Inverse function presentation for a lesson in algebra (Grade 10) on the topic. Reciprocal functions Explanation of new material
Lesson notes on the topic "Inverse functions"
Lesson 1 "Reverse Function"
Target: Form a theoretical apparatus on the topic. Enter
The concept of an invertible function;
The concept of an inverse function;
Formulate and prove a sufficient condition for reversibility
functions;
Basic properties of mutually inverse functions.
Lecture lesson plan
Organizing time.
Actualization of students' knowledge, necessary for the perception of a new topic.
Presentation of new material.
Summing up the lesson.
The course of the lesson-lecture
1. Organizing time.
2. Knowledge update. ( Frontal survey on the topic of the previous lesson.)
For students, a graph of the function is shown on the interactive board (Fig. 1). 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.
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Function properties:
3. Goal setting for students.
At the end of the study, the teacher reports that today at the lesson they will get acquainted with one more property of the 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. Each student has questions in the form of a handout (distributed before the lesson).
Questions:
1. What function is called reversible?
2. What function is called inverse?
3. How are the domains of definition and the sets of values of direct and inverse functions related?
4. Formulate a sufficient condition for the function to be invertible.
5. Is the inverse of an increasing function decreasing or increasing?
6. Is the inverse odd function even or odd?
7. How are the graphs of mutually inverse functions arranged?
4. Presentation of new material.
1) The concept of an invertible function. A sufficient condition for reversibility.
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 (Fig. 2). Thus, a function has a property that is not characteristic of a function: no matter what number from the set of function valuesf ( x ) take it, it is the value of the function at only one point, thus the teacher brings students to the concept of an invertible function.
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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 is calledreversible , if it takes any of its values only at one point of the setX .
Theorem. If the function is monotonic on the setX , then it is reversible.
Proof:
Let the function y=f(x) increases on the setX let it go X 1 ≠x 2 - two points of the setX .
For definiteness, letX 1 < X 2 . Then from whatX 1 < X 2 as the function increases, it follows thatf(x 1 ) < f(x 2 ) .
Thus, different values of the argument correspond to different values of the function, i.e. the function is reversible.
The theorem is proved similarly in the case of a decreasing function.
(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 board shows graphs of functions (Fig. 3, 4) and several analytically specified functions are recorded:
a ) b )
Rice. 3 Fig. four
in ) y=2x+5; G ) y = - + 7.
Comment. The monotonicity of a function, issufficient condition for the existence of an inverse function. But itis not necessary condition.
The teacher gives examples of different situations when the function is not monotonic, but reversible, when the function is not monotonic and not reversible, when it is monotonic and reversible.
2) The concept of an inverse function. Algorithm for compiling an inverse function.
Definition 2. Let the reversible functiony=f(x) defined on the setX and its rangeE(f)=Y . Let's match eachy from Y then the only meaningX, at which f(x)=y. Then we get a function that is defined onY, a X – range of function values. This function is denotedx=f -1 (y), and call reverse with respect to functiony=f(x), .
Then the teacher introduces students to the method of finding the inverse function given analytically.
Algorithm for compiling an inverse function for a function y = f ( x ), .
Make sure the functiony=f(x) reversible on the intervalX .
Express variableX through at from the equation y=f(x), taking into account that.
In the resulting equality, swapX and at. Instead of x=f -1 (y) write y=f -1 (x).
With specific examples, the teacher shows how to use this algorithm.
Example 1 Show what's for a functiony=2x-5
Solution . Linear functiony=2x-5 determined on R, increases by R and its range isR. So the inverse function exists onR . To find its analytical expression, we solve the equationy=2x-5 relatively X ; get. Rename the variables, we get the desired inverse function. It is defined and increases by R.
Example 2 Show what's for a functiony=x 2 , x ≤ 0 exists an inverse function, and find its analytic expression.
Solution . 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, the corresponding conclusion is made about the analytical expression for the inverse function, which has the form.
3) Properties of mutually inverse functions.
Property 1. If a g is the function inverse to f , then and f is the function inverse to g (the functions are mutually inverse), whileD ( g )= E ( f ), E ( g )= D ( f ) .
Property 2. If a function is increasing (decreasing) on the set X, and Y is the range of the function, then the inverse function is increasing (decreasing) on Y.
Property 3. To get a graph of a function that is inverse to a function, it is necessary to transform the graph of the function symmetrically with respect to the straight liney=x .
Property 4. If an odd function is invertible, then its inverse is also odd.
Property 5. If functions f ( x ) and mutually inverse, then it is true for any, and true for any.
Example 3 Plot the inverse function if possible.
Solution. This function has no inverse over its entire domain of definition, since it is not monotonic. Therefore, consider the interval on which the function is monotonic: , hence, there is an inverse. Let's findher . For this, we expressx throughy : . Rename - inverse function. Let's build graphs of functions (Fig. 5) and make sure that they are symmetrical with respect to a straight liney = x .
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Example 4 Find the set of values of each of the mutually inverse functions, if you know that.
Solution. According to property 1 of mutually inverse functions, we have.
5 . Summarizing
Carrying out diagnostic work. The purpose of this work is to determine the level of assimilation of the educational material discussed in the lecture. Students are invited to answer the questions formulated at the beginning of the lecture.
6 . Setting homework.
1. Understand the lecture material, learn the basic definitions and formulations of theorems.
2. Prove the properties of mutually inverse functions.
Lesson 2 A sufficient condition for the invertibility of a function"
Target: to form the ability to apply theoretical knowledge on the topic when solving problems, to consider the main types of problems for studying a function for reversibility, for building an inverse function.
Workshop Lesson Plan:
1. Organizational moment.
2. Actualization of knowledge (frontal work of students).
3. Consolidation of the studied material (problem solving).
4. Summing up the lesson.
5. Statement of homework.
During the classes.
1. Organizing time.
Greeting the teacher, checking the readiness of students for the lesson.
2. Knowledge update. ( front work of students).
Students are asked to complete the following tasks orally:
1. Formulate a sufficient condition for the function to be invertible.
2. Among the functions whose graphs are shown in the figure, indicate those that are reversible.
3. Formulate an algorithm for compiling a function inverse to a given one.
4. Are there functions inverse to data? If yes, find them:
a) ; b ) ; c ) .
5. Are the functions whose graphs are shown in the figure mutually inverse (Fig. 6)? Justify the answer.
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3. Consolidation of the studied material (problem solving).
Consolidation of the studied material consists of two stages:
Individual independent work of students;
Summing up the results of individual work.
At the first stage, students are given cards with tasks that they perform on their own.
Exercise 1.
Is the function reversible over the entire domain of definition? If yes, then find the reverse to it.
a) ; b) ; c).
Task 2.
Are the functions mutually inverse:
a) ;
b ) .
Task 3.
Consider the function on each of the specified intervals, if the function is invertible on this interval, then set its inverse analytically, indicate the domain of definition and range of values:
a ) R ; b ) ; d ) [-2;0].
Task 4.
Prove that the function is irreversible. Find the function inverse to it on the interval and plot its graph.
Task 5.
Plot the function and determine if there is an inverse function for it. If yes, then plot the inverse function on the same drawing and set it analytically:
a ) ; b ) .
At the stage of summing up the results of the individual work of students, the tasks are checked only with fixing the intermediate results. The problems that caused the most difficulties are considered on the board either with the disclosure of the search for solutions, or with a record of the entire solution.
4. Summing up the lesson (reflection).
Students are offered a mini-questionnaire:
What did I like about the lesson?______________________________
What did I not like about the lesson? _____________________________
_________________________________________________________________
Choose one statement that best suits you:
1) I can independently investigate the function for reversibility, build the inverse and I am sure that the result is correct.
2) I can examine the function for reversibility, build the inverse, but I'm not always sure of the correctness of the result, I need the help of my comrades.
3) I practically cannot investigate the function for reversibility, build the inverse, I need additional advice from the teacher.
Where can I apply the acquired knowledge?____________________ _________________________________________________________________
5. Setting homework.
№ 10.3, 10.6(c, d), 10.7(c, d), 10.9(c, d), 10.13(c, d), 10.18.(Mordkovich, A.G. Algebra and the beginning of mathematical analysis. Grade 10. At 2 p.m. Part 2. Task book for students of educational institutions (profile level) / A.G. Mordkovich, P.V. Semenov. - M.: Mnemosyne, 2014. - 384 p.)
Made by Morenshildt I.K. group 1.45.36 Frunzensky district School No. 314 Teacher Koroleva O.P. St. Petersburg 2006 * St. Petersburg CENTER FOR INFORMATION TECHNOLOGIES AND TELECOMMUNICATIONS MUTUAL INVERSE FUNCTIONS
Exponential and logarithmic function Trigonometric functions
Basic Definitions Example Equations Graphs of Inverse Functions Exponential and Logarithmic Functions Sine and Arcsine Functions Cosine and Arccosine Functions Tangent and Arctangent Functions Cotangent and Arccotangent Functions Credits Sources Contents Finish
Reversible function If the function y=f (x) takes each of its values only for one value of x, then this function is called reversible. For such a function, it is possible to express the inverse relationship between the values of the argument and the values of the function.
An example of constructing a function inverse to a given one Special case Given a function y=3x+5 Equation for x Replace x by y Functions (1) and (2) are mutually inverse General case y=f (x) is an invertible function Defined function x= g (y ) Replace x with y y= g(x) The functions y=f(x) and y=g(x) are mutually inverse
Graphs of inverse functions
Exponential and logarithmic functions y=log a x y=a x y=x a>1
Functions sin x and arcsin x Consider the function y=sin x on the segment The function is monotonically increasing. FZF [-1;1]. The function y= arcsin x is the inverse of the function y=sinx . [ - ; ] 2 2
Functions cos x and arccos x Consider the function y=co s x on the segment The function is monotonically decreasing. FZF [-1;1]. The function y=arccos x is the inverse of the function y=co sx .
Functions tg x and arctg x Consider the function y= tg x on the interval The function is monotonically increasing. ORF is the set R . The function y= arctg x is the inverse of the function y= tg x . (- ; ) 2 2
Functions ctg x and arcctg x Consider the function y= ctg x on the interval (0; ). The function is monotonically decreasing. GFZ set R . The inverse is the function y \u003d arcctg x.
Test on the topic "Mutually inverse functions" Question No. 1 Question No. 2 Question No. 3 Question No. 4 Question No. 5 Finish Finish
Question No. 1 Graphs of mutually inverse functions are located in the coordinate system symmetrically with respect to: The origin of coordinates Direct y \u003d x Axes OY Axes OX
Question No. 2 How are the domain of definition of the original and the domain of the inverse function related? Match Independent
Question #3 What is the inverse of a logarithmic function? Power Linear Quadratic Exponential
Question #4 The function y=arcctg x is the inverse of the function y=sin x y= tg x y= ctg x y= cos x
Question #5 The topic “Reciprocal Functions” is Elementary My Favorite Easy Understandable
Hooray! Hooray! Hooray! Well done scientist!
Wrong answer Repeat from the beginning!
Wrong! I'm outraged by your answer!
Sources of Algebra and the Beginnings of Analysis: Proc. for 10-11 cells. general education institutions / Sh.A. Alimov, Yu.M. Kolyagin, Yu.V. Sidorov and others - 12th ed. - M.: Enlightenment, 2004. - 384 p. The study of algebra and the beginning of analysis in grades 10-11: Book. for the teacher / N.E. Fedorova, M.V. Tkachev. - 2nd ed. - M .: Education, 2004. - 205 p. Didactic materials on algebra and the beginnings of analysis for grade 10: A guide for the teacher / B.M. Ivlev, S.M. Sahakyan, S.I. Schwarzburd. - 2nd ed., revised. - M.: Enlightenment, 1998. -143 p. Graphs of inverse trigonometric functions http://chernovskoe.narod.ru/tema13.htm
Topic: "Mutually inverse functions".
Lesson Objectives:
Educational:
Repeat and summarize the knowledge of students on the topic "Function", studied in grade 9. To get acquainted with mutually inverse functions, to study the conditions for the existence of an inverse function and its properties, to learn how to build graphs of inverse functions.
Developing:
To develop the creative and mental activity of students, their intellectual qualities: the ability to "see" the problem.
To form the ability to clearly and clearly express their thoughts, explore, analyze, compare, draw conclusions.
To develop students' interest in independent creativity.
Develop students' spatial imagination.
Educational:
To develop the ability to work with available information in an unusual situation.
Cultivate accuracy and conscientiousness.
Carry out aesthetic education.
Lesson type: combined.
Equipment:
multimedia projector;
application to the lesson: (Presentation.) - on electronic media;
Means of education: computers, softwareexcel, media projector, slide presentation.
Demos: graphs of functions built in one coordinate system.
Forms of organization of educational activities: individual, dialogue, work with the text of the slide, research work in a notebook.
Methods: visual, verbal graphic, research.
During the classes.
1. Introductory speech of the teacher. Installation conversation. Psychological mood of students.
In the lesson, we must repeat and summarize the knowledge on the topic "Function" studied in grade 9, get acquainted with mutually inverse functions, study the conditions for the existence of an inverse function and its properties, learn how to build graphs of inverse functions. We wish each other success and fruitful work.
2. Repetition of the material covered on the topic "Functions and their graphs." Presentation.
Slides 2-10. Frontal work with the class.
3. Learning new material. Educational conversation with elements of research and demonstration (slides 11-24)
Dependency example. Each function value corresponds to one argument value.
For such functions, it is possible to express the inverse relationship between the values of the argument and the values of the function.
Exercise.
Find the domain and range of mutually inverse functions.
4. Consolidation of knowledge.
I. Communication of the topic and purpose of the lesson
II. Repetition and consolidation of the material covered
1. Answers to questions on homework (analysis of unsolved problems).
2. Monitoring the assimilation of the material (independent work).
Option 1
Option 2
Conduct a study of the function and plot its graph:
III. Learning new material
According to the analytical form of the function, for any value of the argument, it is easy to find the corresponding value of the function y. The inverse problem often arises: the value of y is known and it is necessary to find the value of the argument x at which it is achieved.
Example 1
Find the value of the argument x if the value of the functionis equal to: a) 2; b) 7/6; in 1.
From the analytical form of the functionexpress the variable x and get: 4 xy - 2y = 3x + 1 or x(4y - 3) = 2y + 1, whence. Now it's easy to solve the problem:
Function is called the inverse of the function. Since it is customary to denote the function argument with the letter x, and the value of the function with the letter y, the inverse function is written as
Let us give the concepts necessary for studying the topic.
Definition 1. The function y = f(x), x ∈ X is called reversible if it takes any of its values only at one point x of the set X (in other words, if different values of the function correspond to different values of the argument). Otherwise, the function is called irreversible.
Example 2
Function each of its values takes only one point x and is reversible (graph a). Functionhas such values of y (for example, y = 2) that are achieved at two different points x , and is irreversible (plot b).
When considering the topic, the following theorem is useful.
Theorem 1. If the function y = f(x), x ∈ X is monotone on the set X, then it is invertible.
Example 3
Let's go back to the previous example. Functiondecreases (monotone) and is invertible over the entire domain of definition. Functionnonmonotonic and irreversible. However, this function increases on intervals (-∞; -1] and . Therefore, on such intervals, the function is invertible. For example, the function is invertible on the segment x ∈ [-1; one].
Definition 2. Let y = f(x), x ∈ X is an invertible function and E(f) = Y . Let's match each Y the only value of x for which f(x ) = y (i.e., the only root of the equation f(x ) = y with respect to the variable x). Then we get a function that is defined on the set Y (the set X is its range). This function is denoted x - f -1 (y ), y ∈ Y and is called inverse with respect to the function y = f(x), x ∈ X. The figure shows the function y = f (x) and inverse function x \u003d f -1 (y).
Direct and inverse functions have the same monotonicity.
Theorem 2. If the function y = f (x) increases (decreases) on the set X, and Y is its range of values, then the inverse function x = f -1 (y ) increases (decreases) on the set Y .
Example 4
Function decreases on the setand has many meaningsInverse functionalso decreases on the setand has many meaningsObviously, the graphs of the functions and coincide, since these functions lead to the same relationship between the variables x and y: 4xy - 3x - 2y - 1 = 0.
It is customary for us that the argument of the function is denoted by the letter x, the value of the function - by the letter y. Therefore, we will write the inverse function in the form y = f -1 (x) (see example 1).
Theorem 3. Graphs of the function y = f (x) and inverse function y = f-1 are symmetrical to the relative line y = x.
Example 5
For the function y \u003d 2x - 4, we find the inverse function: y + 4 \u003d 2x, whence x \u003d 1/2y + 2. Let us rename x↔ y and write the inverse function as y = 1/2x + 2. Thus, for the function f (x) \u003d 2x - 4 inverse function f -1 (x ) = 1/2x + 2. Let's plot graphs of these functions. It can be seen that the graphs are symmetrical to the relative straight line y \u003d x.
Function f-1(x ) \u003d 1/2x + 2 inverse with respect to the function f (x) \u003d 2x - 4. But the function f (x) = 2x - 4 is the inverse of the function f -1 (x ) = 1/2x + 2. Therefore, the functions f(x) and f-1 (x) is more correct to call reciprocal. In this case, the following equalities are fulfilled: f -1 (f (x)) \u003d x and f (f -1 (x) \u003d x.
IV. test questions
1. Reversible and irreversible functions.
2. Invertibility of a monotone function.
3. Definition of the inverse function.
4. Monotonicity of direct and inverse functions.
5. Graphs of direct and inverse functions.
V. Task in the lesson
§ 3, no. 1 (a, b); 2 (c, d); 3 (a, d); 4 (c, d); 5 (a, c).
VI. Homework
§ 3, no. 1 (c, d); 2 (a, b); 3 (b, c); 4 (a, b); 5 (b, d).
VII. Summing up the lesson
Lesson Objectives:
Educational:
Developing:
Educational:
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"Methodological development of the lesson "Reciprocal functions""
Lesson in grade 10 on the topic "Reciprocal functions"
(according to the program of Alimova Sh.A.)
Lesson type: combined.
Lesson Objectives:
Educational:
Repeat and summarize the knowledge of students on the topic "Function", studied in grade 9.
To get acquainted with mutually inverse functions, to study the conditions for the existence of an inverse function and its properties, to learn how to build graphs of inverse functions.
Developing:
To develop the creative and mental activity of students, their intellectual qualities: the ability to "see" the problem.
To form the ability to clearly and clearly express their thoughts, explore, analyze, compare, draw conclusions.
To develop students' interest in independent creativity.
Develop students' spatial imagination.
Educational:
To develop the ability to work with available information in an unusual situation.
Cultivate accuracy and conscientiousness.
Carry out aesthetic education.
Equipment:
multimedia projector;
application to the lesson: (Presentation.) - on electronic media;
Means of education: computers, Excel program, media projector, slide presentation.
Demos: graphs of functions built in one coordinate system.
Forms of organization of educational activities: individual, dialogue, work with slide text, research work in a notebook.
Methods: visual, verbal, graphic, research.
Lesson steps:
Setting the goal of the lesson and motivation for learning activities. 2 minutes
Repetition of the material covered on the topic "Functions and their graphs". 10 min
The stage of explaining new material.10 min
Operational and executive part. Consolidation stage.10 min
Knowledge control (worksheet with a test on paper)5 minutes
Homework assignment. 1 minute
Reflective-evaluative stage. 2 minutes
During the classes.
1. Introductory speech of the teacher. Installation conversation. Psychological mood of students.
Today's lesson is not quite usual for you: mathematics teacher Elena Semyonovna from Platoshinskaya secondary school, guests are mathematics teachers and methodologists of your school and the education department of the Perm region.
In the lesson, we should repeat and generalize the knowledge of students on the topic “Function”, studied in grade 9, get acquainted with mutually inverse functions, study the conditions for the existence of an inverse function and its properties, learn how to build graphs of inverse functions. We wish each other success and fruitful work.
2. Repetition of the material covered on the topic "Functions and their graphs." Presentation.
Slides 2-10. Frontal work with the class.
3. Learning new material. Educational conversation with elements of research and demonstration (slides 11-24)
4.
Dependency example. Each function value corresponds to one argument value.
For such functions, it is possible to express the inverse relationship between the values of the argument and the values of the function.
Exercise.
Find the domain and range of reciprocal functions.
4. Consolidation of knowledge.
5. Knowledge control.
6. Homework: study pages 46-50, solve No. 132, No. 133, No. 134
7. Reflective-evaluative stage.
During the lesson I learned………………………….
At the lesson I was interested in …………………....
It was difficult ………………………………………….
The knowledge gained in the lesson, I can use …………………………………………