rush to assume that the function
sin
x
is always
;
D
. Regardless of what
function is given, we should always pay attention to the values that
x
can take.
Or consider the function
1
² 4
3
f x
tg
x
x
.
1) It is enough to find the domain of the
1
² 4
3
x
x
function
2)
² 4
3 0,
3,
1
x
x
x
x
is formed.
The domain of the function
1
² 4
3
f x
tg
x
x
is (-∞; ∞), the numbers 3 and 1 are removed
and the following answer is obtained.
Answer
:
; 1 , 1;3 , 3;
D
Let's look at examples from the field of values.
1. Find the domain of the function
F x
kx b
.
1) The example is solved by replacing the expressions
k
and
b
with numbers.
2)
2,
3;
2
3
k
b
y
x
this expression has a solution for all real values of
1,
4;
0,
3;
1,
1.....
x
y
x
y
x
y
It can go on like this.
ISSN:
2181-3906
2023
International scientific journal
«MODERN SCIENCE АND RESEARCH»
VOLUME 2 / ISSUE 5 / UIF:8.2 / MODERNSCIENCE.UZ
733
Answer:
;
A
2. Find the domain of the function
²
F x
ax
bx c
.
1) A, b, c expression is replaced by a numerical value
2)
1,
4,
3
² 4
3
a
b
c
y
x
x
3)
² 4
3
y
x
x
is derived from the expression.
'
2
4
y
x
4) The expression
' 2
4
y
x
is equal to
0.
2
4
0;
2
x
x
5) The expression
2
x
is substituted for the expression
² 4
3
y
x
x
6)
2² 4 2 3;
4 8 3;
1
y
y
y
7) means, after satisfying the condition
0
a
, the answer is
; 1
. If
0
a
, it would be the
opposite, i.e.
1;
Answer:
; 1
A
3. Find the domain of the function
1
F x
x
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
0
x
. And this number 0 is removed from the number line. And the
following response is generated.
Answer:
; 0 0;
4. We try to find the range of values of the expression
1
3
9
y
x
. Based on the reasoning above,
the denominator should not be 0, and we should exclude that number from the answer.
1)
3
9;
3
x
x
x
is formed
Answer:
; 3 3;
5. Find the domain of the function
F x
x
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
,
F x
cosx F x
sinx
.
1) in these functions, which we have seen above,
x
is an unknown number of arbitrary infinite
values.
2) but
x
always has
1;1
values even when it is any infinite number
Answer:
1;1
7. Find the domain of the functions
𝐹(𝑥) = 𝑡𝑔𝑥 𝑎𝑛𝑑 𝐹(𝑥) = 𝑐𝑡𝑔𝑥
. Among the functions, the most
simple way to find the values and the field of determination is precisely these functions.
ISSN:
2181-3906
2023
International scientific journal
«MODERN SCIENCE АND RESEARCH»
VOLUME 2 / ISSUE 5 / UIF:8.2 / MODERNSCIENCE.UZ
734
Answer:
;
D
A
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.
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