Inverse functions giving unique values. §7

Let us assume that we have a certain function y = f (x), which is strictly monotonic (decreasing or increasing) and continuous on the domain of definition x ∈ a; b ; its range of values y ∈ c ; d, and on the interval c; d in this case we will have a function defined x = g (y) with a range of values a ; b. The second function will also be continuous and strictly monotonic. With respect to y = f (x) it will be an inverse function. That is, we can talk about the inverse function x = g (y) when y = f (x) on given interval will either decrease or increase.

These two functions, f and g, will be mutually inverse.

Why do we even need the concept of inverse functions?

We need this to solve the equations y = f (x), which are written precisely using these expressions.

Let's say we need to find a solution to the equation cos (x) = 1 3 . Its solutions will be all points: x = ± a rc c o s 1 3 + 2 π · k, k ∈ Z

For example, the inverse cosine and cosine functions will be inverse to each other.

Let's look at several problems to find functions that are inverse to given ones.

Example 1

Condition: what is the inverse function for y = 3 x + 2?

Solution

The domain of definitions and range of values of the function specified in the condition is the set of all real numbers. Let's try to solve this equation through x, that is, by expressing x through y.

We get x = 1 3 y - 2 3 . This is the inverse function we need, but y will be the argument here, and x will be the function. Let's rearrange them to get a more familiar notation:

Answer: the function y = 1 3 x - 2 3 will be the inverse of y = 3 x + 2.

Both are mutual inverse functions can be plotted as follows:

We see the symmetry of both graphs regarding y = x. This line is the bisector of the first and third quadrants. The result is a proof of one of the properties of mutually inverse functions, which we will discuss later.

Let's take an example in which we need to find the logarithmic function that is the inverse of a given exponential function.

Example 2

Condition: determine which function will be the inverse for y = 2 x.

Solution

For a given function, the domain of definition is all real numbers. The range of values lies in the interval 0; + ∞ . Now we need to express x in terms of y, that is, solve the specified equation in terms of x. We get x = log 2 y. Let's rearrange the variables and get y = log 2 x.

As a result, we have obtained exponential and logarithmic functions, which will be mutually inverse to each other throughout the entire domain of definition.

Answer: y = log 2 x .

On the graph, both functions will look like this:

Basic properties of mutually inverse functions

In this paragraph we list the main properties of the functions y = f (x) and x = g (y), which are mutually inverse.

Definition 1

We already derived the first property earlier: y = f (g (y)) and x = g (f (x)).
The second property follows from the first: the domain of definition y = f (x) will coincide with the range of values of the inverse function x = g (y), and vice versa.
The graphs of functions that are inverse will be symmetrical with respect to y = x.
If y = f (x) is increasing, then x = g (y) will increase, and if y = f (x) is decreasing, then x = g (y) will also decrease.

We advise you to pay close attention to the concepts of domain of definition and domain of meaning of functions and never confuse them. Let's assume that we have two mutually inverse functions y = f (x) = a x and x = g (y) = log a y. According to the first property, y = f (g (y)) = a log a y. This equality will be true only in the case of positive values of y, and for negative values the logarithm is not defined, so do not rush to write down that a log a y = y. Be sure to check and add that this is only true when y is positive.

But the equality x = f (g (x)) = log a a x = x will be true for any real values of x.

Don't forget about this point, especially if you have to work with trigonometric and inverse trigonometric functions. So, a r c sin sin 7 π 3 ≠ 7 π 3, because the arcsine range is π 2; π 2 and 7 π 3 are not included in it. The correct entry will be

a r c sin sin 7 π 3 = a r c sin sin 2 π + π 3 = = a r c sin sin π 3 = π 3

But sin a r c sin 1 3 = 1 3 is a correct equality, i.e. sin (a r c sin x) = x for x ∈ - 1; 1 and a r c sin (sin x) = x for x ∈ - π 2 ; π 2. Always be careful with the range and scope of inverse functions!

Basic mutually inverse functions: power functions

If we have a power function y = x a , then for x > 0 the power function x = y 1 a will also be its inverse. Let's replace the letters and get y = x a and x = y 1 a, respectively.

On the graph they will look like this (cases with positive and negative coefficient a):

Basic mutually inverse functions: exponential and logarithmic

Let's take a, which will be a positive number not equal to 1.

Graphs for functions with a > 1 and a< 1 будут выглядеть так:

Basic mutually inverse functions: trigonometric and inverse trigonometric

If we were to plot the main branch sine and arcsine, it would look like this (shown as the highlighted light area).

Let there be a function y=f(x), X is its domain of definition, Y is its range of values. We know that each x 0  corresponds to a single value y 0 =f(x 0), y 0 Y.

It may turn out that each y (or its part  1) also corresponds to a single x from X.

Then they say that on the region  (or its part  ) the function x=y is defined as the inverse function for the function y=f(x).

For example:

X =(); Y=$

Since this function is decreasing and continuous on the interval $X$, then on the interval $Y=$, which is also decreasing and continuous on this interval (Theorem 1).

Let's calculate $x$:

\ \

Select suitable $x$:

Answer: inverse function $y=-\sqrt(x)$.

Problems on finding inverse functions

In this part we will consider inverse functions for some elementary functions. We will solve problems according to the scheme given above.

Example 2

Find the inverse function for the function $y=x+4$

Let's find $x$ from the equation $y=x+4$:

Example 3

Find the inverse function for the function $y=x^3$

Solution.

Since the function is increasing and continuous over the entire domain of definition, then, according to Theorem 1, it has an inverse continuous and increasing function on it.

Let's find $x$ from the equation $y=x^3$:

Finding suitable values of $x$

The value is suitable in our case (since the domain of definition is all numbers)

Let's redefine the variables, we get that the inverse function has the form

Example 4

Find the inverse function for the function $y=cosx$ on the interval $$

Solution.

Consider the function $y=cosx$ on the set $X=\left$. It is continuous and decreasing on the set $X$ and maps the set $X=\left$ onto the set $Y=[-1,1]$, therefore, by the theorem on the existence of an inverse continuous monotone function, the function $y=cosx$ in the set $ Y$ there is an inverse function, which is also continuous and increasing in the set $Y=[-1,1]$ and maps the set $[-1,1]$ to the set $\left$.

Let's find $x$ from the equation $y=cosx$:

Finding suitable values of $x$

Let's redefine the variables, we get that the inverse function has the form

Example 5

Find the inverse function for the function $y=tgx$ on the interval $\left(-\frac(\pi )(2),\frac(\pi )(2)\right)$.

Solution.

Consider the function $y=tgx$ on the set $X=\left(-\frac(\pi )(2),\frac(\pi )(2)\right)$. It is continuous and increasing on the set $X$ and maps the set $X=\left(-\frac(\pi )(2),\frac(\pi )(2)\right)$ onto the set $Y=R$, therefore, by the theorem on the existence of an inverse continuous monotone function, the function $y=tgx$ in the set $Y$ has an inverse function, which is also continuous and increasing in the set $Y=R$ and maps the set $R$ onto the set $\left(- \frac(\pi )(2),\frac(\pi )(2)\right)$

Let's find $x$ from the equation $y=tgx$:

Finding suitable values of $x$

Let's redefine the variables, we get that the inverse function has the form

Transcript

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

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 material point moves in a straight line in one direction according to the law s f (t), where t is time and s is path, traversable by point during time t Let us note a certain moment

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 (square trinomial) is a function of the form y ax bx c, where abc, given numbers and Quadratic functions at

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 Training and metodology 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. Numerical 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 singular y, then they 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: “Full 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 functions Methods for specifying a function Basic properties of functions Complex function 4 Inverse function Concept of a function Methods for 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 Infinite great 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 to prepare for the EXAM in the discipline "Mathematics" for students correspondence department The derivative of the function y=f() is called: f A) B) f C) f f If in some neighborhood of the point the function

VARIABLES AND CONSTANT QUANTITIES As a result of measuring physical quantities (time, area, volume, mass, speed, etc.) their numeric values. 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'Hôpital'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 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 is linear inequalities with one variable x.. 0 is a quadratic inequality.

SECTION PROBLEMS WITH PARAMETERS Comment Problems with parameters are traditionally complex tasks in structure of the Unified State Exam, requiring from the applicant not only knowledge of all 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 movement S - law of uneven linear motion

Line on a plane General equation of a line. Before entering general equation line on a plane, we 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 University Technical University named after N.E. Bauman Faculty " Basic Sciences» Department of Mathematical Modeling A.H. 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

Mutually inverse functions.

Let the function be strictly monotonic (increasing or decreasing) and continuous on the domain of definition, the domain of values of this function, then on the interval a continuous strictly monotonic function with a domain of values is defined, which is the inverse of .

In other words, it makes sense to talk about an inverse function for a function on a specific interval if it either increases or decreases on this interval.

Functions f And g are called mutually inverse.

Why consider the concept of inverse functions at all?

This is caused by the problem of solving equations. Solutions are written using inverse functions.

Let's consider several examples of finding inverse functions .

Let's start with linear mutually inverse functions.

Find the inverse function for.

This function is linear; its graph is a straight line. This means that the function is monotonic over the entire domain of definition. Therefore, we will look for its inverse function throughout the entire domain of definition.

Let's express x through y (in other words, let’s solve the equation for x ).

- this is the inverse function, though here y - argument, and x is the function of this argument. In order not to break the habits in notation (this is not of fundamental importance), rearranging the letters x And y , will write .

Thus, and are mutually inverse functions.

Here is a graphical illustration of mutually inverse linear functions.

It is obvious that the graphs are symmetrical with respect to a straight line (bisectors of the first and third quarters). This is one of the properties of mutually inverse functions, which will be discussed below.

Find the inverse function.

This function is square; the graph is a parabola with its vertex at a point.

The function increases at and decreases at. This means that you can search for the inverse function for a given one on one of two intervals.

Let, then, and, swapping x and y, we obtain the inverse function on a given interval: .

Find the inverse function.

This function is cubic; the graph is a cubic parabola with its vertex at a point.

The function increases at. This means that one can search for an inverse function for a given one over the entire domain of definition.

, and by swapping x and y, we get the inverse function.

Let's illustrate this on a graph.

Let's list properties of mutually inverse functions And.

And.

From the first property it is clear that the domain of definition of a function coincides with the domain of values of the function and vice versa.

Graphs of mutually inverse functions are symmetrical with respect to a straight line.

If it increases, then it increases; if it decreases, then it decreases.

For a given function, find the inverse function:

For a given function, find the inverse and plot graphs of the given and inverse function:

Find out if there is an inverse function for a given function. If yes, then define the inverse function analytically, plot a graph of the given and inverse function:

Find the domain and range of a function that is the inverse of a function if:

Find the range of values of each of the mutually inverse functions and, if their domains of definition are indicated:

Are functions mutually inverse if:

Find the inverse function of the given one. Construct graphs of these mutually inverse functions on one coordinate system:
Is this function the inverse of itself: Specify the inverse function of this one and plot its graph: