Determination of the inverse function of its properties and graph. Mutually inverse functions

Let the sets $X$ and $Y$ be included in the set of real numbers. Let us introduce the concept of an invertible function .

Definition 1

A function $f:X\to Y$ mapping a set $X$ into a set $Y$ is called invertible if for any elements $x_1,x_2\in X$ it follows from the fact that $x_1\ne x_2$ that $f(x_1 )\ne f(x_2)$.

Now we can introduce the notion of an inverse function.

Definition 2

Let the function $f:X\to Y$ mapping the set $X$ into the set $Y$ be invertible. Then the function $f^(-1):Y\to X$ mapping the set $Y$ into the set $X$ and defined by the condition $f^(-1)\left(y\right)=x$ is called the inverse for $f( x)$.

Let's formulate the theorem:

Theorem 1

Let the function $y=f(x)$ be defined, monotonically increasing (decreasing) and continuous in some interval $X$. Then, in the corresponding interval $Y$ of values ​​of this function, it has an inverse function, which is also monotonically increasing (decreasing) and continuous on the interval $Y$.

Let us now introduce directly the concept of mutually inverse functions.

Definition 3

Within the framework of Definition 2, the functions $f(x)$ and $f^(-1)\left(y\right)$ are called mutually inverse functions.

Properties of mutually inverse functions

Let the functions $y=f(x)$ and $x=g(y)$ be mutually inverse, then

$y=f(g\left(y\right))$ and $x=g(f(x))$

The domain of the function $y=f(x)$ is equal to the domain of the value of the function $\ x=g(y)$. And the domain of the function $x=g(y)$ is equal to the domain of the value of the function $\ y=f(x)$.

The graphs of the functions $y=f(x)$ and $x=g(y)$ are symmetric with respect to the straight line $y=x$.

If one of the functions increases (decreases), then the other function also increases (decreases).

Finding the inverse function

The equation $y=f(x)$ with respect to the variable $x$ is solved.

From the obtained roots, those that belong to the interval $X$ are found.

The found $x$ are assigned to the number $y$.

Example 1

Find the inverse function, for the function $y=x^2$ on the interval $X=[-1,0]$

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).

Calculate $x$:

\ \

Choose the appropriate $x$:

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

Problems for finding inverse functions

In this part, we consider inverse functions for some elementary functions. The tasks will be solved according to the scheme given above.

Example 2

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

Find $x$ from the equation $y=x+4$:

Example 3

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

Decision.

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

Find $x$ from the equation $y=x^3$:

Finding suitable values ​​of $x$

The value in our case is suitable (since the scope is all numbers)

Redefining 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 $$

Decision.

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 increases in the set $Y=[-1,1]$ and maps the set $[-1,1]$ to the set $\left$.

Find $x$ from the equation $y=cosx$:

Finding suitable values ​​of $x$

Redefining 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 $\left(-\frac(\pi )(2),\frac(\pi )(2)\right)$.

Decision.

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 increases in the set $Y=R$ and maps the set $R$ onto the set $\left(- \frac(\pi )(2),\frac(\pi )(2)\right)$

Find $x$ from the equation $y=tgx$:

Finding suitable values ​​of $x$

Redefining the variables, we get that the inverse function has the form

What is an inverse function? How to find the function inverse of a given one?

Definition .

Let the function y=f(x) be defined on the set D and E be the set of its values. Inverse function with respect to function y=f(x) is a function x=g(y), which is defined on the set E and assigns to each y∈E a value x∈D such that f(x)=y.

Thus, the domain of the function y=f(x) is the domain of the inverse function, and the domain of y=f(x) is the domain of the inverse function.

To find the function inverse of the given function y=f(x), one must :

1) In the function formula, instead of y, substitute x, instead of x - y:

2) From the resulting equality, express y in terms of x:

Find the function inverse of the function y=2x-6.

The functions y=2x-6 and y=0.5x+3 are mutually inverse.

Graphs of direct and inverse functions are symmetrical with respect to the direct line y=x(bisectors of I and III coordinate quarters).

y=2x-6 and y=0.5x+3 - . The graph of a linear function is . To draw a straight line, we take two points.



It is possible to uniquely express y in terms of x when the equation x=f(y) has a unique solution. This can be done if the function y=f(x) takes each of its values ​​at a single point of its domain of definition (such a function is called reversible).

Theorem (necessary and sufficient condition for a function to be invertible)

If the function y=f(x) is defined and continuous on a numerical interval, then for the function to be invertible it is necessary and sufficient that f(x) be strictly monotonic.

Moreover, if y=f(x) increases on the interval, then the function inverse to it also increases on this interval; if y=f(x) is decreasing, then the inverse function is also decreasing.

If the reversibility condition is not satisfied over the entire domain of definition, one can single out an interval where the function only increases or only decreases, and on this interval find a function inverse to the given one.

The classic example is . In between

E (y) \u003d [-π / 2; π / 2]

y (-x) \u003d arcsin (-x) \u003d - arcsin x - odd function, the graph is symmetrical about the point O (0; 0).

arcsin x = 0 at x = 0.

arcsin x > 0 at x є (0; 1]

arcsin x< 0 при х є [-1;0)

y \u003d arcsin x increases for any x є [-1; 1]

1 ≤ x 1< х 2 ≤ 1 <=>arcsin x 1< arcsin х 2 – функция возрастающая.

Arc cosine

The cosine function decreases on the segment and takes on all values ​​from -1 to 1. Therefore, for any number a such that |a|1, there is a single root in the equation cosx=a on the segment. This number in is called the arccosine of the number a and is denoted arcos a.

Definition . The arc cosine of the number a, where -1 a 1, is a number from the segment whose cosine is equal to a.

Properties.

1. E(y) =

   y (-x) \u003d arccos (-x) \u003d π - arccos x - the function is neither even nor odd.

   arccos x = 0 at x = 1

   arccos x > 0 at x є [-1; 1)

arccos x< 0 – нет решений

y \u003d arccos x decreases for any x є [-1; 1]

1 ≤ x 1< х 2 ≤ 1 <=>arcsin x 1 ≥ arcsin x 2 - decreasing.

Arctangent

The tangent function increases on the segment -
, therefore, according to the root theorem, the equation tgx \u003d a, where a is any real number, has a unique root x on the interval -. This root is called the arc tangent of the number a and is denoted by arctga.

Definition. Arc tangent of a number aR this number is called x , whose tangent is a.

Properties.

E (y) \u003d (-π / 2; π / 2)

y(-x) \u003d y \u003d arctg (-x) \u003d - arctg x - the function is odd, the graph is symmetrical about the point O (0; 0).

arctg x = 0 at x = 0

The function increases for any x є R

-∞ < х 1 < х 2 < +∞ <=>arctg x 1< arctg х 2

Arc tangent

The cotangent function on the interval (0;) decreases and takes all values ​​from R. Therefore, for any number a in the interval (0;) there is a single root of the equation ctg x = a. This number a is called the arc tangent of the number a and is denoted by arcctg a.

Definition. The arc tangent of a number a, where a R, is such a number from the interval (0;) , whose cotangent is a.

Properties.

E(y) = (0; π)

y (-x) \u003d arcctg (-x) \u003d π - arcctg x - the function is neither even nor odd.

arcctg x = 0- does not exist.

Function y = arcctg x decreases for any х є R

-∞ < х 1 < х 2 < + ∞ <=>arcctg x 1 > arcctg x 2

The function is continuous for any x є R.

2.3 Identity transformations of expressions containing inverse trigonometric functions

Example 1 . Simplify the expression:

a)
where

Decision. Let's put
. Then
and
To find
, we use the relation
We get
But . On this segment, the cosine takes only positive values. Thus,
, i.e
where
.

b)

Decision.

in)

Decision. Let's put
. Then
and
Let us first find, for which we use the formula
, where
Since the cosine takes only positive values ​​on this interval, then
.

Lesson Objectives:

Educational:

• to form knowledge on a new topic in accordance with the program material;
• to study the property of the invertibility of a function and to teach how to find a function inverse to a given one;

Developing:

• develop self-control skills, subject speech;
• master the concept of an inverse function and learn the methods of finding an inverse function;

Educational: to form communicative competence.

Equipment: computer, projector, screen, SMART Board interactive whiteboard, handout (independent work) for group work.

During the classes.

1. Organizational moment.

Target – preparing students for work in the classroom:

Definition of absent,

Attitude of students to work, organization of attention;

Message about the topic and purpose of the lesson.

2. Updating the basic knowledge of students. front poll.

Target - to establish the correctness and awareness of the studied theoretical material, the repetition of the material covered.<Приложение 1 >

A graph of the function is shown on the interactive whiteboard for students. The teacher formulates the task - to consider the graph of the function and list the studied properties of the function. Students list the properties of a function according to the research design. The teacher, to the right of the graph of the function, writes down the named properties with a marker on the interactive whiteboard.

Function properties:

At the end of the study, the teacher reports that today at the lesson they will get acquainted with one more property of a function - reversibility. For a meaningful study of new material, the teacher invites the children to get acquainted with the main questions that students must answer at the end of the lesson. Questions are written on an ordinary board and each student has a handout (distributed before the lesson)

1. What is a reversible function?
2. Is every function reversible?
3. What is the inverse given function?
4. How are the domain of definition and the set of values ​​of a function and its inverse function related?
5. If the function is given analytically, how do you define the inverse function with a formula?
6. If a function is given graphically, how to plot its inverse function?

3. Explanation of new material.

Target - to form knowledge on a new topic in accordance with the program material; to study the property of the invertibility of a function and to teach how to find a function inverse to a given one; develop subject matter.

The teacher conducts a presentation of the material in accordance with the material of the paragraph. On the interactive board, the teacher compares the graphs of two functions whose domains of definition and sets of values ​​are the same, but one of the functions is monotonic and the other is not, thereby bringing students under the concept of an invertible function.

The teacher then formulates the definition of an invertible function and conducts a proof of the invertible function theorem using the graph of the monotonic function on the interactive whiteboard.

Definition 1: The function y=f(x), x X is called reversible, if it takes any of its values ​​only at one point of the set X.

Theorem: If the function y=f(x) is monotone on the set X , then it is invertible.

Proof:

1. Let the function y=f(x) increases by X let it go x 1 ≠ x 2- two points of the set X.
2. For definiteness, let x 1< x 2.
   Then from what x 1< x 2 follows that f(x 1) < f(x 2).
3. Thus, different values ​​of the argument correspond to different values ​​of the function, i.e. the function is reversible.

(During the proof of the theorem, the teacher makes all the necessary explanations on the drawing with a marker)

Before formulating the definition of an inverse function, the teacher asks students to determine which of the proposed functions is reversible? The interactive whiteboard shows graphs of functions and several analytically defined functions are written:

B)

G) y = 2x + 5

D) y = -x 2 + 7

The teacher introduces the definition of an inverse function.

Definition 2: Let an invertible function y=f(x) defined on the set X and E(f)=Y. Let's match each y from Y then the only meaning X, at which f(x)=y. Then we get a function that is defined on Y, a X is the range of the function

This function is denoted x=f -1 (y) and is called the inverse of the function y=f(x).

Students are invited to draw a conclusion about the relationship between the domain of definition and the set of values ​​of inverse functions.

To consider the question of how to find the inverse function of a given, the teacher involved two students. The day before, the children received a task from the teacher to independently analyze the analytical and graphical methods for finding the inverse given function. The teacher acted as a consultant in preparing students for the lesson.

Message from the first student.

Note: the monotonicity of a function is sufficient condition for the existence of an inverse function. But it is not necessary condition.

The student gave examples of various situations when the function is not monotonic, but reversible, when the function is not monotonic and not reversible, when it is monotonic and reversible

Then the student introduces students to the method of finding the inverse function given analytically.

Finding algorithm

1. Make sure the function is monotonic.
2. Express x in terms of y.
3. Rename variables. Instead of x \u003d f -1 (y) they write y \u003d f -1 (x)

Then solves two examples to find the function of the inverse of the given.

Example 1: Show that there is an inverse function for the function y=5x-3 and find its analytical expression.

Decision. The linear function y=5x-3 is defined on R, increases on R, and its range is R. Hence, the inverse function exists on R. To find its analytical expression, we solve the equation y=5x-3 with respect to x; we get This is the desired inverse function. It is defined and increases by R.

Example 2: Show that there is an inverse function for the function y=x 2 , x≤0, and find its analytical expression.

The function is continuous, monotone in its domain of definition, therefore, it is invertible. Having analyzed the domains of definition and the set of values ​​of the function, a corresponding conclusion is made about the analytical expression for the inverse function.

The second student makes a presentation about graphic how to find the inverse function. In the course of his explanation, the student uses the capabilities of the interactive whiteboard.

To get the graph of the function y=f -1 (x), inverse to the function y=f(x), it is necessary to transform the graph of the function y=f(x) symmetrically with respect to the straight line y=x.

During the explanation on the interactive whiteboard, the following task is performed:

Construct a graph of a function and a graph of its inverse function in the same coordinate system. Write down an analytical expression for the inverse function.

4. Primary fixation of the new material.

Target - to establish the correctness and awareness of the understanding of the studied material, to identify gaps in the primary understanding of the material, to correct them.

Students are divided into pairs. They are given sheets with tasks in which they work in pairs. Time to complete the work is limited (5-7 minutes). One pair of students works on the computer, the projector is turned off for this time and the rest of the children cannot see how the students work on the computer.

At the end of the time (it is assumed that the majority of students completed the work), the interactive whiteboard (the projector turns on again) shows the work of the students, where it is clarified during the test that the task was completed in pairs. If necessary, the teacher conducts corrective, explanatory work.

Independent work in pairs<Annex 2 >

5. The result of the lesson. On the questions that were asked before the lecture. Announcement of grades for the lesson.

Homework §10. №№ 10.6(а,c) 10.8-10.9(b) 10.12(b)

Algebra and the beginnings of analysis. Grade 10 In 2 parts for educational institutions (profile level) / A.G. Mordkovich, L.O. Denishcheva, T.A. Koreshkova and others; ed. A.G. Mordkovich, M: Mnemosyne, 2007

Mutually inverse functions.

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

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

Functions f and g are called reciprocal.

Why consider the concept of inverse functions at all?

This is caused by the problem of solving equations. Solutions are just written in terms of inverse functions.

Consider some examples of finding inverse functions .

Let's start with linear mutually inverse functions.

Find the function inverse for.

This function is linear, its graph is a straight line. Hence, the function is monotone on the entire domain of definition. Therefore, we will look for the function inverse to it on the entire domain of definition.

.

Express x through y (in other words, solve the equation for x ).

- this is the inverse function, the truth is here y is an 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.

Let's give a graphical illustration of mutually inverse linear functions.

Obviously, the graphs are symmetrical with respect to the 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 apex at a point.

.

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

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

Find the inverse function.

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

.

The function increases at. This means that it is possible to search for an inverse function for a given one on the entire domain of definition.

, and by interchanging 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.

It can be seen from the first property that the scope of a function coincides with the scope 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 the given and inverse functions: Find out if there is an inverse function for the given function. If yes, then define the inverse function analytically, plot the given and inverse function: Find the domain and range of the function inverse to the function if:

1. Find the range of each of the mutually inverse functions and, if their ranges are given:

Are functions mutually inverse if:

1. Find the function inverse of the given one. Plot on the same coordinate system the graphs of these mutually inverse functions:

   Is this function inverse to itself: Define a function inverse to the given one and plot its graph: