a function is a correspondence between elements of two sets, established according to the rule that each element of one set is associated with some element from another set.
the graph of a function is locus points of the plane, the abscissa (x) and ordinate (y) of which are related by the specified function:
a point is located (or located) on the graph of a function if and only if .
Thus, the function can be adequately described by its graph.
Tabular method. A fairly common one is to specify a table individual values argument and their corresponding function values. This method of defining a function is used when the domain of definition of the function is a discrete finite set.
With the tabular method of specifying a function, it is possible to approximately calculate the values of the function that are not contained in the table, corresponding to intermediate values of the argument. To do this, use the interpolation method.
The advantages of the tabular method of specifying a function are that it makes it possible to determine certain specific values immediately, without additional measurements or calculations. However, in some cases, the table does not define the function completely, but only for some values of the argument and does not provide a visual representation of the nature of the change in the function depending on the change in the argument.
Graphic method. The graph of the function y = f(x) is the set of all points on the plane whose coordinates satisfy the given equation.
The graphical method of specifying a function does not always make it possible to accurately determine the numerical values of the argument. However, he has big advantage before other methods - visibility. In engineering and physics, a graphical method of specifying a function is often used, and a graph is the only way available for this.
In order for the graphical assignment of a function to be completely correct from a mathematical point of view, it is necessary to indicate the exact geometric design of the graph, which is most often specified by an equation. This leads to the following way of specifying a function.
Analytical method. Most often, the law that establishes the connection between argument and function is specified through formulas. This method of specifying a function is called analytical.
This method makes it possible for each numerical value of the argument x to find the corresponding numerical value of the function y exactly or with some accuracy.
If the relationship between x and y is given by a formula resolved with respect to y, i.e. has the form y = f(x), then we say that the function of x is given explicitly.
If the values x and y are related by some equation of the form F(x,y) = 0, i.e. the formula is not resolved for y, which means that the function y = f(x) is given implicitly.
A function can be defined by different formulas in different parts of its domain.
The analytical method is the most common way of specifying functions. Compactness, conciseness, the ability to calculate the value of a function for an arbitrary value of an argument from the domain of definition, the ability to apply the apparatus of mathematical analysis to a given function are the main advantages of the analytical method of specifying a function. The disadvantages include the lack of visibility, which is compensated by the ability to build a graph and the need to perform sometimes very cumbersome calculations.
Verbal method. This method consists in expressing functional dependence in words.
Example 1: function E(x) is the integer part of x. In general, E(x) = [x] denotes the largest integer that does not exceed x. In other words, if x = r + q, where r is an integer (can be negative) and q belongs to the interval = r. The function E(x) = [x] is constant on the interval = r.
Example 2: function y = (x) is the fractional part of a number. More precisely, y =(x) = x - [x], where [x] is the integer part of the number x. This function is defined for all x. If x is an arbitrary number, then represent it as x = r + q (r = [x]), where r is an integer and q lies in the interval .
We see that adding n to the argument x does not change the value of the function.
The smallest non-zero number in n is , so the period is sin 2x .
The argument value at which the function is equal to 0 is called zero (root) functions.
A function may have multiple zeros.
For example, the function y = x (x + 1)(x-3) has three zeros: x = 0, x = - 1, x =3.
Geometrically, the zero of a function is the abscissa of the point of intersection of the function graph with the axis X .
Figure 7 shows a graph of a function with zeros: x = a, x = b and x = c.
If the graph of a function indefinitely approaches a certain line as it moves away from the origin, then this line is called asymptote.
Inverse function
Let a function y=ƒ(x) be given with a domain of definition D and a set of values E. If each value yєE corresponds to a single value xєD, then the function x=φ(y) is defined with a domain of definition E and a set of values D (see Fig. 102 ).
Such a function φ(y) is called the inverse of the function ƒ(x) and is written in the following form: x=j(y)=f -1 (y).The functions y=ƒ(x) and x=φ(y) are said to be mutually inverse. To find the function x=φ(y), inverse to the function y=ƒ (x), it is enough to solve the equation ƒ(x)=y for x (if possible).
1. For the function y=2x the inverse function is the function x=y/2;
2. For the function y=x2 xє the inverse function is x=√y; note that for the function y=x 2 defined on the segment [-1; 1], the inverse does not exist, since one value of y corresponds to two values of x (so, if y = 1/4, then x1 = 1/2, x2 = -1/2).
From the definition of an inverse function it follows that the function y=ƒ(x) has an inverse if and only if the function ƒ(x) specifies a one-to-one correspondence between the sets D and E. It follows that any strictly monotonic function has an inverse. Moreover, if a function increases (decreases), then the inverse function also increases (decreases).
Note that the function y=ƒ(x) and its inverse x=φ(y) are depicted by the same curve, i.e. their graphs coincide. If we agree that, as usual, the independent variable (i.e. argument) is denoted by x, and the dependent variable by y, then the inverse function of the function y=ƒ(x) will be written in the form y=φ(x).
This means that point M 1 (x o;y o) of the curve y=ƒ(x) becomes point M 2 (y o;x o) of the curve y=φ(x). But points M 1 and M 2 are symmetrical with respect to the straight line y=x (see Fig. 103). Therefore, the graphs of the mutually inverse functions y=ƒ(x) and y=φ(x) are symmetrical with respect to the bisector of the first and third coordinate angles.
Complex function
Let the function y=ƒ(u) be defined on the set D, and the function u= φ(x) on the set D 1, and for x D 1 the corresponding value u=φ(x) є D. Then on the set D 1 function u=ƒ(φ(x)), which is called a complex function of x (or a superposition of given functions, or a function of a function).
The variable u=φ(x) is called an intermediate argument of a complex function.
For example, the function y=sin2x is a superposition of two functions y=sinu and u=2x. A complex function can have several intermediate arguments.
4. Basic elementary functions and their graphs.
The following functions are called the main elementary functions.
1) Exponential function y=a x,a>0, a ≠ 1. In Fig. 104 graphs shown exponential functions, corresponding to different degree bases.
2) Power function y=x α, αєR. Examples of graphs power functions, corresponding to various exponents, are provided in the figures
3) Logarithmic function y=log a x, a>0,a≠1; Graphs of logarithmic functions corresponding to different bases are shown in Fig. 106.
4) Trigonometric functions y=sinx, y=cosx, y=tgx, y=ctgx; Charts trigonometric functions have the form shown in Fig. 107.
5) Inverse trigonometric functions y=arcsinx, y=arccosх, y=arctgx, y=arcctgx. In Fig. 108 shows graphs of inverse trigonometric functions.
A function defined by a single formula made up of basic elementary functions and constants using a finite number of arithmetic operations (addition, subtraction, multiplication, division) and operations of taking a function from a function, is called an elementary function.
Examples of elementary functions are the functions
Examples of non-elementary functions are the functions
5. Concepts of limit of sequence and function. Properties of limits.
Function limit (limit value of function) at a given point, limiting the domain of definition of a function, is the value to which the value of the function under consideration tends as its argument tends to a given point.
In mathematics limit of the sequence elements of a metric space or topological space are an element of the same space that has the property of “attracting” elements of a given sequence. The limit of a sequence of elements of a topological space is a point such that each neighborhood of it contains all elements of the sequence, starting from a certain number. In a metric space, neighborhoods are defined through the distance function, so the concept of a limit is formulated in the language of distances. Historically, the first was the concept of the limit of a numerical sequence, arising in mathematical analysis, where it serves as the basis for a system of approximations and is widely used in the construction of differential and integral calculus.
Designation:
(reads: the limit of the x-nth sequence as en tends to infinity is a)
The property of a sequence having a limit is called convergence: if a sequence has a limit, then it is said that this sequence converges; otherwise (if the sequence has no limit) the sequence is said to be diverges. In a Hausdorff space and, in particular, a metric space, every subsequence of a convergent sequence converges, and its limit coincides with the limit of the original sequence. In other words, a sequence of elements of a Hausdorff space cannot have two different limits. It may, however, turn out that the sequence has no limit, but there is a subsequence (of the given sequence) that has a limit. If a convergent subsequence can be identified from any sequence of points in space, then the given space is said to have the property of sequential compactness (or, simply, compactness, if compactness is defined exclusively in terms of sequences).
The concept of a limit of a sequence is directly related to the concept of a limit point (set): if a set has a limit point, then there is a sequence of elements of this set converging to this point.
Definition
Let a topological space and a sequence be given. Then, if there is an element such that
where is an open set containing , then it is called the limit of the sequence. If the space is metric, then the limit can be defined using the metric: if there is an element such that
where is the metric, it is called the limit.
· If the space is equipped with an anti-discrete topology, then the limit of any sequence will be any element of the space.
6. Limit of a function at a point. One-sided limits.
Function of one variable. Determination of the limit of a function at a point according to Cauchy. Number b called the limit of the function at = f(x) at X, striving for A(or at the point A), if for any positive number there is a positive number such that for all x ≠ a, such that | x – a | < , выполняется неравенство
| f(x) – a | < .
Determination of the limit of a function at a point according to Heine. Number b called the limit of the function at = f(x) at X, striving for A(or at the point A), if for any sequence ( x n ), converging to A(aiming for A, having a limit number A), and at any value n x n ≠ A, subsequence ( y n= f(x n)) converges to b.
These definitions assume that the function at = f(x) is defined in some neighborhood of the point A, except, perhaps, the point itself A.
The Cauchy and Heine definitions of the limit of a function at a point are equivalent: if the number b serves as a limit for one of them, then this is also true for the second.
The specified limit is indicated as follows:
Geometrically, the existence of a limit of a function at a point according to Cauchy means that for any number > 0 it is possible to indicate on the coordinate plane such a rectangle with base 2 > 0, height 2 and center at point ( A; b) that all points of the graph of a given function on the interval ( A– ; A+ ), with the possible exception of the point M(A; f(A)), lie in this rectangle
One-sided limit in mathematical analysis, the limit of a numerical function, implying “approaching” the limit point on one side. Such limits are called accordingly left-hand limit(or limit to the left) And right-hand limit (limit to the right). Let on some number set be given numeric function and the number is the limit point of the domain of definition. There are different definitions for the one-sided limits of a function at a point, but they are all equivalent.
>>Mathematics: Methods of specifying a function
Methods for specifying a function
By giving various examples of functions in the previous paragraph, we have somewhat impoverished the very concept of function.
After all, defining a function means specifying a rule that allows you to calculate the corresponding value y from an arbitrarily chosen value x from B(0. Most often, this rule is associated with a formula or several formulas - this method of specifying a function is usually called analytical. All functions discussed in § 7, were given analytically.Meanwhile, there are other ways to define a function, which will be discussed in this section.
If the function was specified analytically and we managed to construct a graph of the function, then we have actually moved from the analytical method of specifying the function to the graphical one. The reverse transition is not always possible. As a rule, this is a rather difficult but interesting task.
Not every line on the coordinate plane can be considered as a graph of some function. For example, a circle defined by the equation x 2 + y 2 - 9 (Fig. 51) is not a graph of a function, since any straight line x = a, where | a |<3, пересекает эту линию в д в у х точках (а для задания функции таких точек должно быть не более одной, т.е. прямая х = а должна пересекать линию F только в одной точке либо вообще не должна ее пересекать).
At the same time, if this circle is cut into two parts - the upper semicircle (Fig. 52) and the lower semicircle (Fig. 53), then each of the semicircles can be considered a graph of some function, and in both cases it is easy to switch from the graphical method of specifying the function to analytical.
From the equation x 2 + y 2 = 9 we find y 2 = 9 - x 2 and further 
The graph of the function is the upper semicircle of the circle x 2 + y 2 = 9 (Fig. 52), and the graph of the function is the lower semicircle of the circle x 2 + y 2 = 9 (Fig. 53).
This example allows us to draw attention to one significant circumstance. Look at the graph of the function (Fig. 52). It is immediately clear that D(f) = [-3, 3]. And if we were talking about finding the domain of definition analytically given function Then we would have to, as we did in § 7, spend time and effort solving the inequality. That is why they usually try to work simultaneously with both analytical and graphical methods of specifying functions. However, after two years of studying algebra at school, you have already become accustomed to this.
In addition to analytical and graphical, in practice, a tabular method of specifying a function is used. With this method, a table is provided that indicates the values of the function (sometimes exact, sometimes approximate) for a finite set of argument values. Examples of tabular functions can be tables of squares of numbers, cubes of numbers, square roots, etc.
In many cases, table specification of a function is convenient. It allows you to find the value of a function for the argument values available in the table without any calculations.
Analytical, graphical, tabular - naitabular, simpler, and therefore the most popular verbal task functions, these methods are quite sufficient for our needs. In fact, in mathematics there are quite a few different ways to define a function, but we will introduce you to only one more method, which is used in very peculiar situations. We are talking about the verbal method, when the rule for specifying a function is described in words. Let's give examples.
Example 1.
The function y = f(x) is defined on the set of all non-negative numbers using next rule: each number x > 0 is assigned to the first decimal place in the decimal notation of the number x. If, say, x = 2.534, then f(x) = 5 (the first decimal place is the number 5); if x = 13.002, then f(x) = 0; if then, writing 0.6666... as an infinite decimal fraction, we find f(x) = 6. What is the value of f(15)? It is equal to 0, since 15 = 15,000..., and we see that the first decimal place after the decimal point is 0 (in fact, the equality 15 = 14,999... is also true, but mathematicians have agreed not to consider infinite periodic decimal fractions with a period 9).
Any non-negative number x can be written as decimal(finite or infinite), and therefore for each value of x we can find a certain value of the first decimal place, so we can talk about a function, albeit a somewhat unusual one. This function 
Example 2.
The function y = f(x) is defined on the set of all real numbers using the following rule: each number x is associated with the largest of all integers that do not exceed x. In other words, the function y = f(x) is determined by the following conditions:
a) f(x) - an integer;
b) f(x)< х (поскольку f(х) не превосходит х);
c) f(x) + 1 > x (since f(x) is the largest integer not exceeding x, which means f(x) + 1 is already greater than r). If, say, x = 2.534, then f(x) = 2, since, firstly, 2 is an integer, and secondly, 2< 2,534 и, в-третьих, следующее целое число 3 уже больше, чем 2,534. Если х = 47, то /(х) = 47, поскольку, во-первых, 47 - целое число, во-вторых, 47< 47 (точнее, 47 = 47) и, в-третьих, следующее за числом 47 целое число 48 уже больше, чем 47. А чему равно значение f(-0,(23))? Оно равно -1. Проверяйте: -1 - наибольшее из всех целых чисел, которые не превосходят числа -0,232323....
This function has (set of integers).
The function discussed in example 2 is called the integer part of a number; for the integer part of the number x, use the notation [x]. For example, = 2, = 47, [-0,(23)] = -1. The graph of the function y = [x] looks very peculiar (Fig. 54).
Let us make a number of explanatory remarks regarding specifying a function by an analytical expression or formula, which plays an extremely important role in mathematical analysis.
1° First of all, what analytical operations or actions can be included in these formulas? In the first place here are all the operations studied in elementary algebra and trigonometry: arithmetic operations, exponentiation (and root extraction), logarithm, transition from angles to their trigonometric quantities and back [see. below 48 - 51]. However, and this is important to emphasize, as our information on analysis develops, other operations will be added to their number, first of all, the passage to the limit, with which the reader is already familiar from Chapter I.
Thus, full content The term “analytic expression” or “formula” will be revealed only gradually.
2° The second remark relates to the scope of defining a function by an analytical expression or formula.
Each analytical expression containing an argument x has, so to speak, a natural scope: this is the set of all those values of x for which it retains meaning, that is, it has a well-defined, finite, real value. Let's explain this using simple examples.
So, for the expression, such a region will be the entire set of real numbers. For expression, this area will be reduced to a closed interval beyond which its value ceases to be real. On the contrary, the expression will have to include an open interval as a natural area of application because at the ends its denominator turns to 0. Sometimes the range of values for which the expression retains its meaning consists of isolated intervals: for this there will be intervals for - intervals, etc.
As a final example, consider the sum of an infinite geometric progression
If then, as we know, this limit exists and matters. When the limit is either equal or does not exist at all. Thus, for the given analytical expression, the natural domain of application would be the open interval
In the subsequent presentation, we will have to consider both more complex and more general analytical expressions, and we will more than once be engaged in the study of the properties of functions specified by such an expression in the entire area where it retains its meaning, i.e., in the study of the analytical apparatus itself.
However, another state of affairs is also possible, to which we consider it necessary to draw the reader’s attention in advance. Let us imagine that some specific question in which the variable x is essentially limited to the range of variation of X has led to the consideration of a function that can be expressed analytically. Although it may happen that this expression has meaning outside the region X, it is, of course, still impossible to go beyond it. Here the analytical expression plays a subordinate, auxiliary role.
For example, if, studying the free fall of a heavy point from a height above the surface of the earth, we resort to the formula
It would be absurd to consider negative values of t or values greater than this because, as is easy to see, at the point the point will already fall to the ground. And this despite the fact that the expression itself retains the meaning for all real ones.
3° It may happen that a function is not determined by the same formula for all values of the argument, but for some - by one formula, and for others - by another. An example of such a function in the interval is the function defined by the following three formulas:
and finally, if .
Let us also mention the Dirichlet function (P. G. Lejeune-Dinchlet), which is defined as follows:
Finally, together with Kronecker (L. Kroneckcf), we will consider the function, which he called “signum” and denoted by
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Function %%y = f(x), x \in X%% is given in an explicit analytical way, if given a formula indicating the sequence of mathematical operations that must be performed with the argument %%x%% in order to obtain the value %%f(x)%% of this function.
So, for example, in physics with uniform acceleration straight motion the speed of a body is determined by the formula %%v = v_0 + a t%%, and the formula for moving %%s%% a body with uniformly accelerated motion over a period of time from %%0%% to %%t%% is written as: %% s = s_0 + v_0 t + \frac(a t^2)(2) %%.
Sometimes the function in question can be specified by several formulas acting on various areas the domain of its definition in which the argument of the function changes. For example: $$ y = \begin(cases) x ^ 2,~ if~x< 0, \\ \sqrt{x},~ если~x \geq 0. \end{cases} $$
Functions of this type are sometimes called composite or piecewise specified. An example of such a function is %%y = |x|%%
If a function is specified in an explicit analytical way using a formula, but the domain of definition of the function in the form of a set %%D%% is not specified, then by %%D%% we will always mean the set of values of the argument %%x%% for which this formula makes sense . So for the function %%y = x^2%% the domain of definition is the set %%D = \mathbb(R) = (-\infty, +\infty)%%, since the argument %%x%% can take any values on number line. And for the function %%y = \frac(1)(\sqrt(1 - x^2))%% the domain of definition will be the set of values %%x%% satisfying the inequality %%1 - x^2 > 0%%, t .e. %%D = (-1, 1)%%.
Note that the explicit analytical method of specifying a function is quite compact (the formula, as a rule, takes up little space), is easy to reproduce (the formula is not difficult to write) and is most suitable for performing mathematical operations and transformations on functions.
Some of these operations - algebraic (addition, multiplication, etc.) - are well known from the school mathematics course, others (differentiation, integration) will be studied in the future. However, this method is not always clear, since the nature of the function’s dependence on the argument is not always clear, and sometimes cumbersome calculations are required to find the function values (if they are necessary).
Function %%y = f(x)%% defined in an implicit analytical way, if given the relation $$F(x,y) = 0, ~~~~~~~~~~(1)$$ connecting the values of the function %%y%% and the argument %%x%%. If you specify argument values, then to find the value of %%y%% corresponding to a specific value of %%x%%, you need to solve the equation %%(1)%% for %%y%% at this specific value of %%x%%.
For given value%%x%% the equation %%(1)%% may have no solution or have more than one solution. In the first case, the specified value %%x%% does not belong to the domain of definition of the implicitly specified function, and in the second case it specifies multivalued function, which has more than one value for a given argument value.
Note that if the equation %%(1)%% can be explicitly resolved with respect to %%y = f(x)%%, then we obtain the same function, but already specified in an explicit analytical way. So, the equation %%x + y^5 - 1 = 0%%
and the equality %%y = \sqrt(1 - x)%% define the same function.
When the dependence of %%y%% on %%x%% is not given directly, but instead the dependences of both variables %%x%% and %%y%% on some third auxiliary variable %%t%% are given in the form
$$ \begin(cases) x = \varphi(t),\\ y = \psi(t), \end(cases) ~~~t \in T \subseteq \mathbb(R), ~~~~~ ~~~~~(2) $$what they talk about parametric method of specifying the function;
then the auxiliary variable %%t%% is called a parameter.
If it is possible to eliminate the parameter %%t%% from the equations %%(2)%%, then we arrive at a function defined by the explicit or implicit analytical dependence of %%y%% on %%x%%. For example, from the relations $$ \begin(cases) x = 2 t + 5, \\ y = 4 t + 12, \end(cases), ~~~t \in \mathbb(R), $$ except for the % parameter %t%% we obtain the dependence %%y = 2 x + 2%%, which defines a straight line in the %%xOy%% plane.
Example graphic task functions
The above examples show that the analytical method of specifying a function corresponds to its graphic image, which can be considered as a convenient and visual form of describing a function. Sometimes used graphic method specifying a function when the dependence of %%y%% on %%x%% is specified by a line on the plane %%xOy%%. However, despite all the clarity, it loses in accuracy, since the values of the argument and the corresponding function values can be obtained from the graph only approximately. The resulting error depends on the scale and accuracy of measurement of the abscissa and ordinate of individual points on the graph. In the future, we will assign the function graph only the role of illustrating the behavior of the function and therefore we will limit ourselves to constructing “sketches” of graphs that reflect the main features of the functions.
Note tabular method function assignments, when some argument values and the corresponding function values are placed in a table in a certain order. This is how well-known tables of trigonometric functions, tables of logarithms, etc. are constructed. The relationship between quantities measured at experimental studies, observations, tests.
The disadvantage of this method is that it is impossible to directly determine function values for argument values not included in the table. If there is confidence that the argument values not presented in the table belong to the domain of definition of the function in question, then the corresponding function values can be approximately calculated using interpolation and extrapolation.
| x | 3 | 5.1 | 10 | 12.5 |
| y | 9 | 23 | 80 | 110 |
The function can be set algorithmic(or software) in a way that is widely used in computer calculations.
Finally, it can be noted descriptive(or verbal) a way to specify a function, when the rule for matching the function values to the argument values is expressed in words.
For example, the function %%[x] = m~\forall (x \in )
