The section contains reference material on the main elementary functions and their properties. A classification of elementary functions is given. Below are links to subsections that discuss the properties of specific functions - graphs, formulas, derivatives, antiderivatives (integrals), series expansions, expressions through complex variables.
ContentReference 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) = xp,
where p is the exponent. It depends on the base of the degree x. Back to 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.
Inverse function- 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,
.
Basic elementary functions are: constant function (constant), root n-th degree, power function, exponential, logarithmic function, trigonometric and inverse trigonometric functions.
Permanent function.
A constant function is defined on the set of all real numbers formula, where C– some real number. Constant function associates each actual value of the independent variable x same value of the dependent variable y- meaning WITH. A constant function is also called a constant.
The graph of a constant function is a straight line parallel to the x-axis and passing through the point with coordinates (0,C). For example, let's show graphs of constant functions y=5,y=-2 and , which in the figure below correspond to the black, red and blue lines, respectively.
Properties of a constant function.
Domain: the entire set of real numbers.
The constant function is even.
Range of values: set consisting of singular WITH.
A constant function is non-increasing and non-decreasing (that’s why it’s constant).
It makes no sense to talk about convexity and concavity of a constant.
There are no asymptotes.
The function passes through the point (0,C) coordinate plane.
Root of the nth degree.
Let's consider the basic elementary function, which is given by the formula, where n – natural number, greater than one.
The nth root, n is an even number.
Let's start with the root function n-th power for even values of the root exponent n.
As an example, here is a picture with images of function graphs 
and , they correspond to black, red and blue lines.
The graphs of even-degree root functions have a similar appearance for other values of the exponent.
Properties of the root functionn -th power for evenn .
The nth root, n is an odd number.
Root function n-th power with an odd root exponent n is defined on the entire set of real numbers. For example, here are the function graphs 
and , they correspond to black, red and blue curves.
1) Function domain and function range.
The domain of a function is the set of all valid valid argument values x(variable x), for which the function y = f(x) determined. The range of a function is the set of all real values y, which the function accepts.
In elementary mathematics, functions are studied only on the set of real numbers.
2) Function zeros.
Function zero is the value of the argument at which the value of the function is equal to zero.
3) Intervals of constant sign of a function.
Intervals of constant sign of a function are sets of argument values on which the function values are only positive or only negative.
4) Monotonicity of the function.
An increasing function (in a certain interval) is a function in which a larger value of the argument from this interval corresponds to a larger value of the function.
A decreasing function (in a certain interval) is a function in which a larger value of the argument from this interval corresponds to a smaller value of the function.
5) Even (odd) function.
An even function is a function whose domain of definition is symmetrical with respect to the origin and for any X from the domain of definition the equality f(-x) = f(x).
The graph of an even function is symmetrical about the ordinate. X An odd function is a function whose domain of definition is symmetrical with respect to the origin and for any from the domain of definition the equality is true f(-x) = - f(x ). Schedule
odd function.
symmetrical about the origin.
6) Limited and unlimited functions.
A function is called bounded if there is a positive number M such that |f(x)| ≤ M for all values of x. If such a number does not exist, then the function is unlimited.
19. Basic elementary functions, their properties and graphs. Application of functions in economics.
Basic elementary functions. Their properties and graphs
1. Linear function.
Linear function is called a function of the form , where x is a variable, a and b are real numbers.
Number A called the slope of the line, it is equal to the tangent of the angle of inclination of this line to the positive direction of the x-axis. The graph of a linear function is a straight line. It is defined by two points.
Properties of a Linear Function
1. Domain of definition - the set of all real numbers: D(y)=R
2. The set of values is the set of all real numbers: E(y)=R
3. The function takes a zero value when or.
4. The function increases (decreases) over the entire domain of definition.
5. A linear function is continuous over the entire domain of definition, differentiable and .
2. Quadratic function.
A function of the form, where x is a variable, coefficients a, b, c are real numbers, is called quadratic
Knowledge basic elementary functions, their properties and graphs no less important than knowing the multiplication tables. They are like the foundation, everything is based on them, everything is built from them and everything comes down to them.
In this article we will list all the main elementary functions, provide their graphs and give without conclusion or proof properties of basic elementary functions according to the scheme:
- behavior of the function at the boundaries of the domain of definition, vertical asymptotes(if necessary, see the article classification of breakpoints of a function);
- even and odd;
- intervals of convexity (convexity upward) and concavity (convexity downward), inflection points (if necessary, see the article convexity of a function, direction of convexity, inflection points, conditions of convexity and inflection);
- inclined and horizontal asymptotes;
- singular points of functions;
- special properties of some functions (for example, the smallest positive period of trigonometric functions).
If you are interested in or, then you can go to these sections of the theory.
Basic elementary functions are: constant function (constant), nth root, power function, exponential, logarithmic function, trigonometric and inverse trigonometric functions.
Page navigation.
Permanent function.
A constant function is defined on the set of all real numbers by the formula , where C is some real number. A constant function associates each real value of the independent variable x with the same value of the dependent variable y - the value C. A constant function is also called a constant.
The graph of a constant function is a straight line parallel to the x-axis and passing through the point with coordinates (0,C). As an example, we will show graphs of constant functions y=5, y=-2 and, which in the figure below correspond to the black, red and blue lines, respectively.
Properties of a constant function.
- Domain: the entire set of real numbers.
- The constant function is even.
- Range of values: a set consisting of the singular number C.
- A constant function is non-increasing and non-decreasing (that’s why it’s constant).
- It makes no sense to talk about convexity and concavity of a constant.
- There are no asymptotes.
- The function passes through the point (0,C) of the coordinate plane.
Root of the nth degree.
Let's consider the basic elementary function, which is given by the formula , where n is a natural number greater than one.
Root of the nth degree, n is an even number.
Let's start with the nth root function for even values of the root exponent n.
As an example, here is a picture with images of function graphs 
and , they correspond to black, red and blue lines.
The graphs of functions root have a similar appearance. even degree for other values of the indicator.
Properties of the nth root function for even n.
The nth root, n is an odd number.
The nth root function with an odd root exponent n is defined on the entire set of real numbers. For example, here are the function graphs 
and , they correspond to black, red and blue curves.
For other odd values of the root exponent, the function graphs will have a similar appearance.
Properties of the nth root function for odd n.
Power function.
The power function is given by a formula of the form .
Let's consider the form of graphs of a power function and the properties of a power function depending on the value of the exponent.
Let's start with a power function with an integer exponent a. In this case, the appearance of the graphs of power functions and the properties of the functions depend on the evenness or oddness of the exponent, as well as on its sign. Therefore, we will first consider power functions for odd positive values of the exponent a, then for even positive exponents, then for odd negative exponents, and finally, for even negative a.
The properties of power functions with fractional and irrational exponents (as well as the type of graphs of such power functions) depend on the value of the exponent a. We will consider them, firstly, for a from zero to one, secondly, for a greater than one, thirdly, for a from minus one to zero, fourthly, for a less than minus one.
At the end of this section, for completeness, we will describe a power function with zero exponent.
Power function with odd positive exponent.
Let's consider a power function with an odd positive exponent, that is, with a = 1,3,5,....
The figure below shows graphs of power functions – black line, – blue line, – red line, – green line. For a=1 we have linear function y=x.
Properties of a power function with an odd positive exponent.
Power function with even positive exponent.
Let's consider a power function with an even positive exponent, that is, for a = 2,4,6,....
As an example, we give graphs of power functions – black line, – blue line, – red line. For a=2 we have quadratic function, whose graph is quadratic parabola.
Properties of a power function with an even positive exponent.
Power function with odd negative exponent.
Look at the graphs of the power function for odd negative values of the exponent, that is, for a = -1, -3, -5,....
The figure shows graphs of power functions as examples - black line, - blue line, - red line, - green line. For a=-1 we have inverse proportionality, whose graph is hyperbola.
Properties of a power function with an odd negative exponent.
Power function with even negative exponent.
Let's move on to the power function for a=-2,-4,-6,….
The figure shows graphs of power functions – black line, – blue line, – red line.
Properties of a power function with an even negative exponent.
A power function with a rational or irrational exponent whose value is greater than zero and less than one.
Note! If a is a positive fraction with an odd denominator, then some authors consider the domain of definition of the power function to be the interval. It is stipulated that the exponent a is an irreducible fraction. Now the authors of many textbooks on algebra and principles of analysis DO NOT DEFINE power functions with an exponent in the form of a fraction with an odd denominator for negative values of the argument. We will adhere to precisely this view, that is, we will consider the set to be the domains of definition of power functions with fractional positive exponents. We recommend that students find out your teacher's opinion on this subtle point in order to avoid disagreements.
Let us consider a power function with a rational or irrational exponent a, and .
Let us present graphs of power functions for a=11/12 (black line), a=5/7 (red line), (blue line), a=2/5 (green line).
A power function with a non-integer rational or irrational exponent greater than one.
Let us consider a power function with a non-integer rational or irrational exponent a, and .
Let us present graphs of power functions given by the formulas 
(black, red, blue and green lines respectively).
For other values of the exponent a, the graphs of the function will have a similar appearance.
Properties of the power function at .
A power function with a real exponent that is greater than minus one and less than zero.
Note! If a is a negative fraction with an odd denominator, then some authors consider the domain of definition of a power function to be the interval 
. It is stipulated that the exponent a is an irreducible fraction. Now the authors of many textbooks on algebra and principles of analysis DO NOT DEFINE power functions with an exponent in the form of a fraction with an odd denominator for negative values of the argument. We will adhere to precisely this view, that is, we will consider the domains of definition of power functions with fractional fractional negative exponents to be a set, respectively. We recommend that students find out your teacher's opinion on this subtle point in order to avoid disagreements.
Let's move on to the power function, kgod.
In order to have a good idea of the form of graphs of power functions for , we give examples of graphs of functions 
(black, red, blue and green curves, respectively).
Properties of a power function with exponent a, .
A power function with a non-integer real exponent that is less than minus one.
Let us give examples of graphs of power functions for 
, they are depicted by black, red, blue and green lines, respectively.
Properties of a power function with a non-integer negative exponent less than minus one.
When a = 0, we have a function - this is a straight line from which the point (0;1) is excluded (it was agreed not to attach any significance to the expression 0 0).
Exponential function.
One of the main elementary functions is the exponential function.
Schedule exponential function, where he receives different kind depending on the value of the base a. Let's figure this out.
First, consider the case when the base of the exponential function takes a value from zero to one, that is, .
As an example, we present graphs of the exponential function for a = 1/2 – blue line, a = 5/6 – red line. The graphs of the exponential function have a similar appearance for other values of the base from the interval.
Properties of an exponential function with a base less than one.
Let us move on to the case when the base of the exponential function is greater than one, that is, .
As an illustration, we present graphs of exponential functions - blue line and - red line. For other values of the base greater than one, the graphs of the exponential function will have a similar appearance.
Properties of an exponential function with a base greater than one.
Logarithmic function.
The next basic elementary function is the logarithmic function, where , . The logarithmic function is defined only for positive values of the argument, that is, for .
The graph of a logarithmic function takes different forms depending on the value of the base a.
Considering functions of a complex variable, Liouville defined elementary functions somewhat more broadly. Elementary function y variable x- analytical function, which can be represented as an algebraic function of x and functions 
, and is the logarithm or exponent of some algebraic function g 1 from x
.
For example, sin( x) - algebraic function of e ix .
Without limiting the generality of the consideration, we can assume that the functions are algebraically independent, that is, if algebraic equation runs for everyone x, then all coefficients of the polynomial 
are equal to zero.
Differentiation of elementary functions
Where z 1 "(z) equals or g 1 " / g 1 or z 1 g 1" depending on whether it is a logarithm z 1 or exponential, etc. In practice, it is convenient to use a derivative table.
Integrating Elementary Functions
Liouville's theorem is the basis for creating algorithms for symbolic integration of elementary functions, implemented, for example, in
Calculation of limits
Liouville's theory does not apply to the calculation of limits. It is not known whether there is an algorithm that, given a sequence given by an elementary formula, gives an answer whether it has a limit or not. For example, the question is open whether the sequence converges.
Literature
- J. Liouville. Mémoire sur l'intégration d'une classe de fonctions transcendantes// J. Reine Angew. Math. Bd. 13, p. 93-118. (1835)
- J.F. Ritt. Integration in Finite Terms. N.-Y., 1949 // http://lib.homelinux.org
- A. G. Khovansky. Topological Galois theory: solvability and unsolvability of equations in finite form Ch. 1. M, 2007
Notes
Wikimedia Foundation.
- 2010.
- Elementary excitation
Elementary outcome
See what “Elementary function” is in other dictionaries: elementary function - A function that, if divided into smaller functions, cannot be uniquely defined in the digital transmission hierarchy. Therefore, from the network's point of view it is indivisible (ITU T G.806).
Topics: telecommunications, basic concepts EN adaptation functionA... Technical Translator's Guide - A function that, if divided into smaller functions, cannot be uniquely defined in the digital transmission hierarchy. Therefore, from the network's point of view it is indivisible (ITU T G.806).
