The section contains reference material on the main elementary functions and their properties. The classification is given elementary functions. Below are links to subsections that discuss the properties of specific functions - graphs, formulas, derivatives, antiderivatives (integrals), series expansions, expressions through complex variables.

Reference pages for basic functions

Classification of elementary functions

Algebraic function is a function that satisfies the equation:
,
where is a polynomial in the dependent variable y and the independent variable x. It can be written as:
,
where are polynomials.

Algebraic functions are divided into polynomials (entire rational functions), rational functions and irrational functions.

Entire rational function, which is also called polynomial or polynomial, is obtained from the variable x and a finite number of numbers using the arithmetic operations of addition (subtraction) and multiplication. After opening the brackets, the polynomial is reduced to canonical form:
.

Fractional rational function, or simply rational function, is obtained from the variable x and a finite number of numbers using the arithmetic operations of addition (subtraction), multiplication and division. The rational function can be reduced to the form
,
where and are polynomials.

Irrational function is an algebraic function that is not rational. As a rule, an irrational function is understood as roots and their compositions with rational functions. A root of degree n is defined as the solution to the equation
.
It is designated as follows:
.

Transcendental functions are called non-algebraic functions. These are exponential, trigonometric, hyperbolic and their inverse functions.

Overview of basic elementary functions

All elementary functions can be represented as a finite number of addition, subtraction, multiplication and division operations performed on an expression of the form:
z t .
Inverse functions can also be expressed in terms of logarithms. The basic elementary functions are listed below.

Power function :
y(x) = x p ,
where p is the exponent. It depends on the base of the degree x.
Back to power function is also a power function:
.
For an integer non-negative value of the exponent p, it is a polynomial. For an integer value p - a rational function. With a rational meaning - an irrational function.

Transcendental functions

Exponential function :
y(x) = a x ,
where a is the base of the degree. It depends on the exponent x.
The inverse function is the logarithm to base a:
x = log a y.

Exponent, e to the x power:
y(x) = e x ,
This is an exponential function whose derivative is equal to the function itself:
.
The base of the exponent is the number e:
≈ 2,718281828459045... .
The inverse function is the natural logarithm - the logarithm to the base of the number e:
x = ln y ≡ log e y.

Trigonometric functions:
Sine: ;
Cosine: ;
Tangent: ;
Cotangent: ;
Here i is the imaginary unit, i 2 = -1.

Inverse trigonometric functions:
Arcsine: x = arcsin y, ;
Arc cosine: x = arccos y, ;
Arctangent: x = arctan y, ;
Arc tangent: x = arcctg y, .

The function y=x^2 is called a quadratic function. The graph of a quadratic function is a parabola. General form The parabola is shown in the figure below.

Quadratic function

Fig 1. General view of the parabola

As can be seen from the graph, it is symmetrical about the Oy axis. The Oy axis is called the axis of symmetry of the parabola. This means that if you draw a straight line on the graph parallel to the Ox axis above this axis. Then it will intersect the parabola at two points. The distance from these points to the Oy axis will be the same.

The axis of symmetry divides the graph of a parabola into two parts. These parts are called branches of the parabola. And the point of a parabola that lies on the axis of symmetry is called the vertex of the parabola. That is, the axis of symmetry passes through the vertex of the parabola. The coordinates of this point are (0;0).

Basic properties of a quadratic function

1. At x =0, y=0, and y>0 at x0

2. The quadratic function reaches its minimum value at its vertex. Ymin at x=0; It should also be noted that the function does not have a maximum value.

3. The function decreases on the interval (-∞;0] and increases on the interval ; inequality a<x<b – interval and is denoted by () ; inequalities and - half-intervals and are denoted by and respectively. You also often have to deal with infinite intervals and half-intervals: , , , and . It’s convenient to call them all at intervals .

Interval, i.e. set of points satisfying the inequality (where ), is called the -neighborhood of the point a.

The concept of function. Basic properties of a function

If each element x sets X a single element is matched y sets Y, then they say that on the set X given function y=f(x). Wherein x called independent variable or argument, A y – dependent variable or function, A f denotes the law of correspondence. A bunch of X called domain of definition functions, and a set Y – range of values functions.

There are several ways to specify functions.


1) Analytical method - the function is given by a formula of the form y=f(x).

2) Tabular method - the function is specified by a table containing the argument values ​​and the corresponding function values y=f(x).

3) Graphical method - depicting a graph of a function, i.e. set of points ( x; y) coordinate plane, the abscissas of which represent the values ​​of the argument, and the ordinates represent the corresponding values ​​of the function y=f(x).

4) Verbal method - a function is described by the rule for its composition. For example, the Dirichlet function takes the value 1 if x is a rational number and 0 if x– irrational number.

The following main properties of functions are distinguished.

1 Even and odd Function y=f(x) is called even, if for any values x from its domain of definition is satisfied f(–x)=f(x), And odd, If f(–x)=–f(x). If none of the listed equalities is satisfied, then y=f(x) is called general function. The graph of an even function is symmetrical about the axis Oy, and the graph of the odd function is symmetrical about the origin.

2 Monotony Function y=f(x) is called increasing (decreasing) on the interval X, if a larger argument value from this interval corresponds to a larger (smaller) function value. Let x 1 ,x 2 Î X, x 2 >x 1 . Then the function increases on the interval X, If f(x 2)>f(x 1), and decreases if f(x 2)<f(x 1).

Along with increasing and decreasing functions, non-decreasing and non-increasing functions are considered. The function is called non-decreasing (non-increasing), if at x 1 ,x 2 Î X, x 2 >x 1 inequality holds f(x 2)≥f(x 1) (f(x 2)≤f(x 1)).

Increasing and decreasing functions, as well as non-increasing and non-decreasing functions are called monotonic.

3 Limited Function y=f(x) is called bounded on the interval X, if there is such a positive number M>0, what | f(x)|≤M for anyone xÎ X. Otherwise the function is said to be unbounded X.

4 Frequency Function y=f(x) is called periodic with a period T≠0, if for any x from the domain of the function f(x+T)=f(x). In what follows, by period we mean the smallest positive period of a function.

The function is called explicit, if it is given by a formula of the form y=f(x). If the function is given by the equation F(x, y)=0, not allowed relative to the dependent variable y, then it is called implicit.

Let y=f(x) is a function of the independent variable defined on the set X with range Y. Let's match each one yÎ Y single meaning xÎ X, at which f(x)=y.Then the resulting function x=φ (y), defined on the set Y with range X, called reverse and is designated y=f –1 (x). The graphs of mutually inverse functions are symmetrical with respect to the bisector of the first and third coordinate quarters.

Let the function y=f(u) is a function of a variable u, defined on the set U with range Y, and the variable u in turn is a function u=φ (x), defined on the set X with range U. Then given on the set X function y=f(φ (x)) is called complex function (composition of functions, superposition of functions, function of a function).

Elementary functions

The main elementary functions include:

From the basic elementary functions, new functions can be obtained using algebraic operations and superposition of functions.

Functions constructed from basic elementary functions using a finite number of algebraic operations and a finite number of superposition operations are called elementary.

Algebraic is a function in which a finite number of algebraic operations are performed on the argument. Algebraic functions include:

· an entire rational function (polynomial or polynomial)

· fractional-rational function (ratio of two polynomials)

· irrational function (if the operations on the argument include extracting the root).

Any non-algebraic function is called transcendental. Transcendental functions include exponential, logarithmic, trigonometric, and inverse trigonometric functions.

Russian gymnasium

ABSTRACT

Completed

student of class 10 “F” Burmistrov Sergey

Supervisor

mathematic teacher

Yulina O.A.

Nizhny Novgorod


Function and its properties

Function- variable dependence at from variable x , if each value X matches a single value at .

Variable x- independent variable or argument.

Variable y- dependent variable

Function value- meaning at, corresponding to the specified value X .

The scope of the function is all the values ​​that the independent variable takes.

Function range (set of values) - all the values ​​that the function accepts.

The function is even- if for anyone X f(x)=f(-x)

The function is odd- if for anyone X from the domain of definition of the function the equality f(-x)=-f(x)

Increasing function- if for any x 1 And x 2, such that x 1 < x 2, the inequality holds f( x 1 ) x 2 )

Decreasing function- if for any x 1 And x 2, such that x 1 < x 2, the inequality holds f( x 1 )>f( x 2 )

Methods for specifying a function

¨ To define a function, you need to specify a way in which, for each argument value, the corresponding function value can be found. The most common way to specify a function is using a formula at =f(x), Where f(x)- expression with a variable X. In this case, they say that the function is given by a formula or that the function is given analytically.

¨ In practice it is often used tabular way to specify a function. With this method, a table is provided indicating the function values ​​​​for the argument values ​​​​available in the table. Examples table job functions are table of squares, table of cubes.

Types of functions and their properties

1) Constant function- function given by formula y= b , Where b- some number. The graph of the constant function y=b is a straight line parallel to the abscissa axis and passing through the point (0;b) on the ordinate axis

2) Direct proportionality - function given by formula y= kx , where k¹0. Number k called proportionality factor .

Function properties y=kx :

1. The domain of a function is the set of all real numbers

2. y=kx- odd function

3. When k>0 the function increases, and when k<0 убывает на всей числовой прямой

3)Linear function- function, which is given by the formula y=kx+b, Where k And b - real numbers. If in particular k=0, then we get a constant function y=b; If b=0, then we get direct proportionality y=kx .

Function properties y=kx+b :

1. Domain - the set of all real numbers

2. Function y=kx+b general form, i.e. neither even nor odd.

3. When k>0 the function increases, and when k<0 убывает на всей числовой прямой

The graph of the function is straight .

4)Inverse proportionality- function given by formula y=k /X, where k¹0 Number k called coefficient of inverse proportionality.

Function properties y=k / x:

1. Domain - the set of all real numbers except zero

2. y=k / x - odd function

3. If k>0, then the function decreases on the interval (0;+¥) and on the interval (-¥;0). If k<0, то функция возрастает на промежутке (-¥;0) и на промежутке (0;+¥).

The graph of the function is hyperbola .

5)Function y=x2

Function properties y=x2:

2. y=x2 - even function

3. On the interval the function decreases

The graph of the function is parabola .

6)Function y=x 3

Function properties y=x 3:

1. Domain of definition - the entire number line

2. y=x 3 - odd function

3. The function increases along the entire number line

The graph of the function is cubic parabola

7)Power function with natural exponent - function given by formula y=x n, Where n- natural number. When n=1 we obtain the function y=x, its properties are discussed in paragraph 2. For n=2;3 we obtain the functions y=x 2 ; y=x 3 . Their properties are discussed above.

Let n be an arbitrary even number greater than two: 4,6,8... In this case, the function y=x n has the same properties as the function y=x 2. The graph of the function resembles a parabola y=x 2, only the branches of the graph for |x|>1 rise steeper the larger n, and for |x|<1 тем “теснее прижимаются” к оси Х, чем больше n.

Let n be an arbitrary odd number greater than three: 5,7,9... In this case, the function y=x n has the same properties as the function y=x 3 . The graph of the function resembles a cubic parabola.

8)Power function with a negative integer exponent - function given by formula y=x -n , Where n- natural number. For n=1 we obtain y=1/x; the properties of this function are discussed in paragraph 4.

Let n be an odd number greater than one: 3,5,7... In this case, the function y=x -n has basically the same properties as the function y=1/x.

Let n be an even number, for example n=2.

Function properties y=x -2 :

1. The function is defined for all x¹0

2. y=x -2 - even function

3. The function decreases by (0;+¥) and increases by (-¥;0).

Any functions with even n greater than two have the same properties.

9)Function y= Ö X

Function properties y= Ö X :

1. Domain of definition - ray)