Inverse function. The concept of an inverse function How to construct an inverse function of a given example

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) will either decrease or increase over a given interval.

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 mutually 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).

Lesson objectives:

Educational:

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

Developmental:

develop self-control skills, substantive speech;
master the concept of inverse function and learn methods for finding the inverse function;

Educational: to develop communicative competence.

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

During the classes.

1. Organizational moment.

Target – preparing students for work in class:

Definition of absentees,

Getting students in the mood for work, organizing attention;

State the topic and purpose of the lesson.

2. Updating students’ basic knowledge. Frontal survey.

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

A graph of a function is shown on the interactive whiteboard for students. The teacher formulates a task - consider the graph of a function and list the studied properties of the function. Students list the properties of a function in accordance with 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 board.

Function properties:

At the end of the study, the teacher reports that today in the lesson they will become acquainted with another property of a function - reversibility. To meaningfully study new material, the teacher invites the children to get acquainted with the main questions that students must answer at the end of the lesson. The questions are written on a regular board and each student has them as handouts (distributed before the lesson)

Which function is called invertible?
Is any function invertible?
What function is called the inverse of a datum?
How are the domain of definition and the set of values of a function and its inverse related?
If a function is given analytically, how can one define the inverse function by a formula?
If a function is given graphically, how to graph its inverse function?

3. Explanation of new material.

Target - generate knowledge on a new topic in accordance with the program material; study the property of reversibility of a function and teach how to find the inverse function of a given one; develop substantive speech.

The teacher presents the material in accordance with the material in the paragraph. On the interactive whiteboard, 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 introducing students to 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 a 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 a function y=f(x) is monotonic on a set X, then it is invertible.

Proof:

Let the function y=f(x) increases by X let it go x 1 ≠x 2- two points of the set X.
To be specific, let x 1< x 2.
Then from the fact that x 1< x 2 follows that f(x 1) < f(x 2).
Thus, different values of the argument correspond to different values of the function, i.e. the function is invertible.

(As the proof of the theorem progresses, the teacher uses a marker to make all the necessary explanations on the drawing)

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

G) y = 2x + 5

D) y = -x 2 + 7

The teacher introduces the definition of an inverse function.

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

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

Students are asked to draw a conclusion about the connection between the domain of definition and the set of values of inverse functions.

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

Message from the first student.

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

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

The student then introduces students to a method for finding the inverse function given analytically.

Finding algorithm

Make sure the function is monotonic.
Express the variable x in terms of y.
Rename variables. Instead of x=f -1 (y) write y=f -1 (x)

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

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

Solution. The linear function y=5x-3 is defined on R, increases on R, and its range of values is R. This means that the inverse function exists on R. To find its analytical expression, solve the equation y=5x-3 for x; we get This is the required inverse function. It is defined and increasing on R.

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

The function is continuous, monotonic in its domain of definition, therefore, it is invertible. Having analyzed the domains of definition and sets 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 method of finding the inverse function. During his explanation, the student uses the capabilities of the interactive whiteboard.

To obtain a 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 consolidation of new material.

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

Students are divided into pairs. They are given sheets of tasks, in which they do the work in pairs. The time to complete the work is limited (5-7 minutes). One pair of students works on the computer, the projector turns off during 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 have completed the work), the students’ work is shown on the interactive board (the projector is turned on again), where it is determined during the check whether the task was completed correctly in pairs. If necessary, the teacher carries out corrective and explanatory work.

Independent work in pairs<Appendix 2 >

5. Lesson summary. Regarding the questions that were asked before the lecture. Announcement of grades for the lesson.

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

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

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

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 definition of the function y=f(x) is the domain of values of its inverse function, and the domain of values y=f(x) is the domain of definition of the inverse function.

To find the inverse function of a given function y=f(x), you need :

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

2) From the resulting equality, express y through x:

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

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

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

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

It is possible to express y unambiguously in terms of x in the case 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 in its domain of definition (such a function is called reversible).

Theorem (necessary and sufficient condition for the invertibility of a function)

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

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

A classic example is . In between)