Definition
Function y=f (x) the law (rule, mapping) is called, 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 function scope.
Set of elements y ∈ Y, which have preimages in the set X , is called set of function values(or range).

Domain functions are sometimes called set of definitions or set of tasks functions.

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

The mapping f itself is called function characteristic.

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

The character denoting the characteristic can be the same as the character of the function value element. That is, you can write it like this: At the same time, it is worth remembering that y is an element from the set of function values, and is a rule according to which element x is associated with element y .

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

Private value of the function name the value of the function for the selected (private) value of its argument.

The graph of the function f is called a set of pairs.

Complex functions

Definition
Let the functions and be given. Moreover, the domain of the function f contains a set of values ​​of the function g. Then each element t from the domain 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 is sometimes referred to as:

In mathematical analysis, it is generally accepted that if the characteristic of a function is denoted by one letter or symbol, then it sets 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 and are considered distinct. Let's take an example from physics. Suppose we are considering the dependence of the momentum on the coordinate . And let we have the dependence of the coordinate on time . Then the dependence of momentum on time is a complex function. But for brevity, it is denoted as follows:. With this approach, and are different functions. With the same values ​​of the arguments, they can give different values. In mathematics, this notation is not accepted. If a reduction is required, then a new characteristic must be entered. For example . Then it is clearly seen that and are different functions.

Valid Functions

The scope of the function and the set of its values ​​can be any sets.
For example, numerical sequences are functions whose domain of definition 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.
The boolean expression is a function. Its domain of definition is the set of real numbers (or any set in which the operation of comparison 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 the 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 such a number M that the following inequality holds for all:
.

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

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

top face or exact upper bound real, bounded from above, the function is called the smallest of the numbers that limits the range of its values ​​from above. That is, this is a number s for which, for all and for any , there is such an argument, the value of the function of which exceeds s′ : .
The upper bound of the function can be denoted as follows:
.

The upper bound of a function unbounded from above

bottom face or precise lower bound real, bounded from below, the function is called the largest of the numbers that limits the range of its values ​​from below. That is, this is a number i for which for all and for any , there is such an argument , the value of the function from which is less than i′ : .
The lower bound of a function can be denoted as follows:
.

The lower bound of a function unbounded from below is the point at infinity.

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 all .
This function has top and bottom faces:
.
But it has no maximum and minimum.

If we consider the same function on the interval , then it is bounded above and below on this set, has upper and lower bounds, and has a maximum and a minimum:
for all ;
;
.

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 marked with different colors. Each branch is a feature.

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 an infinite number of 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 satisfies infinitely many values ​​of y (see figure).

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

Instead of condition (2), one 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 the others. Many of these functions are multivalued function. And the function determined from (1) under condition (2.n) is branch of a multivalued function.

This is a collection of functions defined on some set.

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

single valued function is a function.

References:
O.I. Demons. Lectures on mathematical analysis. Part 1. Moscow, 2004.
L.D. Kudryavtsev. Course of mathematical analysis. Volume 1. Moscow, 2003.
CM. Nikolsky. Course of mathematical analysis. Volume 1. Moscow, 1983.

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 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 of solving which will be required in other problems of 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 people remember the children's rhyme: "stone, scissors, paper", and 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 path, then 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:

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 the 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 the .

A little more complicated is the situation when the denominator occupied the 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 turned out to be negative, which means that there are no real roots, and our function is defined on the entire number axis.

Answer: domain:

Example 4

Find the scope of a function

This is a do-it-yourself example. 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

The square root function 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 pay 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 comments in detail on the logic of reasoning + the second solution method 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 function with a square root 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 a natural logarithm, which comes across in about 99 examples out of 100. If a certain function contains a logarithm, then its domain of definition should include only those x values ​​\u200b\u200bthat 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.

Useful 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 you have one more "practical application" of the 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:

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 - ∞; +∞[ .

Example 1. Find the scope of a function y = 2 .

Solution. The scope of the function is not specified, which means that, by virtue of the above definition, the natural domain of definition is meant. Expression f(x) = 2 is defined for any real values x, therefore, this function is defined on the entire set R real numbers.

Therefore, in the drawing above, the number line is shaded all the way from minus infinity to plus infinity.

Scope of the root n th degree

In the case when the function is given by the formula and n- natural number:

Example 2. Find the scope of a function .

Solution. As follows from the definition, the root of an even degree makes sense if the radical expression is non-negative, that is, if - 1 ≤ x≤ 1 . Therefore, the scope of this function is [- 1; one] .

The shaded area of ​​the number line in the drawing above is the area of ​​definition of this function.

Power function domain

Domain of a power function with an integer exponent

if a- positive, then the domain of definition of the function is the set of all real numbers, that is, ]- ∞; + ∞[ ;

if a- negative, then the domain of definition of the function is the set ]- ∞; 0[ ∪ ]0 ;+ ∞[ , that is, the entire number line except for zero.

In the corresponding drawing, the entire numerical line is shaded from above, and the point corresponding to zero is punched out (it is not included in the function definition area).

Example 3. Find the scope of a function .

Solution. The first term is an integer power of x equal to 3, and the power of x in the second term can be represented as a unit - also an integer. Therefore, the domain of this function is the entire real line, that is, ]- ∞; +∞[ .

Domain of a power function with a fractional exponent

In the case when the function is given by the formula:

if - is positive, then the domain of the function is the set 0; +∞[ .

Example 4. Find the scope of a function .

Solution. Both terms in the function expression are power functions with positive fractional exponents. Therefore, the domain of this function is the set - ∞; +∞[ .

Domain of definition of exponential and logarithmic functions

Domain of the exponential function

In the case when the function is given by the formula, the domain of the function is the entire number line, that is, ]- ∞; +∞[ .

The domain of the logarithmic function

The logarithmic function is defined under the condition that its argument is positive, that is, its domain of definition is the set ]0; +∞[ .

Find the scope of the function yourself and then see the solution

Domain of definition of trigonometric functions

Function scope y= cos( x) is also a set R real numbers.

Function scope y= tg( x) - lots of R real numbers other than numbers .

Function scope y=ctg( x) - lots of R real numbers other than numbers.

Example 8. Find the scope of a function .

Solution. The external function is a decimal logarithm, and the conditions for the domain of definition of a logarithmic function in general apply to its domain of definition. That is, its argument must be positive. The argument here is the sine of "x". Turning an imaginary compass around a circle, we see that the condition sin x> 0 is violated when "x" is equal to zero, "pi", two, multiplied by "pi" and generally equal to the product of the number "pi" and any even or odd integer.

Thus, the domain of definition of this function is given by the expression

,

where k is an integer.

Domain of inverse trigonometric functions

Function scope y= arcsin( x) - set [-1; one] .

Function scope y= arccos( x) - also the set [-1; one] .

Function scope y= arctan( x) - lots of R real numbers.

Function scope y= arcctg( x) is also a set R real numbers.

Example 9. Find the scope of a function .

Solution. Let's solve the inequality:

Thus, we obtain the domain of definition of this function - the segment [- 4; 4] .

Example 10. Find the scope of a function .

Solution. Let's solve two inequalities:

Solution of the first inequality:

Solution of the second inequality:

Thus, we obtain the domain of definition of this function - the segment.

Fraction domain

If the function is given by a fractional expression in which the variable is in the denominator of the fraction, then the domain of the function is the set R real numbers other than x for which the denominator of the fraction vanishes.

Example 11. Find the scope of a function .

Solution. Solving the equality to zero of the denominator of the fraction, we find the domain of definition of this function - the set] - ∞; - 2[ ∪ ]- 2 ;+ ∞[ .