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1 Mutually inverse functions Two functions f and g are called mutually inverse if the formulas y=f(x) and x=g(y) express the same relationship between the variables x and y, i.e. if the equality y=f(x) is true if and only if the equality x=g(y) is true: y=f(x) x=g(y) If two functions f and g are mutually inverse, then g is called the inverse function for f and, conversely, f is the inverse function for g. For example, y=10 x and x=lgy are mutually inverse functions. Condition for the existence of a mutually inverse function A function f has an inverse if, from the relation y=f(x), the variable x can be uniquely expressed through y. There are functions for which it is impossible to unambiguously express the argument through the given value of the function. For example: 1. y= x. For a given positive number y, there are two values of the argument x such that x = y. For example, if y=2, then x=2 or x= - 2. This means that it is impossible to express x unambiguously through y. Therefore, this function does not have a reciprocal. 2. y=x 2. x=, x= - 3. y=sinx. For a given value of y (y 1), there are infinitely many values of x such that y=sinx. The function y=f(x) has an inverse if every straight line y=y 0 intersects the graph of the function y=f(x) at no more than one point (it may not intersect the graph at all if y 0 does not belong to the range of values of the function f) . This condition can be formulated differently: the equation f(x)=y 0 for each y 0 has at most one solution. The condition that a function has an inverse is certainly satisfied if the function is strictly increasing or strictly decreasing. If f is strictly increasing, then for two different values of the argument it takes on different values, since a larger value of the argument corresponds to a larger value of the function. Consequently, the equation f(x)=y for a strictly monotone function has at most one solution. The exponential function y=a x is strictly monotonic, so it has an inverse logarithmic function. Many functions do not have inverses. If for some b the equation f(x)=b has more than one solution, then the function y=f(x) does not have an inverse. On a graph, this means that the line y=b intersects the graph of the function at more than one point. For example, y=x 2 ; y=sinx; y=tgx.
2 The ambiguity of the solution to the equation f(x) = b can be dealt with by reducing the domain of definition of the function f so that its range of values does not change, but so that it takes each value once. For example, y=x 2, x 0; y=sinx, ; y=tgx,. The general rule for finding the inverse function for a function: 1. solving the equation for x, we find; 2. Changing the designations of the variable x to y, and y to x, we obtain the inverse function of the given one. Properties of mutually inverse functions Identities Let f and g be mutually inverse functions. This means that the equalities y=f(x) and x=g(y) are equivalent: f(g(y))=y and g(f(x))=x. For example, 1. Let f be an exponential function and g a logarithmic function. We get: i. 2. The functions y=x2, x0 and y= are mutually inverse. We have two identities: and for x 0. Domain of definition Let f and g be mutually inverse functions. The domain of the function f coincides with the domain of the function g, and, conversely, the domain of the function f coincides with the domain of the function g. Example. The domain of definition of the exponential function is the entire numerical axis R, and its range of values is the set of all positive numbers. For a logarithmic function it is the opposite: the domain of definition is the set of all positive numbers, and the range of values is the entire set of R. Monotonicity If one of the mutually inverse functions is strictly increasing, then the other is strictly increasing. Proof. Let x 1 and x 2 be two numbers lying in the domain of definition of the function g, and x 1 3 Graphs of mutually inverse functions Theorem. Let f and g be mutually inverse functions. The graphs of the functions y=f(x) and x=g(y) are symmetrical to each other with respect to the bisector of the angle how. Proof. By the definition of mutually inverse functions, the formulas y=f(x) and x=g(y) express the same dependence between the variables x and y, which means that this dependence is depicted by the same graph of some curve C. Curve C is a graph functions y=f(x). Let's take an arbitrary point P(a; b) C. This means that b=f(a) and at the same time a=g(b). Let us construct a point Q symmetrical to the point P relative to the bisector of the angle xy. Point Q will have coordinates (b; a). Since a=g(b), then point Q belongs to the graph of the function y=g(x): indeed, for x=b, the value of y=a is equal to g(x). Thus, all points symmetrical to the points of the curve C relative to the indicated straight line lie on the graph of the function y=g(x). Examples of functions whose graphs are mutually inverse: y=e x and y=lnx; y=x 2 (x 0) and y= ; y=2x 4 and y= +2. 4 Derivative of an inverse function Let f and g be mutually inverse functions. The graphs of the functions y=f(x) and x=g(y) are symmetrical to each other with respect to the bisector of the angle how. Let's take the point x=a and calculate the value of one of the functions at this point: f(a)=b. Then, by definition of the inverse function, g(b)=a. The points (a; f(a))=(a; b) and (b; g(b))=(b; a) are symmetrical about the straight line l. Since the curves are symmetrical, the tangents to them are symmetrical with respect to the straight line l. From symmetry, the angle of one of the lines with the x-axis is equal to the angle of the other line with the y-axis. If a straight line forms an angle α with the x-axis, then its angular coefficient is equal to k 1 =tgα; then the second straight line has an angular coefficient k 2 =tg(α)=ctgα=. Thus, the angular coefficients of lines symmetrical with respect to straight line l are mutually inverse, i.e. k 2 =, or k 1 k 2 =1. Moving on to derivatives and taking into account that the slope of the tangent is the value of the derivative at the point of contact, we conclude: The values of the derivatives of mutually inverse functions at the corresponding points are mutually inverse, i.e. Example 1. Prove that the function f(x) = x 3, reversible. Solution. y=f(x)=x 3. The inverse function will be the function y=g(x)=. Let's find the derivative of the function g:. Those. =. Task 1. Prove that the function given by the formula is invertible 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 5 Example 2. Find the inverse function of the function y=2x+1. Solution. The function y=2x+1 is increasing, therefore it has an inverse. Let's express x through y: we get.. Moving on to generally accepted notations, Answer: Task 2. Find inverse functions for these functions 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) Lecture 20 THEOREM ABOUT THE DERIVATIVE COMPLEX FUNCTION. Let y = f(u), and u= u(x). We obtain a function y depending on the argument x: y = f(u(x)). The last function is called a function from a function or a complex function. Chapter 9 Degrees Degree with an integer exponent. 0 = 0; 0 = ; 0 = 0. > 0 > 0 ; > >.. >. If it's even, then ()< (). Например, () 0 = 0 < 0 = = () 0. Если нечетно, то () >(). For example, () = > = = (), so What we will study: Lesson on the topic: Study of a function for monotonicity. Decreasing and increasing functions. Relationship between derivative and monotonicity of a function. Two important theorems about monotonicity. Examples. Guys, we The linear equation a x = b has: a unique solution, for a 0; an infinite set of solutions, with a = 0, b = 0; has no solutions, for a = 0, b 0. The quadratic equation ax 2 + bx + c = 0 has: two different 6 Problems leading to the concept of derivative Let a material point move along a straight line in one direction according to the law s f (t), where t is time, and s is the path traversed by the point in time t Let us note a certain point Bank of tasks on the topic “DERIVATIVE” MATHEMATICS Grade 11 (basic) Students should know/understand: The concept of derivative. Definition of derivative. Theorems and rules for finding derivatives of a sum, difference, product Geometric meaning of the derivative Consider the graph of the function y=f(x) and the tangent at the point P 0 (x 0 ; f(x 0)). Let's find the slope of the tangent to the graph at this point. Angle of inclination of the tangent P 0 Quadratic function in various problems Dikhtyar MB Basic information A quadratic function (quadratic trinomial) is a function of the form y ax bx c, where abc, given numbers and Quadratic functions y THE CONCEPT OF A DERIVATIVE FUNCTION Let us have a function defined on a set X and let a point X be an interior point of those points for which there is a neighborhood X. Take any point and denote it by called Lecture 5 Derivatives of basic elementary functions Abstract: Physical and geometric interpretations of the derivative of a function of one variable are given. Examples of differentiation of functions and rules are considered. 1 SA Lavrenchenko Lecture 12 Inverse functions 1 The concept of an inverse function Definition 11 A function is called one-to-one if it does not take any value more than once, those of which follow when Department of Mathematics and Computer Science Elements of Higher Mathematics Educational and methodological complex for secondary vocational education students studying using distance technologies Module Differential calculus Compiled by: Chapter 5 Study of functions using the Taylor formula Local extremum of a function Definition Function = f (reaches a local maximum (minimum) at point c, if it is possible to specify a δ > such that its increment MODULE “Application of continuity and derivative. Application of the derivative to the study of functions." Application of continuity.. Interval method.. Tangent to the graph. Lagrange's formula. 4. Application of derivative Lecture 9. Derivatives and differentials of higher orders, their properties. Extremum points of the function. Theorems of Fermat and Rolle. Let the function y be differentiable on some interval [b]. In this case, its derivative Department of Mathematics and Computer Science Mathematical analysis Educational and methodological complex for higher education students studying using distance technologies Module 4 Derivative applications Compiled by: Associate Professor Chapter 1. Limits and continuity 1. Number sets 1 0. Real numbers From school mathematics you know natural N integers Z rational Q and real R numbers Natural and integer numbers Lecture 19 DERIVATIVE AND ITS APPLICATIONS. DEFINITION OF DERIVATIVE. Let us have some function y=f(x), defined on some interval. For each value of the argument x from this interval, the function y=f(x) Differential calculus Basic concepts and formulas Definition 1 The derivative of a function at a point is the limit of the ratio of the increment of the function to the increment of the argument, provided that the increment of the argument Topic 8. Exponential and logarithmic functions. 1. Exponential function, its graph and properties In practice, the functions y=2 x,y=10 x,y=(1 2x),y=(0,1) x, etc. are often used, i.e. the function of the form y=a x, 44 Example Find the total derivative of a complex function = sin v cos w where v = ln + 1 w= 1 Using formula (9) d v w v w = v w d sin cos + cos cos + 1 sin sin 1 Now find the total differential of the complex function f Tasks for independent solution. Find the domain of the function 6x. Find the tangent of the angle of inclination to the x-axis of the tangent passing through point M (;) of the graph of the function. Find the tangent of the angle Topic Numerical function, its properties and graph Concept of a numerical function Domain of definition and set of values of a function Let a numerical set X be given A rule that associates each number X with a unique Lecture 23 CONVEX AND CONCAVENESS OF THE GRAPH OF AN INFLECTION POINT FUNCTION The graph of the function y=f(x) is called convex on the interval (a; b) if it is located below any of its tangents on this interval Graph Topic Theory of limits Practical lesson Number sequences Definition of a number sequence Bounded and unbounded sequences Monotonic sequences Infinitesimal Numerical functions and numerical sequences D. V. Lytkina NPP, I semester D. V. Lytkina (SibGUTI) mathematical analysis of NPP, I semester 1 / 35 Contents 1 Numerical function Concept of function Numerical functions. Bank of tasks on the topic “DERIVATIVE” MATHEMATICS class (profile) Students should know/understand: The concept of derivative. Definition of derivative. Theorems and rules for finding derivatives of a sum, difference, product Â. À. DANGER MEASUREMENT: THE FRAMEWORK OF THE FRAMEWORK. RESUME TEACHING MANUAL FOR SPO - edition, corrected and supplemented by the Russian Academy of Sciences synonymous A.V. Zemlyanko Mathematics. Algebra and principles of analysis Voronezh CONTENTS TOPIC 1. BASIC PROPERTIES OF A FUNCTION... 6 1.1. Numerical function... 6 1.2. Graph of a function... 9 1.3. Converting function graphs... Subject. Function. Methods of assignment. Implicit function. Inverse function. Classification of functions Elements of set theory. Basic concepts One of the basic concepts of modern mathematics is the concept of set. Let a numerical set D R be given. If each number x D is associated with a single number y, then we say that a numerical function is given on the set D: y = f (x), x D. The set D is called Functions of several variables 11. Definition of a function of several variables. Limit and continuity of the FNP 1. Definition of a function of several variables DEFINITION. Let X = ( 1 n i X i R ) U R. Function MATHEMATICS FOR ALL Y.L. Kalinovsky Contents 1 Graphs of functions. Part I................................... 5 1.1 Introduction 5 1.1.1 The concept of set... ........................................... 5 1.1. Practical work 6 Topic: “Complete study of functions. Plotting graphs" Purpose of the work: learn to explore functions according to a general scheme and construct graphs. As a result of completing the work, the student must: Chapter 8 Functions and graphs Variables and dependencies between them. Two quantities are called directly proportional if their ratio is constant, that is, if =, where is a constant number that does not change with changes LECTURE 2. Operations with subspaces, the number of bases, the number of bases and the number of subspaces of dimension k. Main results of Lecture 2. 1) U V, U + V, dim(u + V). 2) Counting the number of planes in F 4 2. Question 5. Function, methods of assignment. Examples of elementary functions and their graphics. Let two arbitrary sets X and Y be given. A function is a rule by which each element from the set X can be found Lecture 4 NUMERICAL FUNCTIONS OF A REAL VARIABLE Concept of a function Methods of specifying a function Basic properties of functions Complex function 4 Inverse function Concept of a function Methods of specifying a function Let D Lectures Chapter Functions of several variables Basic concepts Some functions of several variables are well known Let's give a few examples To calculate the area of a triangle, Heron's formula S is known Continuity of functions Continuity of a function at a point One-sided limits Definition A number A is called the limit of a function f(x) from the left as x tends to a if for any number there exists such a number Research work Mathematics “Application of extremal properties of a function for solving equations” Completed by: Elena Gudkova, student of 11th grade “G” MBOU secondary school “Anninsky Lyceum” urban settlement. Anna Head: Federal Agency for Education ----- ST. PETERSBURG STATE POLYTECHNIC UNIVERSITY AI Surygin EF Izotova OA Novikova TA Chaikina MATHEMATICS Elementary functions and their graphs Educational FUNCTIONS OF SEVERAL VARIABLES The functions of one independent variable do not cover all the dependencies that exist in nature. Therefore, it is natural to expand the well-known concept of functional dependence and introduce Function Concept of a function Methods for specifying a function Characteristics of a function Inverse function Limit of a function Limit of a function at a point One-sided limits Limit of a function at x Infinitely large function 4 Lecture Section Differential calculus of functions of one and several variables Function of real argument Real numbers Positive integers are called natural numbers Add to natural numbers Sergey A Belyaev page 1 Mathematical minimum Part 1 Theoretical 1 Is the definition correct? The least common multiple of two integers is the smallest number that is divisible by each of the given numbers Section 2 Theory of limits Topic Number sequences Definition of a number sequence 2 Bounded and unbounded sequences 3 Monotone sequences 4 Infinitesimal and Differentiation of an implicitly given function Consider the function (,) = C (C = const) This equation defines the implicit function () Suppose we solved this equation and found the explicit expression = () Now we can Test tasks for preparing for the EXAM in the discipline "Mathematics" for correspondence students. The derivative of the function y=f() is called: f A) B) f C) f f If in some neighborhood of a point the function VARIABLES AND CONSTANT QUANTITIES As a result of measuring physical quantities (time, area, volume, mass, speed, etc.), their numerical values are determined. Mathematics deals with quantities, distracted Mathematical analysis Section: Introduction to analysis Topic: Concept of function (basic definitions, classification, basic characteristics of behavior) Lecturer Rozhkova S.V. 2012 Literature Piskunov N.S. Differential Lesson 7 Mean value theorems. L'Hopital's rule 7. Theorems on the mean Theorems on the mean are three theorems: Rolle, Lagrange and Cauchy, each of which generalizes the previous one. These theorems are also called Lecture prepared by Associate Professor Musina MV Continuity of a function Let the function y = f(x) be defined at the point x and in some neighborhood of this point The function y = f(x) is called continuous at the point x if it exists DIFFERENTIATION OF FUNCTIONS OF ONE VARIABLE The concept of a derivative, its geometric and physical meaning. Problems leading to the concept of a derivative. Determination of the Tangent S to the line y f (x) at the point A x; f ( 13. Partial derivatives of higher orders Let = have and are defined on D O. The functions and are also called first-order partial derivatives of a function or first partial derivatives of a function. and in general Ministry of Education of the Republic of Belarus EDUCATIONAL INSTITUTION "GRODNO STATE UNIVERSITY NAMED AFTER YANKA KUPALA" Yu.Yu. Gnezdovsky, V.N. Gorbuzov, P.F. Pronevich EXPONENTARIAL AND LOGARITHMIC Lecture Chapter Sets and operations on them The concept of a set The concept of a set refers to the most primary concepts of mathematics that are not defined through simpler ones. A set is understood as a collection Lecture 8 Differentiation of a complex function Consider a complex function t t t f where ϕ t t t t t t t f t t t t t t t t t Theorem Let the functions be differentiable at some point N t t t and the function f be differentiable Lecture 3 Extremum of a function of several variables Let a function of several variables u = f (x, x) be defined in the domain D, and the point x (x, x) = belongs to this domain The function u = f (x, x) has Question. Inequalities, system of linear inequalities Let's consider expressions that contain an inequality sign and a variable:. >, - +x are linear inequalities with one variable x.. 0 is a quadratic inequality. SECTION PROBLEMS WITH PARAMETERS Comment Problems with parameters are traditionally complex tasks in the structure of the Unified State Examination, requiring the applicant not only to master all the methods and techniques for solving various 2.2.7. Application of differential to approximate calculations. The differential of the function y = depends on x and is the main part of the increment of x. You can also use the formula: dy d Then the absolute error is: Chapter 6 Differential calculus of a function of one variable Problems leading to the concept of derivative Problem about the speed of non-uniform rectilinear motion S - the law of non-uniform rectilinear motion Line on a plane General equation of a line. Before introducing the general equation of a straight line on a plane, let us introduce the general definition of a line. Definition. An equation of the form F(x,y)=0 (1) is called the line equation L COMMITTEE OF GENERAL AND PROFESSIONAL EDUCATION OF THE LENINGRAD REGION STATE BUDGET PROFESSIONAL EDUCATIONAL INSTITUTION OF THE LENINGRAD REGION “VOLKHOV ALUMINUM COLLEGE” Methodological Derivative and differentiation rules Let the function y = f receive an increment y f 0 f 0 corresponding to the increment of the argument 0 Definition If there is a limit on the ratio of the increment of the function y to the caller Moscow State Technical University named after N.E. Bauman Faculty of Fundamental Sciences Department of Mathematical Modeling A.N. Kaviakovykov, A.P. Kremenko INVERSE FUNCTIONS Problems in which inverse functions are involved are found in various branches of mathematics and in its applications. An important area of mathematics is inverse problems in the theory of integral System of problems on the topic “Tangent Equation” Determine the sign of the slope of the tangent drawn to the graph of the function y f (), at points with abscissas a, b, c a) b) Indicate the points at which the derivative Corresponding expressions that reverse each other. To understand what this means, it's worth looking at a specific example. Let's say we have y = cos(x). If you take the cosine from the argument, you can find the value of y. Obviously, for this you need to have X. But what if the game was initially given? This is where it comes to the heart of the matter. To solve the problem, you need to use the inverse function. In our case it is arccosine. After all the transformations we get: x = arccos(y). That is, to find a function inverse to a given one, it is enough to simply express an argument from it. But this only works if the resulting result has a single meaning (more on this later). In general terms, this fact can be written as follows: f(x) = y, g(y) = x. Let f be a function whose domain is the set X and whose domain is the set Y. Then, if there exists a g whose domains perform opposite tasks, then f is invertible. Moreover, in this case g is unique, which means that there is exactly one function that satisfies this property (no more, no less). Then it is called the inverse function, and in writing it is denoted as follows: g(x) = f -1 (x). In other words, they can be thought of as a binary relation. Reversibility occurs only when one element of the set corresponds to one value from another. The inverse function does not always exist. To do this, each element y є Y must correspond to at most one x є X. Then f is called one-to-one or injection. If f -1 belongs to Y, then each element of this set must correspond to some x ∈ X. Functions with this property are called surjections. It holds by definition if Y is an image of f, but this is not always the case. To be inverse, a function must be both an injection and a surjection. Such expressions are called bijections. The function is defined on )
Definition
Example: square and root functions
