Build a function

We bring to your attention a service for plotting function graphs online, all rights to which belong to the company Desmos. Use the left column to enter functions. You can enter it manually or with virtual keyboard at the bottom of the window. To enlarge the chart window, you can hide both the left column and the virtual keyboard.

Benefits of online charting

• Visual display of introduced functions
• Building very complex graphs
• Plotting implicitly defined graphs (e.g. ellipse x^2/9+y^2/16=1)
• The ability to save charts and get a link to them, which becomes available to everyone on the Internet
• Scale control, line color
• The ability to plot graphs by points, the use of constants
• Construction of several graphs of functions at the same time
• Plotting in polar coordinates (use r and θ(\theta))

With us it is easy to build graphs of varying complexity online. The construction is done instantly. The service is in demand for finding intersection points of functions, for displaying graphs for their further movement to word document as illustrations in solving problems, for analyzing the behavioral features of function graphs. The optimal browser for working with charts on this page of the site is Google Chrome. When using other browsers, correct operation is not guaranteed.

Let's get acquainted with the concept of superposition (or imposition) of functions, which consists in the fact that instead of an argument of a given function, some function of another argument is substituted. For example, the superposition of functions gives a function; similarly, functions are obtained

In general terms, suppose that a function is defined in some domain and a function is defined in a domain and all its values ​​are contained in the domain Then the variable z, as they say, through y, is itself a function of

Given a given from, first find the value y from Y corresponding to it (according to the rule characterized by the sign), and then set the corresponding value y (according to the rule,

characterized by a sign and its value is considered corresponding to the chosen x. The resulting function from a function or a complex function is the result of a superposition of functions

The assumption that the values ​​of a function do not go beyond the region Y in which the function is defined is very significant: if it is omitted, then absurdity may result. For example, assuming we can consider only those values ​​of x for which otherwise the expression would not make sense.

We consider it useful here to emphasize that the characterization of a function as complex is not connected with the nature of the functional dependence of z on x, but only with the way this dependence is specified. For example, let for y in for Then

Here the function turned out to be given as a complex function.

Now that the concept of superposition of functions has been fully elucidated, we can accurately characterize the simplest of those classes of functions that are studied in analysis: these are, first of all, the elementary functions listed above and then all those that are obtained from them using four arithmetic operations and superpositions , successively applied a finite number of times. They say about them that they are expressed through elementary ones in the final form; sometimes they are all also called elementary.

Subsequently, having mastered a more complex analytical apparatus (infinite series, integrals), we will also get acquainted with other functions that also play an important role in analysis, but already go beyond the class of elementary functions.

Let there be 2 functions:

: A→B and g: D→F

Let the domain D of the function g be included in the domain of the function f (DB). Then one can define new feature – superposition (composition, complex function) functions f and g: z= g( (x)).

Examples. f(x)=x 2 , g(x)=e x . f:R→R, g:R→R .

(g(x))=e 2x , g((x))=.

Definition

Let two functions. Then their composition is the function defined by the equality:

Composition Properties

The composition is associative:

If a F= id X- identity mapping on X, that is

.

If a G= id Y- identity mapping on Y, that is

.

Additional properties

Countable and uncountable sets.

Two finite sets consist of an equal number of elements if a one-to-one correspondence can be established between these sets. The number of elements of a finite set is the cardinality of the set.

For an infinite set, one can establish a one-to-one correspondence between the entire set and its part.

The simplest of the infinite sets is the set N.

Definition. The sets A and B are called equivalent(AB) if a one-to-one correspondence can be established between them.

If two finite sets are equivalent, then they consist of the same number of elements.

If the equivalent sets A and B are arbitrary, then they say that A and B have the same power. (power = equivalence).

For finite sets, the concept of cardinality coincides with the concept of the number of elements in a set.

Definition. The set is called countable if it is possible to establish a one-to-one correspondence between it and the set of natural numbers. (That is, a countable set is infinite, equivalent to the set N).

(Ie, all elements of a countable set can be enumerated).

Equivalence relationship properties.

1) AA - reflexivity.

2) AB, then BA - symmetry.

3) AB and BC, then AC is transitivity.

Examples.

1) n→2n, 2,4,6,… - even natural numbers

2) n→2n-1, 1,3,5,… are odd natural numbers.

Properties of countable sets.

1. Infinite subsets of a countable set are countable.

Proof. Because A is countable, then A: x 1, x 2, ... - displayed A in N.

ВА, В: →1,→2,… - assigned to each element В a natural number, i.e. mapped B to N. Therefore, B is countable. Ch.t.d.

2. The union of a finite (countable) system of countable sets is countable.

Examples.

1. The set of integers Z is countable, because set Z can be represented as the union of countable sets A and B, where A: 0,1,2,.. and B: -1,-2,-3,…

2. Many orderly pairs ((m,n): m,nZ) (i.e. (1,3)≠(3,1)).

3 (!) . The set of rational numbers is countable.

Q=. One can establish a one-to-one correspondence between the set of irreducible fractions Q and the set of ordered pairs:

That. the set Q is equivalent to the set ((p,q))((m,n)).

The set ((m,n)) - the set of all ordered pairs - is countable. Consequently, the set ((p,q)) is also countable, and hence Q is countable.

Definition. An irrational number is an arbitrary infinite decimal non-periodic fraction, i.e.  0 , 1  2 …

The set of all decimal fractions form the set real (real) numbers.

The set of irrational numbers is uncountable.

Theorem 1. The set of real numbers from the interval (0,1) is an uncountable set.

Proof. Assume the contrary, i.e. that all numbers in the interval (0,1) can be enumerated. Then, writing these numbers as infinite decimal fractions, we get the sequence:

x 1 \u003d 0, a 11 a 12 ... a 1n ...

x 2 \u003d 0,a 21 a 22 ... a 2n ...

…………………..

x n =0,a n 1 a n 2 …a nn …

……………………

Consider now a real number x=0,b 1 b 2 ... b n ..., where b 1 is any number other than a 11, (0 and 9), b 2 - any number other than a 22, (0 and 9) ,…, b n - any digit other than a nn , (0 and 9).

That. x(0,1), but xx i (i=1,…,n) because otherwise, b i =a ii . They came to a contradiction. Ch.t.d.

Theorem 2. Any interval of the real axis is an uncountable set.

Theorem 3. The set of real (real) numbers is uncountable.

Any set that is equivalent to the set of real numbers is said to be continuum powers(lat. continuum - continuous, continuous).

Example. Let us show that the interval has the cardinality of the continuum.

The function y \u003d tg x: → R displays the interval on the entire number line (graph).

Topic: “Function: concept, methods of assignment, main characteristics. Inverse function. Superposition of functions."

Epigraph of the lesson:

“Study something and not think about

learned - absolutely useless.

Thinking about something without studying

preliminary subject of thought

Confucius.

The purpose and psychological and pedagogical tasks of the lesson:

1) General educational (normative) goal: repeat with students the definition and properties of a function. Introduce the concept of superposition of functions.

2) Tasks of mathematical development of students: on non-standard educational and mathematical material, continue the development of the mental experience of students, the meaningful cognitive structure of their mathematical intelligence, including the ability to logical-deductive and inductive, analytical and synthetic reversible thinking, to algebraic and figurative-graphic thinking, to meaningful generalization and concretization, to reflection and independence as a metacognitive ability of students; to continue the development of the culture of written and oral speech as psychological mechanisms of educational and mathematical intelligence.

3) Educational tasks: to continue the personal education of students of cognitive interest in mathematics, responsibility, sense of duty, academic independence, communicative ability to cooperate with a group, teacher, classmates; autological ability for competitive educational and mathematical activity, striving for its high and highest results (acmeic motive).

Lesson type: learning new material; according to the criterion of the leading mathematical content - a practical lesson; according to the criterion of the type of information interaction between students and the teacher - a lesson of cooperation.

Lesson equipment:

1. Educational literature:

1) Kudryavtsev of mathematical analysis: Proc. for students of universities and universities. In 3 vols. T. 3. - 2nd ed., Revised. and additional - M .: Higher. school, 1989. - 352 p. : ill.

2) Demidovich problems and exercises in mathematical analysis. – 9th ed. - M .: Publishing house "Nauka", 1977.

2. Illustrations.

During the classes.

1. Announcement of the topic and the main educational goal of the lesson; stimulation of a sense of duty, responsibility, cognitive interest of students in preparation for the session.

2. Repetition of material on questions.

a) Define the function.

One of the basic mathematical concepts is the concept of a function. The concept of a function is associated with establishing a relationship between the elements of two sets.

Let two non-empty sets and be given. A match f that matches each element with one and only one element is called function and written y = f(x). They also say that the function f displays set to set.

https://pandia.ru/text/79/018/images/image003_18.gif" width="63" height="27">.gif" width="59" height="26"> is called set of values function f and is denoted by E(f).

b) Numeric functions. Function graph. Ways to set functions.

Let a function be given.

If the elements of the sets and are real numbers, then the function f is called numerical function . The variable x is called argument or an independent variable, and y is function or dependent variable(from x). Regarding the quantities x and y themselves, they are said to be in functional dependence.

Function Graph y = f(x) is the set of all points of the Oxy plane, for each of which x is the value of the argument, and y is the corresponding value of the function.

To define a function y = f(x), you need to specify a rule that allows, knowing x, to find the corresponding value of y.

There are three most common ways to define a function: analytical, tabular, graphic.

Analytical method: The function is specified as one or more formulas or equations.

For example:

If the domain of the function y = f(x) is not specified, then it is assumed that it coincides with the set of all values ​​of the argument for which the corresponding formula makes sense.

The analytical method of defining a function is the most perfect, since it is accompanied by methods of mathematical analysis that allow you to fully explore the function y = f(x).

Graphical way: Sets the graph of the function.

The advantage of a graphic task is its visibility, the disadvantage is its inaccuracy.

Tabular way: A function is specified by a table of a series of argument values ​​and corresponding function values. For example, the well-known tables of values trigonometric functions, logarithmic tables.

c) The main characteristics of the function.

1. The function y = f(x) defined on the set D is called even , if the conditions are met and f(-x) = f(x); odd , if the conditions are met and f(-x) = -f(x).

The graph of an even function is symmetrical about the Oy axis, and an odd function is symmetrical about the origin. For example, are even functions; and y = sinx, https://pandia.ru/text/79/018/images/image014_3.gif" width="73" height="29"> are general functions, i.e. neither even nor odd .

2. Let the function y = f(x) be defined on the set D and let . If for any values ​​of the arguments, the inequality implies the inequality: , then the function is called increasing on the set ; if , then the function is called non-decreasing on https://pandia.ru/text/79/018/images/image021_1.gif" width="117" height="28 src="> then the function is called. waning on the ; - non-increasing .

Increasing, non-increasing, decreasing and non-decreasing functions on the set https://pandia.ru/text/79/018/images/image023_0.gif" width="13" height="13">D value (x+T)D and the equality f(x+T) = f(x) holds.

To plot a periodic function of period T, it suffices to plot it on any segment of length T and periodically extend it to the entire domain of definition.

We note the main properties of a periodic function.

1) The algebraic sum of periodic functions having the same period T is a periodic function with period T.

2) If the function f(x) has period T, then the function f(ax) has period T/a.

d) Inverse function.

Let the function y = f(x) be given with the domain of definition D and the set of values ​​E..gif" width="48" height="22">, then the function x = z(y) with the domain of definition E and the set of values ​​D Such a function z(y) is called reverse to the function f(x) and is written in the following form: . The functions y = f(x) and x = z(y) are said to be mutually inverse. To find the function x = z(y) inverse to the function y = f(x), it suffices to solve the equation f(x) = y with respect to x.

Examples:

1. For a function y = 2x, the inverse function is the function x = ½ y;

2. For function the inverse function is the function .

It follows from the definition of an inverse function that a function y = f(x) has an inverse if and only if f(x) defines a one-to-one correspondence between the sets D and E. It follows that any a strictly monotonic function has an inverse . Moreover, if the function increases (decreases), then the inverse function also increases (decreases).

3. Learning new material.

Complicated function.

Let the function y = f(u) be defined on the set D, and the function u = z(x) on the set , and for the corresponding value . Then the set has a function u = f(z(x)), which is called complex function from x (or superposition given functions, or function from function ).

The variable u = z(x) is called intermediate argument complex function.

For example, the function y = sin2x is a superposition of two functions y = sinu and u = 2x. A complex function can have multiple intermediate arguments.

4. Solution of several examples at the blackboard.

5. Conclusion of the lesson.

1) theoretical and applied results practical session; differentiated assessment of the level of mental experience of students; the level of their assimilation of the topic, competence, quality of oral and written mathematical speech; the level of manifested creativity; level of independence and reflection; level of initiative, cognitive interest in certain methods of mathematical thinking; levels of cooperation, intellectual competitiveness, striving for high performance educational and mathematical activities, etc.;

2) announcement of reasoned marks, lesson points.

Superposition of functions

A superposition of functions f1, …, fm is a function f obtained by substituting these functions into each other and renaming variables.

Let there be two mappings and, moreover, a non-empty set. Then a superposition or composition of functions is a function defined by equality for any.

The domain of definition of a superposition is a set.

The function is called the outer, and the inner function for superposition.

Functions presented as a composition of "simpler" ones are called complex functions.

Examples of the use of superposition are: solving a system of equations by the substitution method; finding the derivative of a function; finding the value of an algebraic expression by substituting the values ​​of given variables into it.

Recursive functions

Recursion is such a way of defining a function, in which the values ​​of the function being defined for arbitrary values ​​of the arguments are expressed in a known way through the values ​​of the function being defined for smaller values ​​of the arguments.

Primitive recursive function

The definition of the concept of a primitive recursive function is inductive. It consists of specifying a class of basic primitive recursive functions and two operators (superposition and primitive recursion) that allow building new primitive recursive functions based on existing ones.

The basic primitive recursive functions include the following three types of functions:

Zero function-- function no arguments, always returning 0 .

A one-variable succession function that assigns any natural number to the immediately following natural number.

Functions, where, from n variables, which assign to any ordered set of natural numbers a number from this set.

The substitution and primitive recursion operators are defined as follows:

Superposition operator (sometimes a substitution operator). Let be a function of m variables, and be an ordered set of functions of non-variables each. Then the result of the superposition of functions into a function is a function of variables that associates a number with any ordered set of natural numbers.

Primitive recursion operator. Let be a function of n variables, and be a function of variables. Then the result of applying the primitive recursion operator to a pair of functions is the function of the type variable;

In this definition, a variable can be understood as an iteration counter, -- as original function at the beginning of the iterative process, issuing a certain sequence of functions of variables, starting with, and -- as an operator that accepts as input variables, the iteration step number, the function at this iteration step, and returning the function at the next iteration step.

The set of primitive recursive functions is the minimum set containing all basic functions and closed under the specified substitution and primitive recursion operators.

In terms of imperative programming -- primitive recursive functions correspond to program blocks that use only arithmetic operations, as well as conditional operator and an arithmetic loop operator (a loop operator in which the number of iterations is known at the start of the loop). If the programmer starts using the while loop operator, in which the number of iterations is not known in advance and, in principle, can be infinite, then he passes into the class of partially recursive functions.

Let us point out a number of well-known arithmetic functions that are primitively recursive.

The function of adding two natural numbers () can be considered as a primitive recursive function of two variables, obtained by applying the primitive recursion operator to the functions and, the second of which is obtained by substituting the main function into the main function:

The multiplication of two natural numbers can be considered as a primitive recursive function of two variables, obtained as a result of applying the primitive recursion operator to the functions and, the second of which is obtained by substituting the main functions and into the addition function:

The symmetric difference (the absolute value of the difference) of two natural numbers () can be considered as a primitive recursive function of two variables, obtained by applying the following substitutions and primitive recursions: