To define a function means to establish a rule (law) with the help of which, based on the given values of the independent variable, we find the corresponding function values. Let's look at different ways to define a function.
This entry defines the temperature T as a function of time t:T=f(t). The advantages of the tabular method of specifying a function are that it makes it possible to determine one or another specific values functions immediately, without additional changes or calculations. Disadvantages: does not define the function completely, but only for some argument values; does not provide a visual representation of the nature of the change in the function with a change in the argument.
2. Graphic method.Schedule function y=f(x) is the set of all points of the plane whose coordinates satisfy this equation. This can be some curve, in particular a straight line, or a set of points on a plane.
The advantage is clarity, the disadvantage is that it is not possible to accurately determine the values of the argument. In engineering and physics, it is often the only available way to specify a function, for example, when using recording instruments that automatically record changes in one quantity relative to another (barograph, thermograph, etc.).
3. Analytical method. Using this method, the function is specified analytically, using a formula. 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.
In the analytical method, a function can be specified by several different formulas. For example, the function
defined in the domain [- 
, 15] using three formulas.
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 they say that the function of x is given explicitly, for example. If the values of x and y are related by some equation of the form F(x,y) = 0, i.e. the formula is not resolved with respect to y, then the function is said to be specified implicitly. For example,. Note that not every explicit function can be represented in the form y =f(x), on the contrary, any explicit function can always be represented as an implicit one: 
. Another type of analytical specification of a function is parametric, when the argument x and function y are functions of a third quantity - parameter t: 
, Where 
, T – some interval. This method is widely used in mechanics and geometry.
The analytical method is the most common way to define a function. Compactness, the ability to apply mathematical analysis to a given function, and the ability to calculate function values for any argument values are its main advantages.
4. Verbal method. This method is that functional dependence expressed in words. For example, the function E(x) is the integer part of the number x, the Dirichlet function, the Riemann function, n!, r(n) is the number of divisors of the natural number n.
5. Semi-graphic method. Here, the function values are represented as segments, and the argument values are represented as numbers placed at the ends of the segments indicating the function values. So, for example, a thermometer has a scale with equal divisions with numbers on them. These numbers are the values of the argument (temperature). They stand in the place that determines the graphical elongation of the mercury column (function value) due to its volumetric expansion as a result of temperature changes.
One of classical definitions The concept of “function” is considered to be a definition based on correspondences. Let us present a number of such definitions.
Definition 1
A relationship in which each value of the independent variable corresponds to a single value of the dependent variable is called function.
Definition 2
Let two non-empty sets $X$ and $Y$ be given. A correspondence $f$ that matches each $x\in X$ with one and only one $y\in Y$ Is called function($f:X → Y$).
Definition 3
Let $M$ and $N$ be two arbitrary number sets. A function $f$ is said to be defined on $M$, taking values from $N$, if each element $x\in X$ is associated with one and only one element from $N$.
The following definition is given through the concept variable size. A variable quantity is a quantity that is this study takes on different numeric values.
Definition 4
Let $M$ be the set of values of the variable $x$. Then, if each value $x\in M$ corresponds to one specific value of another variable $y$ is a function of the value $x$ defined on the set $M$.
Definition 5
Let $X$ and $Y$ be some number sets. A function is a set $f$ of ordered pairs of numbers $(x,\ y)$ such that $x\in X$, $y\in Y$ and each $x$ is included in one and only one pair of this set, and each $y$ is in at least one pair.
Definition 6
Any set $f=\(\left(x,\ y\right)\)$ of ordered pairs $\left(x,\ y\right)$ such that for any pairs $\left(x",\ y" \right)\in f$ and $\left(x"",\ y""\right)\in f$ from the condition $y"≠ y""$ it follows that $x"≠x""$ is called a function or display.
Definition 7
A function $f:X → Y$ is a set of $f$ ordered pairs $\left(x,\ y\right)\in X\times Y$ such that for any element $x\in X$ there is a unique element $y\in Y$ such that $\left(x,\ y\right)\in f$, that is, the function is a tuple of objects $\left(f,\ X,\ Y\right)$.
In these definitions
$x$ is the independent variable.
$y$ is the dependent variable.
All possible values The variable $x$ is called the domain of the function, and all possible values of the variable $y$ are called the domain of the function.
Analytical method of specifying a function
For this method we need the concept of an analytical expression.
Definition 8
Analytical expression is called the product of all possible mathematical operations over any numbers and variables.
The analytical way to specify a function is to specify it using an analytical expression.
Example 1
$y=x^2+7x-3$, $y=\frac(x+5)(x+2)$, $y=cos5x$.
Pros:
- Using formulas we can determine the value of a function for any certain value variable $x$;
- Functions defined in this way can be studied using the apparatus of mathematical analysis.
Minuses:
- Low visibility.
- Sometimes you have to make very cumbersome calculations.
Tabular method of specifying a function
This method of assignment consists of writing down the values of the dependent variable for several values of the independent variable. All this is entered into the table.
Example 2
Picture 1.
Plus: For any value of the independent variable $x$, which is entered into the table, the corresponding value of the function $y$ is immediately known.
Minuses:
- Most often, no complete task functions;
- Low visibility.
Function and ways to set it.
To define a function means to establish a rule (law) with the help of which, given the values of the independent variable, one should find the corresponding function values. Let's look at some ways to specify functions.
Tabular method. A fairly common one is to specify a table of individual argument values 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, it has a big advantage over 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.
To graphic task function was quite correct from a mathematical point of view, it is necessary to indicate the exact geometric construction of the graph, which, most often, is given 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 its corresponding numerical value functions 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.
The function can be defined different formulas on different areas areas of your assignment.
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) - fraction numbers. 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 - arbitrary number, then presenting it in the form x = r + q (r = [x]), where r is an integer and q lies in the interval denote the largest of the integers 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 the opposite. 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 are mutually inverse functions y=ƒ(x) and y=φ(x) are symmetrical with respect to the bisector of the first and third coordinate angles.
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 superposition specified 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 various reasons degrees.
2) Power function y=x α, αєR. Examples of graphs power functions, corresponding various indicators degrees provided in the pictures
3) Logarithmic function y=log a x, a>0,a≠1;Graphs logarithmic functions, corresponding to various bases, are shown in Fig. 106.
4) Trigonometric functions y=sinx, y=cosx, y=tgx, y=ctgx; Graphs of trigonometric functions have the form shown in Fig. 107.
5) Reverse 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 constant with the help finite number 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 is an element of the same space that has the property of “attracting” elements 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 limit number 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 given sequence converges; V otherwise(if the sequence has no limit) they say that the sequence 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 from any sequence of points in space a convergent subsequence can be identified, then we say that given space has 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 such positive number 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 Cauchy point means that for any number > 0 we can point to 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 it be for some time numerical set given numeric function and the number is the limit point of the domain of definition. Exist various definitions for one-sided limits of the function at the point, but they are all equivalent.