There are an infinite number of functions in mathematics. And each has its own character.) To work with a wide variety of functions you need single approach. Otherwise, what kind of mathematics is this?!) And there is such an approach!

When working with any function, we present it with a standard set of questions. And the first, the most important question- This domain of definition of the function. Sometimes this area is called the set of valid argument values, the area for specifying a function, etc.

What is the domain of a function? How to find it? These questions often seem complex and incomprehensible... Although, in fact, everything is extremely simple. You can see for yourself by reading this page. Let's go?)

Well, what can I say... Just respect.) Yes! The natural domain of a function (which is discussed here) matches with ODZ of expressions included in the function. Accordingly, they are searched according to the same rules.

Now let’s look at a not entirely natural domain of definition.)

Additional restrictions on the scope of a function.

Here we will talk about the restrictions that are imposed by the task. Those. the task contains some additional conditions, which were invented by the compiler. Or the restrictions emerge from the very method of defining the function.

As for the restrictions in the task, everything is simple. Usually, there is no need to look for anything, everything is already said in the task. Let me remind you that the restrictions written by the author of the task do not cancel fundamental limitations of mathematics. You just need to remember to take into account the conditions of the task.

For example, this task:

Find the domain of a function:

on the set of positive numbers.

We found the natural domain of definition of this function above. This area:

D(f)=( -∞ ; -1) ∪ (-1; 2] ∪ ∪

In the verbal method of specifying a function, you need to carefully read the condition and find restrictions on the Xs there. Sometimes the eyes look for formulas, but the words whistle past the consciousness yes...) Example from the previous lesson:

The function is specified by the condition: each value of the natural argument x is associated with the sum of the digits that make up the value of x.

It should be noted here that we are talking only about the natural values ​​of X. Then D(f) instantly recorded:

D(f): x ∈ N

As you can see, the domain of a function is not such a complicated concept. Finding this region comes down to examining the function, writing a system of inequalities, and solving this system. Of course, there are all kinds of systems, simple and complex. But...

I'll open it little secret. Sometimes a function for which you need to find the domain of definition looks simply intimidating. I want to turn pale and cry.) But as soon as I write down the system of inequalities... And, suddenly, the system turns out to be elementary! Moreover, often, the more terrible the function, the simpler the system...

Moral: the eyes fear, the head decides!)

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

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

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

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

The mapping f itself is called characteristic of the function.

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

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

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

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

Graph of function f called a set of pairs.

Complex functions

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

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

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

Valid functions

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

Numerical functions play an important role in mathematical analysis.

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

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

Maximum and minimum

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

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

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

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

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

The upper bound of an upper bounded function

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

The infimum of a lower bounded function is infinite remote point.

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

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

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

Monotonic functions

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

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

Multivalued functions

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

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

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

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

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

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

Multivalued function branch is one of the functions included in multivalued function.

Single-valued function is a function.

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

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Any expression with a variable has its own range of valid values, where it exists. ODZ must always be taken into account when making decisions. If it is absent, you may get an incorrect result.

This article will show how to correctly find ODZ and use examples. The importance of indicating the DZ when making a decision will also be discussed.

Yandex.RTB R-A-339285-1

Valid and invalid variable values

This definition is related to the allowed values ​​of the variable. When we introduce the definition, let's see what result it will lead to.

Starting in 7th grade, we begin to work with numbers and numerical expressions. Initial definitions with variables move on to the meaning of expressions with selected variables.

When there are expressions with selected variables, some of them may not satisfy. For example, an expression of the form 1: a, if a = 0, then it does not make sense, since it is impossible to divide by zero. That is, the expression must have values ​​that are suitable in any case and will give an answer. In other words, they make sense with the existing variables.

Definition 1

If there is an expression with variables, then it makes sense only if the value can be calculated by substituting them.

Definition 2

If there is an expression with variables, then it does not make sense when, when substituting them, the value cannot be calculated.

That is, this implies a complete definition

Definition 3

Existing admissible variables are those values ​​for which the expression makes sense. And if it doesn’t make sense, then they are considered unacceptable.

To clarify the above: if there is more than one variable, then there may be a pair of suitable values.

Example 1

For example, consider an expression of the form 1 x - y + z, where there are three variables. Otherwise, you can write it as x = 0, y = 1, z = 2, while another entry has the form (0, 1, 2). These values ​​are called valid, which means that the value of the expression can be found. We get that 1 0 - 1 + 2 = 1 1 = 1. From this we see that (1, 1, 2) are unacceptable. The substitution results in division by zero, that is, 1 1 - 2 + 1 = 1 0.

What is ODZ?

Range of acceptable values ​​– important element when evaluating algebraic expressions. Therefore, it is worth paying attention to this when making calculations.

Definition 4

ODZ area is the set of values ​​allowed for a given expression.

Let's look at an example expression.

Example 2

If we have an expression of the form 5 z - 3, then the ODZ has the form (− ∞, 3) ∪ (3, + ∞) . This is the range of valid values ​​that satisfies the variable z for a given expression.

If there are expressions of the form z x - y, then it is clear that x ≠ y, z takes any value. This is called ODZ expressions. It must be taken into account so as not to obtain division by zero when substituting.

The range of permissible values ​​and the range of definition have the same meaning. Only the second of them is used for expressions, and the first is used for equations or inequalities. With the help of DL, the expression or inequality makes sense. The domain of definition of the function coincides with the range of permissible values ​​of the variable x for the expression f (x).

How to find ODZ? Examples, solutions

Finding ODZ means finding everything valid values, suitable for given function or inequality. Failure to meet these conditions may result in incorrect results. To find the ODZ, it is often necessary to go through transformations in a given expression.

There are expressions where their calculation is impossible:

• if there is division by zero;
• taking the root of a negative number;
• the presence of a negative integer indicator – only for positive numbers;
• calculating the logarithm of a negative number;
• domain of definition of tangent π 2 + π · k, k ∈ Z and cotangent π · k, k ∈ Z;
• finding the value of the arcsine and arccosine of a number for a value not belonging to [ - 1 ; 1].

All this shows how important it is to have ODZ.

Example 3

Find the ODZ expression x 3 + 2 x y − 4 .

Solution

Any number can be cubed. This expression does not have a fraction, so the values ​​of x and y can be any. That is, ODZ is any number.

Answer: x and y – any values.

Example 4

Find the ODZ of the expression 1 3 - x + 1 0.

Solution

It can be seen that there is one fraction where the denominator is zero. This means that for any value of x we ​​will get division by zero. This means that we can conclude that this expression is considered undefined, that is, it does not have any additional liability.

Answer: ∅ .

Example 5

Find the ODZ of the given expression x + 2 · y + 3 - 5 · x.

Solution

The presence of a square root means that this expression must be greater than or equal to zero. If it is negative, it has no meaning. This means that it is necessary to write an inequality of the form x + 2 · y + 3 ≥ 0. That is, this is the desired range of acceptable values.

Answer: set of x and y, where x + 2 y + 3 ≥ 0.

Example 6

Determine the ODZ expression of the form 1 x + 1 - 1 + log x + 8 (x 2 + 3) .

Solution

By condition, we have a fraction, so its denominator should not be equal to zero. We get that x + 1 - 1 ≠ 0. The radical expression always makes sense when greater than or equal to zero, that is, x + 1 ≥ 0. Since it has a logarithm, its expression must be strictly positive, that is, x 2 + 3 > 0. The base of the logarithm must also have a positive value and different from 1, then we add the conditions x + 8 > 0 and x + 8 ≠ 1. It follows that the desired ODZ will take the form:

x + 1 - 1 ≠ 0, x + 1 ≥ 0, x 2 + 3 > 0, x + 8 > 0, x + 8 ≠ 1

In other words, it is called a system of inequalities with one variable. The solution will lead to the following ODZ notation [ − 1, 0) ∪ (0, + ∞) .

Answer: [ − 1 , 0) ∪ (0 , + ∞)

Why is it important to consider DPD when driving change?

During identity transformations, it is important to find the ODZ. There are cases when the existence of ODZ does not occur. To understand whether a given expression has a solution, you need to compare the VA of the variables of the original expression and the VA of the resulting one.

Identity transformations:

• may not affect DL;
• may lead to the expansion or addition of DZ;
• can narrow the DZ.

Let's look at an example.

Example 7

If we have an expression of the form x 2 + x + 3 · x, then its ODZ is defined over the entire domain of definition. Even when bringing similar terms and simplifying the expression, the ODZ does not change.

Example 8

If we take the example of the expression x + 3 x − 3 x, then things are different. We have a fractional expression. And we know that division by zero is unacceptable. Then the ODZ has the form (− ∞, 0) ∪ (0, + ∞) . It can be seen that zero is not a solution, so we add it with a parenthesis.

Let's consider an example with the presence of a radical expression.

Example 9

If there is x - 1 · x - 3, then you should pay attention to the ODZ, since it must be written as an inequality (x − 1) · (x − 3) ≥ 0. It is possible to solve by the interval method, then we find that the ODZ will take the form (− ∞, 1 ] ∪ [ 3 , + ∞) . After transforming x - 1 · x - 3 and applying the property of roots, we have that the ODZ can be supplemented and everything can be written in the form of a system of inequalities of the form x - 1 ≥ 0, x - 3 ≥ 0. When solving it, we find that [ 3 , + ∞) . This means that the ODZ is completely written as follows: (− ∞, 1 ] ∪ [ 3 , + ∞) .

Transformations that narrow the DZ must be avoided.

Example 10

Let's consider an example of the expression x - 1 · x - 3, when x = - 1. When substituting, we get that - 1 - 1 · - 1 - 3 = 8 = 2 2 . If we transform this expression and bring it to the form x - 1 · x - 3, then when calculating we find that 2 - 1 · 2 - 3 the expression makes no sense, since the radical expression should not be negative.

It is necessary to adhere to identical transformations that the ODZ will not change.

If there are examples that expand on it, then it should be added to the DL.

Example 11

Let's look at the example of a fraction of the form x x 3 + x. If we cancel by x, then we get that 1 x 2 + 1. Then the ODZ expands and becomes (− ∞ 0) ∪ (0 , + ∞) . Moreover, when calculating, we already work with the second simplified fraction.

In the presence of logarithms, the situation is slightly different.

Example 12

If there is an expression of the form ln x + ln (x + 3), it is replaced by ln (x · (x + 3)), based on the property of the logarithm. From this we can see that ODZ from (0 , + ∞) to (− ∞ , − 3) ∪ (0 , + ∞) . Therefore for ADL definitions ln (x · (x + 3)) it is necessary to carry out calculations on the ODZ, that is, the (0 , + ∞) set.

When solving, it is always necessary to pay attention to the structure and type of the expression given by the condition. If the definition area is found correctly, the result will be positive.

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How ?
Examples of solutions

If something is missing somewhere, it means there is something somewhere

We continue to study the “Functions and Graphs” section, and the next station on our journey is. Active discussion this concept began in the article about sets and continued in the first lesson about function graphs, where I looked at elementary functions, and, in particular, their domains of definition. Therefore, I recommend that dummies start with the basics of the topic, since I will not dwell again on some basic points.

It is assumed that the reader knows the domain of definition of the following functions: linear, quadratic, cubic functions, polynomials, exponential, sine, cosine. They are defined on (the set of all real numbers). For tangents, arcsines, so be it, I forgive you =) - rarer graphs are not immediately remembered.

The domain of definition seems to be a simple thing, and it arises logical question, what will the article be about? On this lesson I will consider common problems of finding the domain of definition of a function. Moreover, we will repeat inequalities with one variable, the solution skills of which will also be required in other problems of higher mathematics. The material, by the way, is all school material, so it will be useful not only for students, but also for students. The information, of course, does not pretend to be encyclopedic, but here are not far-fetched “dead” examples, but roasted chestnuts, which are taken from real practical works.

Let's start with a quick dive into the topic. Briefly about the main thing: we are talking about a function of one variable. Its domain of definition is many meanings of "x", for which exist meanings of "players". Let's look at a hypothetical example:

The domain of definition of this function is a union of intervals:
(for those who have forgotten: - unification icon). In other words, if you take any value of “x” from the interval , or from , or from , then for each such “x” there will be a value “y”.

Roughly speaking, where the domain of definition is, there is a graph of the function. But the half-interval and the “tse” point are not included in the definition area and there is no graph there.

How to find the domain of a function? Many people remember the children's rhyme: “rock, scissors, paper,” and in in this case it can be safely paraphrased: “root, fraction and logarithm.” Thus, if you come across a fraction, root or logarithm on your life’s path, you should immediately be very, very wary! Tangent, cotangent, arcsine, arccosine are much less common, and we will also talk about them. But first, sketches from the life of ants:

Domain of a function that contains a fraction

Suppose we are 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 won’t dwell on the most simple functions like etc., since everyone perfectly sees points that are not included in their domain of definition. Let's look at more meaningful fractions:

Example 1

Find the domain of a function

Solution: There is nothing special in the numerator, but the denominator must be non-zero. Let's set it equal to zero and try to find the “bad” points:

The resulting equation has two roots: . Data values are not in the scope of the function. Indeed, substitute or into the function and you will see that the denominator goes to zero.

Answer: scope of definition:

The entry reads like this: “the domain of definition is all real numbers with the exception of the set consisting of values " Let me remind you that the backslash symbol in mathematics means logical subtraction, and braces– many. The answer can be equivalently written as a union of three intervals:

Whoever likes it.

At points function tolerates endless breaks, and the straight lines given by the equations are vertical asymptotes for the graph of this function. However, this is a slightly different topic, and I will not focus on this further.

Example 2

Find the domain of a function

The task is essentially oral and many of you will almost immediately find the area of ​​definition. The answer is at the end of the lesson.

Will a fraction always be “bad”? No. For example, a function is defined on the entire number line. No matter what value of “x” we take, the denominator will not go to zero, moreover, it will always be positive: . Thus, the scope of this function is: .

All functions like defined and continuous on .

The situation is a little more complicated when the denominator is occupied by a quadratic trinomial:

Example 3

Find the domain of a function

Solution: Let's try to find the points at which the denominator goes to zero. For this we will decide quadratic equation:

The discriminant turned out to be negative, which means there are no real roots, and our function is defined on the entire number axis.

Answer: scope of definition:

Example 4

Find the domain of a function

This is an example for independent decision. The solution and answer are at the end of the lesson. I advise you not to be lazy with simple problems, since misunderstandings will accumulate with further examples.

Domain of a function with a root

Function with square root 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 positive even degree: , however, the root is already of the 4th degree in function studies I don't remember.

Example 5

Find the domain of a function

Solution: the radical expression must be non-negative:

Before continuing with the solution, let me remind you of the basic rules for working with inequalities, known from school.

I pay special attention! Now we are considering 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) The terms can be transferred from part to part by changing their (the terms) signs.

2) Both sides of the inequality can be multiplied by a positive number.

3) If both sides of the inequality are multiplied by negative number, then you need to change sign of inequality itself. For example, if there was “more”, then it will become “less”; if it was “less than or equal”, then it will become “greater than or equal”.

In the inequality we move the “three” to right side with change of sign (rule No. 1):

Let's multiply both sides of the inequality by –1 (rule No. 3):

Let's multiply both sides of the inequality by (rule No. 2):

Answer: scope of definition:

The answer can also be written in an equivalent phrase: “the function is defined at .”
Geometrically, the definition area is depicted by shading the corresponding intervals on the abscissa axis. In this case:

I remind you once again geometric meaning domain of definition – graph of a function exists only in the shaded area and is absent at .

In most cases, a purely analytical determination of the domain of definition is suitable, but when the function is very complicated, you should draw an axis and make notes.

Example 6

Find the domain of a function

This is an example for you to solve on your own.

When there is a square binomial or trinomial under the square root, the situation becomes a little more complicated, and now we will analyze in detail the solution technique:

Example 7

Find the domain 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 factor the quadratic trinomial:

The discriminant is positive, we are looking for 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 located above the axis (the inequality we need).

Since the coefficient is , the branches of the parabola point upward. From the above it follows that the inequality is satisfied on the intervals (the branches of the parabola go upward to infinity), and the vertex of the parabola is located on the interval below the x-axis, which corresponds to the inequality:

! Note: If you don't fully understand the explanations, please draw the second axis and the entire parabola! It is advisable to return to the article and manual Hot formulas for school mathematics course.

Please note that the points themselves are removed (not included in the solution), since our inequality is strict.

Answer: scope of definition:

In general, many inequalities (including the one considered) are solved by the universal interval method, known again from the school curriculum. But in the cases of square binomials and trinomials, in my opinion, it is much more convenient and faster to analyze the location of the parabola relative to the axis. And we will analyze the main method - the interval method - in detail in the article. Function zeros. Constancy intervals.

Example 8

Find the domain of a function

This is an example for you to solve on your own. The sample comments in detail on the logic of reasoning + the second method of solution and another important transformation of inequality, without knowledge of which the student will be limping on one leg..., ...hmm... I guess I got excited about the leg, rather, on one toe. 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 ​​of “x” and “ka”: , therefore also and .

Here's a less obvious example: . Here the discriminant is negative (the parabola does not intersect the x-axis), while the branches of the parabola are directed upward, hence the domain of definition: .

The opposite question: can the domain of definition 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: (empty set icon). Such a function is not defined at all (of course, the graph is also illusory).

WITH odd roots etc. everything is much better - here radical expression can be negative. For example, a function is defined on the entire number line. However, the function has a single point that is still not included in the domain of definition, since the denominator is set to zero. For the same reason for the function points are excluded.

Domain of a function with a logarithm

The third common function is the logarithm. As an example, I will draw the natural logarithm, which occurs in approximately 99 examples out of 100. If a certain function contains a logarithm, then its domain of definition should include only those values ​​of “x” that satisfy the inequality. If the logarithm is in the denominator: , then additionally a condition is imposed (since ).

Example 9

Find the domain of a function

Solution: in accordance with the above, we will compose and solve the system:

Graphic solution for dummies:

Answer: scope of definition:

I’ll dwell on one more technical point - I don’t have the scale indicated and the divisions along the axis are not marked. The question arises: how to make such drawings in a notebook on checkered paper? Should the distance between points be measured by 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 domain of a function

To solve the problem, you can use the method of the previous paragraph - analyze how the parabola is located relative to the x-axis. The answer is at the end of the lesson.

As you can see, in the realm of logarithms everything is very similar to the situation with square roots: the function (square trinomial from Example No. 7) is defined on the intervals, and the function (square binomial from Example No. 6) on the interval . It’s embarrassing to even say, type functions are defined on the entire number line.

Useful information : interesting typical function, it is defined on the entire number line except the point. According to the property of the logarithm, the “two” can be multiplied outside the logarithm, but in order for the function not to change, the “x” must be enclosed under the modulus sign: . Here's another one for you" practical application» 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 modulus sign and it is enough to use parentheses: .

To avoid repetition, let's complicate the task:

Example 11

Find the domain of a function

Solution: in this function we have both the root and the logarithm.

The radical 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 system solution must satisfy to everyone condition.

By examining the location of the parabola relative to the axis, we come to the conclusion that the inequality is satisfied by the interval (blue shading):

The inequality obviously corresponds to the “red” half-interval.

Since both conditions must be met simultaneously, then the solution to the system is the intersection of these intervals. "Common interests" are met at half-time.

Answer: scope of definition:

The typical inequality, as demonstrated in Example No. 8, is not difficult to resolve analytically.

The found domain will not change for “similar functions”, e.g. 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 matan theory about this verbal... oh... there are theorems.

Example 12

Find the domain of a function

This is an example for you to solve on your own. Using a drawing is quite appropriate, since the function is not the simplest.

A couple more examples to reinforce the material:

Example 13

Find the domain of a function

Solution: let’s compose and solve the system:

All actions have already been discussed throughout the article. Let us depict the interval corresponding to the inequality on the number line and, according to the second condition, eliminate two points:

The meaning turned out to be completely irrelevant.

Answer: domain of definition

A little math pun on a variation of the 13th example:

Example 14

Find the domain of a function

This is an example for you to solve on your own. Those who missed it are out of luck ;-)

The final section of the lesson is devoted to more rare, but also “working” functions:

Function Definition Areas
with tangents, cotangents, arcsines, arccosines

If some function includes , then from its domain of definition excluded points , Where Z– a 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: .

Let's not kill too much:

Example 15

Find the domain of a function

Solution: in this case they will not be included in the scope of definition next points:

Let's throw the "two" of the left side into the denominator of the right side:

As a result :

Answer: scope of definition: .

In principle, the answer can be written as a combination of an infinite number of intervals, but the construction will be very cumbersome:

The analytical solution is completely consistent with geometric transformation of the graph: if the argument of a function is multiplied by 2, then its graph will shrink to the axis twice. Notice how the function's period has been halved, and break points doubled in frequency. Tachycardia.

A similar story with cotangent. If some function includes , then the points are excluded from its domain of definition. In particular, for the automatic burst function we shoot the following values:

In other words: