Functionally graphical method for solving equations and inequalities. Topic: "Exponential function

Topic: "Exponential function. Functional-graphical methods for solving equations, inequalities, systems"

Target : consider the problems of ZNO using functional-graphic methods using the exponential function as an example y \u003d a x, a > 0, a1

Lesson objectives:

repeat the property of monotonicity and boundedness of the exponential function;

repeat the algorithm for plotting function graphs using transformations;

find set of values and set of function definitions according to the formula and using a chart;

solve exponential equations, inequalities and systems using graphs and function properties.

work with graphs of functions containing a module;

consider the graphs of a complex function and their range of values;

During the classes:

1. Introductory speech of the teacher. Motivation for studying this topic

slide 1 Exponential function. “Functional-graphical methods for solving equations and inequalities”

The functional-graphical method is based on the use of graphic illustrations, the application of function properties and allows solving many mathematical problems.

Slide 2-3 Goals and lesson tasks.

Today we will consider the problems of ZNO of different levels of complexity using functional-graphic methods using the example of the exponential function y = a x, a > o, a1. With the help of a graphic program, we will perform illustrations for the tasks.

slide 4 Why is it important to know the properties of an exponential function?

According to the law of the exponential function, all life on Earth would multiply if there were favorable conditions for this, i.e. there were no natural enemies and there was plenty of food. Proof of this is the spread of rabbits in Australia, which were not there before. It was enough to release a couple of individuals, as after a while their offspring became a national disaster.

In nature, technology and economics, there are numerous processes in the course of which the value of a quantity changes by the same number of times, i.e. according to the law of exponential function. These processes are called processes organic growth or organic decay.

For example, bacteria growth under ideal conditions corresponds to the process of organic growth; radioactive decay– the process of organic attenuation.

obeys the laws of organic growth contribution growth at the savings bank hemoglobin recovery in the blood of a donor or an injured person who has lost a lot of blood.

Give your examples

Application in real life (dose of medication).

Medication dose notification:

Everyone knows that the pills recommended by the doctor for treatment must be taken several times a day, otherwise they will be ineffective. The need for repeated administration of the drug to maintain a constant concentration in the blood is caused by the destruction of the drug in the body. The figure shows how, in most cases, the concentration of drugs in the blood of a person or animal changes after a single injection. Slide 5.

The decrease in drug concentration can be approximated by an exponent whose exponent contains time. Obviously, the rate of destruction of the drug in the body should be proportional to the intensity of metabolic processes.

One tragic case is known, which occurred due to ignorance of this dependence. From a scientific point of view, the drug LSD, which causes peculiar hallucinations in normal people, is very interesting for psychiatrists and neurophysiologists. Some researchers decided to study the reaction of the elephant to this drug. To do this, they took the amount of LSD that infuriates cats and multiplied it by the number of times the mass of an elephant is greater than the mass of a cat, believing that the dose of the drug administered should be directly proportional to the mass of the animal. The introduction of such a dose of LSD to an elephant led to his death in 5 minutes, from which the authors concluded that elephants have an increased sensitivity to this drug. A review of this work that appeared later in the press called it an "elephant-like mistake" by the authors of the experiment.

2. Actualization of students' knowledge.

What does it mean to learn a function? (formulate a definition, describe properties, build a graph)

What is the exponential function? Give an example.

What are the main properties of an exponential function?

Scope (Limitation)

domain

monotonicity (ascending-decreasing condition)

slide 6 . Specify the set of function values (according to the finished drawing)

Slide 7. Name the condition for the increase and decrease of the function and correlate the formula of the function with its graph

slide 8. According to the finished drawing, describe the algorithm for plotting function graphs

Slide a) y \u003d 3 x + 2

b) y \u003d 3 x-2 - 2

3.Diagnostic independent work (using a PC).

The class is divided into two groups. The main part of the class is doing test tasks. Strong students perform more difficult tasks.

Independent work in the programpower point(for the main part

Independent work (for the strong part of the class)

Slide 9. Write down the algorithm for plotting a graph of a function, name its domain of definition, range of value, intervals of increase, decrease.

slide 10. Match the formula of a function with its graph

Students check their answers without correcting mistakes, hand over independent work to the teacher

Slides 11-21. Checking the test for the main part

4. Learning a new topic. Application of the functional-graphical method for solving equations, inequalities, systems, determining the range of complex functions

Slides 22-23. Functionally graphical way to solve equations

To solve an equation of the form f (x) \u003d g (x) by the functional-graphic method, you need:

Construct graphs of functions y=f(x) and y=g(x) in one coordinate system.

Determine the coordinates of the intersection point of the graphs of these functions.

Write down the answer.

SOLUTION OF EQUATIONS

Slide 24-25.

Does the equation have a root, and if so, is it positive or negative?

SLIDE 26

5. Implementation of practical work.

SOLUTION OF EQUATIONS. SLIDES 27-30

This equation can be solved graphically. Students are invited to complete the task, and then answer the question: “Is it necessary to build graphs of functions to solve this equation?”. Answer: “The function is increasing on the entire domain of definition, and the function is decreasing. Therefore, the graphs of such functions have at most one intersection point, which means that the equation has at most one root. By selection, we find that .

Solve equation 3 x = (x-1) 2 + 3

Solution: we apply the functional method of solving equations:

because this system has a unique solution, then by selection we find x = 1

SOLUTION OF INEQUALITIES. Slides 31-33

G Graphical methods make it possible to solve inequalities containing different functions. To do this, after plotting the graphs of the functions on the left and right sides of the inequality and determining the abscissa of the intersection point of the graphs, it is necessary to determine the interval on which all points of one of the graphs lie above (below0 points of the second.

Solve the inequality:

a) cos x 1 + 3 x

Solution:

Answer: ( ; )

Solve graphically inequality.

(The graph of the exponential function lies above the function written on the right side of the equation).

Answer: x>2. O

.
Answer: x>0.

The exponential function contains the sign of the modulus in the exponent. slide 34-35

Let's repeat the definition of the module.

(writing on the board)

Make notes in your notebook:

1).

2).

A graphic illustration is presented on the slide. Explain how the graphs are built.

E(y)=(0;1]

To solve this equation, you need to remember the boundedness property of the exponential function. The function takes values > 1, a - 1 < > 1, so equality is possible only if both sides of the equation are simultaneously equal to 1. Hence, Solving this system, we find that X = 0.

.Finding the range of values of a complex function. Slides 36-37.

Using the ability to build a graph of a quadratic function, determine sequentially the coordinates of the top of the parabola, find the range.

, is the vertex of the parabola.

Question: determine the nature of the monotonicity of the function.

The exponential function y \u003d 16 t increases, since 16>1.

At the smallest value of the function index

The graph illustrates our conclusion.

The idea of a graphical method for solving an equation is simple. It is necessary to build graphs of the functions contained in both parts of the equation and find the abscissas of the intersection points. But plotting some functions is difficult. It is not always necessary to resort to the construction of graphs. Such equations can be solved by the root selection method, using the properties of monotonicity and boundedness of functions. This allows you to quickly solve the tasks offered when passing the exam.

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Municipal educational institution

"Gymnasium No. 24"

Functionally - graphical method

Solutions of equations.

Prepared by the teacher

Danilina Olga Sergeevna

Magadan 2007

« Functional - graphical method for solving equations "

The purpose of the lesson: to form the ability to solve equations of a certain type by a functional-graphical method, using the properties of limitedness and monotonicity of functions

Lesson structure:

Introductory speech of the teacher, introduction to the topic of the lesson, goal setting

Actualization of previously acquired knowledge necessary for mastering the topic of the lesson

Presentation by the moderators, which includes a presentation of new material with examples of solving various types of equations

Work in groups, with the aim of primary consolidation of the studied

Conducting a game on the model of the game: “What? Where? When?"

Summing up the lesson.

In the opening speech, the teacher shares his experience of getting to know the new method. speaks of the need for its development, its significance, the possibility of acquiring skills for a more rational solution of equations
Actualization of knowledge:: increase and decrease of functions, examples, properties of monotonicity and limitedness of functions.
Presentation of a new topic using slides presenting theoretical material with examples of solutions to equations. (See Appendix).
Work in groups: Each group is given cards with tasks, sample solutions and assignments. The students leading the lesson - consultants control the progress of the tasks, if necessary, come to the rescue. In their work, working in groups can use computers that are configured for a special program that allows you to build graphs of functions. Due to this, in difficult situations, a computer can be used as a tool for prompting or as an opportunity to demonstrate the correctness of the solution and the correctness of the chosen method.
Protection by a representative of a group of completed tasks, using a multimedia board, which demonstrates the solution of equations by a graphical method to confirm the correctness of the completed task. RA
Conducting the game. For each group, a question is heard from the monitor screen, pre-recorded by different teachers of the school, a minute is given for discussion, after which the guys must give their well-founded answer. After that, from the newly turned on screen, the teacher, who previously asked the question, presents a variant of his answer. Such repeated repetition of reasoning on the newly studied topic, especially pronounced correctly by different people, achieves the most favorable conditions for mastering the new topic. (See appendix.)
Summing up: Identification of the best “five of experts, the best player.

Questions to the class;

What did you learn in today's lesson

What equations can be solved by the selection method

What properties of functions are used in this case.

Questions to the participants of the game:

Dear experts, in one minute find the root of this equation and prove that it is the only one.

Answer: The sum of two increasing functions is an increasing function. y \u003d - monotonically increases, therefore the equation has one root, because the graph of this function intersects with the straight line y=3 once. For x=1, we get the correct equality. Answer: x=1

Dear experts, after one minute, name the functions that are contained in both parts of the inequality and find the root of this equation.

Answer: y \u003d is an exponential function that increases on the set of real numbers. y=6 - x is a linear function, it decreases monotonically on the set of real numbers. So the graphs of the functions intersect at one point, the equation has one root. For x=2, we get the correct equality. Answer: x=2

3. Dear experts, you already know that the equation has a single root x=3. After one minute, answer for what values of x, the inequality is true.

Answer: the inequality is satisfied for x Є, because on this interval, the graph of the function y \u003d is located below the graph of the function y \u003d

4. Dear experts, many find it difficult to solve the equation. In one minute, find the root of this equation and prove that it is unique.

Answer: the root of the equation x \u003d -3 is the only one, because the left side of the equation contains a decreasing function, and the right side is increasing, which means that the graphs of the functions intersect at one point and the equation has a single root.

5. Dear experts, I have a difficult question for you. You can easily find the root of the equation. Prove that he is the only one. Answer: x=1 is the only root.

Functionally - a graphical method for solving equations.

________________________________________________________________________

The purpose of the lesson: Learn to solve equations by the substitution method, using the properties of monotonicity and boundedness of functions.

_________________________________________________________________________

Reference material

A function is called increasing (decreasing) on a set X if on this set, as the argument increases (decreases), the value of the function increases (decreases).

Example 1:

are increasing functions

Example 2:

are decreasing functions

Reference material

2. The sum of two increasing functions is an increasing function.

Example:

3. The sum of two decreasing functions is a decreasing function.

Target: consider the problems of ZNO using functional-graphic methods using the exponential function as an example y \u003d a x, a > 0, a1

Lesson objectives:

repeat the property of monotonicity and boundedness of the exponential function;
repeat the algorithm for plotting function graphs using transformations;
find a set of values and a set of definitions of a function by the form of a formula and using a graph;
solve exponential equations, inequalities and systems using graphs and function properties.
work with graphs of functions containing a module;
consider the graphs of a complex function and their range of values;

During the classes:

1. Introductory speech of the teacher. Motivation for studying this topic

slide 1 Exponential function. “Functional-graphical methods for solving equations and inequalities”

The functional-graphical method is based on the use of graphic illustrations, the application of function properties and allows solving many mathematical problems.

slide 2 Tasks for the lesson

slide 3 Why is it important to know the properties of an exponential function?

According to the law of the exponential function, all life on Earth would multiply if there were favorable conditions for this, i.e. there were no natural enemies and there was plenty of food. Proof of this is the spread of rabbits in Australia, which were not there before. It was enough to release a couple of individuals, as after a while their offspring became a national disaster.
In nature, technology and economics, there are numerous processes in the course of which the value of a quantity changes by the same number of times, i.e. according to the law of exponential function. These processes are called processes organic growth or organic decay.
For example, bacteria growth under ideal conditions corresponds to the process of organic growth; radioactive decay– the process of organic attenuation.
obeys the laws of organic growth contribution growth at the savings bank hemoglobin recovery in the blood of a donor or an injured person who has lost a lot of blood.
Give your examples
Application in real life (dose of medication).

Medication dose notification:

2. Actualization of students' knowledge.

What does it mean to learn a function? (formulate a definition, describe properties, build a graph)
What is the exponential function? Give an example.
What are the main properties of an exponential function?
Scope (Limitation)
domain
monotonicity (ascending-decreasing condition)
slide 5 . Specify the set of function values (according to the finished drawing)

slide 6. Name the condition for the increase and decrease of the function and correlate the formula of the function with its graph

Slide 7. According to the finished drawing, describe the algorithm for plotting function graphs

Slide a) y \u003d 3 x + 2

b) y \u003d 3 x-2 - 2

3.Diagnostic independent work (using a PC).

The class is divided into two groups. The main part of the class is doing test tasks. Strong students perform more difficult tasks.

Independent work in the programpower point(for the main part of the class according to the type of test tasks from ZNO with a closed answer form)
1. Which exponential function is increasing?
2. Find the scope of the function.
3. Find the range of the function.
4. The graph of the function is obtained from the graph of the exponential function by parallel translation along the axis ... by .. units ...
5. According to the finished drawing, determine the scope and scope of the function
6. Determine the value of a for which the exponential function passes through the point.
7. Which figure shows the graph of an exponential function with a base greater than one.
8. Match the graph of the function with the formula.
9. The graphical solution of which inequality is shown in the figure.
10. solve the inequality graphically (according to the finished drawing)
Independent work (for the strong part of the class)

slide 8. Write down the algorithm for plotting a graph of a function, name its domain of definition, range of value, intervals of increase, decrease.

slide 9. Match the formula of a function with its graph

)

Students check their answers without correcting mistakes, hand over independent work to the teacher

Slide 10. Answers to test tasks

1) D 2) B 3) C 4) A

5) D 6) C 7) B 8) 1-D 2-A 3-C 4-B

9) A 10)(2;+ )

Slide 11 (check task 8)

The figure shows graphs of exponential functions. Match the graph of the function with the formula.

4. Learning a new topic. Application of the functional-graphical method for solving equations, inequalities, systems, determining the range of complex functions

Slide 12. Functionally graphical way to solve equations

To solve an equation of the form f (x) \u003d g (x) by the functional-graphic method, you need:

Construct graphs of functions y=f(x) and y=g(x) in one coordinate system.

Determine the coordinates of the intersection point of the graphs of these functions.

Write down the answer.

TASK №1 SOLUTION OF EQUATIONS

Slide 13.

Does the equation have a root, and if so, is it positive or negative?

6 x \u003d 1/6

(4/3) x = 4

SLIDE 14

5. Implementation of practical work.

slide 15.

Solve the equation:

3x = (x-1) 2 + 3

slide 16. .Solution: we apply the functional method of solving equations:

because this system has a unique solution, then by selection we find x = 1

TASK № 2 SOLUTION OF INEQUALITIES

Graphical methods make it possible to solve inequalities containing different functions. To do this, after plotting the graphs of the functions on the left and right sides of the inequality and determining the abscissa of the intersection point of the graphs, it is necessary to determine the interval on which all points of one of the graphs lie above (below0 points of the second.

Solve the inequality:

slide 17.

a) cos x 1 + 3 x

slide 1 8. Solution:

Answer: ( ; )

Solve graphically inequality.

Slide 19.

(The graph of the exponential function lies above the function written on the right side of the equation).

Answer: x>2. O

.
Answer: x>0.

TASK №3 The exponential function contains the sign of the modulus in the exponent.

Let's repeat the definition of the module.

(writing on the board)

slide 20.

Make notes in your notebook:

1).

2).

A graphic illustration is presented on the slide. Explain how the graphs are built.

Slide 21.

To solve this equation, you need to remember the boundedness property of the exponential function. The function takes values > 1, a - 1 > 1, so equality is possible only if both sides of the equation are simultaneously equal to 1. Hence, Solving this system, we find that X = 0.

TASK 4. Finding the range of complex functions.

slide 22.

Using the ability to build a graph of a quadratic function, determine sequentially the coordinates of the top of the parabola, find the range.

slide 23.

, is the vertex of the parabola.

Question: determine the nature of the monotonicity of the function.

The exponential function y \u003d 16 t increases, since 16>1.

Sections: Maths

Class: 11

Systematize, generalize, expand the knowledge, skills of students related to the use of functional-graphical method for solving equations
Development of skills for solving equations by the functional-graphic method.
Formation of logical thinking, the ability to think independently and outside the box.
Develop communication skills in group work.
Carry out productive interaction in the group to achieve the maximum overall result.
Practicing the ability to listen to a friend. Analyze his answer and ask questions.

To conduct this lesson, groups of children were organized in the class, who got to remember a certain method for solving equations, select 5-8 equations, solve them and prepare a presentation.

Equipment: Computer, projector. Presentation .

The presentations of the guys were inserted into the presentation of the teacher, but they have a different background.

During the classes

Today in the lesson we will remember the functional-graphical method for solving equations, consider when it is applied, what difficulties may arise in solving and we will choose methods for solving equations.

Recall the basic methods for solving equations.(slide number 2)

The first group analyzes the graphical method.

The second group talks about the majorant method.

The majorant method is a method for finding boundedness of a function.

Majorization - finding the points of restriction of the function. M - majorant.

If we have f(x) = g(x) and the ODZ is known, and if

.#1 Solve the equation:

x = 4 - solution of the equation.

#2 Solve the equation

Solution: Estimate the right and left sides of the equation:

a) , because , a ;

b) , because .

Evaluation of the parts of the equation shows that the left side is not less than, and the right is not more than two for any admissible values of the variable x. Therefore, this equation is equivalent to the system

The first equation of the system has only one root x=-2. Substituting this value into the second equation, we obtain the correct numerical equality:

Answer: x=-2.

The third group explains the use of the uniqueness of the root theorem.

If one of the functions (F(x)) is decreasing and the other (G(x)) is increasing on some domain of definition, then the equation F(x)=G(x) has at most one solution.

#1 Solve the equation

Solution: the domain of the given equation is x>0. We investigate the monotonicity of the function . The first one is decreasing (since it is a logarithmic function with a base greater than zero but less than one), and the second one is increasing (it is a linear function with a positive coefficient at x). The selection easily finds the root of the equation x=3, which is the only solution to this equation.

Answer: x=3.

The teacher reminds. where the monotonicity of a function is also used in solving equations.

A) - From an equation of the form h(f(x))=h(g(x)) we pass to an equation of the form f(x)=g(x)

When the function is monotonic

#5 sin (4x+?/6) = sin 3x

WRONG! (periodic function). And then we pronounce the correct answer.

WRONG! (Even degree) And then we pronounce the correct answer:

B) The method of using functional equations.

Theorem. If the function y = f(x) is an increasing (or decreasing) function on the domain of admissible values of the equation f(g(x)) = f(h(x)), then the equations f(g(x)) = f(h( x)) and g(x)=f(x) are equivalent.

#1 Solve the equation:

Consider the functional equation f(2x+1) = f(-x), where f(x) = f()

Find the derivative

Determine its sign.

Because the derivative is always positive, then the function is increasing on the entire number line, then we pass to the equation

Solve the equation. X 6 -|13 + 12x| 3= 27cos x 2- 27cos(13 + 12x).

1) the equation is reduced to the form

x6 - 27cos x2 = |13 + 12x|3 - 27cos(13 + 12x),

f(x2) = f(13 + 12x),

where f(t) = |t|3-27st;

2) The function f is even and for t > 0 has the following derivative

f"(t)= therefore f"(t)> 0 for all

Consequently, the function f increases on the positive semiaxis, which means that it takes each of its values at exactly two points that are symmetric with respect to zero. This equation is equivalent to

the following set:

Answer: -1, 13, -6+?/23.

Tasks for solving in the lesson. Answer

Reflection.

1. What have you learned?

2. Which method do you do best?

House assignment: Choose 2 equations for each method and solve them.

In a standard school mathematics course, the properties of functions are mainly used to plot their graphs. The functional method for solving equations is used if and only if the equation F(x) = G(x) as a result of transformations or change of variables cannot be reduced to one or another standard equation that has a certain solution algorithm.

Unlike the graphical method, knowledge of the properties of functions allows you to find the exact roots of the equation, without the need to plot function graphs. Using the properties of functions contributes to the rationalization of the solution of equations.

The paper considers the following properties of a function: the scope of the function; function range; monotonicity properties of a function; convexity properties of a function; properties of even and odd functions.

Purpose of the work: to carry out some classification of non-standard equations according to the use of general properties of functions, to describe the essence of each property, to give recommendations on its use, instructions for use.

All work is accompanied by the solution of specific problems proposed at the Unified State Examination of various years.

Chapter 1. Using the concept of function scope.

Let us introduce some key definitions.

The domain of the function y = f(x) is the set of values of the variable x for which the function makes sense.

Let an equation f(x) = g(x) be given, where f(x) and g(x) are elementary functions defined on the sets D1, D2. Then the domain D of admissible values of the equation will be the set consisting of those values of x that belong to both sets, that is, D = D1 ∩ D2. It is clear that when the set D is empty (D= ∅), then the equation has no solutions. (Appendix No. 1).

1. arcsin(x+2) +2x- x2 = x-2.

ODZ:-1 =0⇔-3

Answer: There are no solutions.

2. (x2-4x+3 +1) log5x5 + 1x(8x-2x2-6 + 1) = 0.

ODZ: х2-4х+3>=0,х>0.8х-2х2-6>=0⇔х∈(-infinity;1∪ 3;infinity),х>01

Check: x = 1.

(1-4+3 +1)log515 + (8-2-6 + 1) = 0,

0 = 0 is correct.

x = 3. (9-12+3+1)log535 +13(24-18-6+1) = 0, log535 +13 = 0 is wrong.

It often turns out to be sufficient to consider not the entire domain of a function, but only its subset, on which the function takes values that satisfy certain conditions (for example, only non-negative values).

1. x+27-x(x-9 +1) = 1.

ODZ: x-9>=0, x>=9.

For x>=9 x+2>0, 7-x 0, thus, the product of the three factors on the left side of the equation is negative, and the right side of the equation is positive, which means that the equation has no solutions.

Answer: ∅.

2. 3-x2+ x+2 = x-2.

ODZ: 3-x2>=0,x+2>=0,⇔ 3-x(3+x)>=0,x>=-2,⇔ -3=-2,⇔

On the set of admissible values, the left side of the equation is positive, and the right side is negative, which means that the equation has no solutions.

Answer: There are no solutions.

Chapter 2. Using the concept of the range of a function.

The range of the function y = f(x) is the set of values of the variable y with admissible values of the variable x.

A function y = f(x) is called bounded from below (respectively, from above) on the set X if there exists such a number M that the inequality fx>=M is satisfied on X (respectively, fx

A function y = f(x) is called bounded on a given interval (contained in its domain of definition) if there exists such a number M > 0 that for all values of the argument belonging to this interval, the inequality f(x)

Let the equation f(x) = g(x) be given, where g(x) are elementary functions defined on the sets D1, D2. Let us designate the range of these functions as E1 and E2, respectively. If x1 is a solution to the equation, then the numerical equality f(x1) = g(x1) will hold, where f(x1) is the value of the function f(x) at x = x1, and g(x1) is the value of the function g(x) at x = x1. Hence, if the equation has a solution, then the ranges of the functions f(x) and g(x) have common elements (Е1∩Е2 !=∅). If the sets E1 and E2 do not contain such common elements, then the equation has no solutions.

Basic inequalities are used to evaluate expressions. (Appendix No. 2).

Let the equation f(x) = g(x) be given. If f(x)>=0 and g(x)

1. x2+2xsinxy+1=0.

Solution. There is a unit on the left side, which means that you can use the basic trigonometric identity: x2+ 2xsinxy+ sin2xy+cos2xy=0.

The sum of the first three terms is a perfect square:

(x+sinxy)2+cos2xy=0.

Therefore, on the left side, the sum of squares, it is equal to zero when the expressions in the squares are simultaneously equal to zero. Let's write down the system: cosxy=0,x+sinxy=0.

If cosxy=0, then sinxy= +-1, so this system is equivalent to the combination of two systems: x+1=0,cosxy=0 or x-1=0,cosxy=0.

Their solutions are pairs of numbers x=1, y = PI 2 + PIm, m∈Z, and x=-1, y = PI 2 + PIm, m∈Z.

Answer: x=1, y = PI 2 + PIm, m∈Z, and x=-1, y = PI 2 + PIm, m∈Z.

If on the interval X the largest value of one of the functions y = f(x), y = g(x) is equal to A and the smallest value of the other function is also equal to A, then the equation f(x) = g(x) is equivalent on the interval X to the system of equations fx=A, gx=A.

1. Find all values of a for which the equation has a solution

2cos222x-x2=a+3sin(22x-x2+1).

After replacing t= 22x-x2, we arrive at the equation cos(2t+PI3)=a-12.

The function t=2m is increasing, which means that it reaches its maximum value at the maximum value of m. But m=2x - x has the largest value equal to 1. Then tmax = 22·1-1=2. Thus, the set of values of the function t= 22x-x2 is the interval (0;2, and the function cos(2t+PI3) is the interval -1;0.5). Therefore, the original equation has a solution for those and only those values of a that satisfy the inequalities -1Answer: -12. Solve the equation (log23)x+a+2 = (log94)x2+a2-6a-5.

Taking advantage of the obvious inequalities

Answer: x= - 5+32 if a=1+32 and x=-5+32 if a= 1-32.

Other equations can be considered in more detail. (Appendix No. 3).

Chapter 3. Using the monotonicity property of a function.

The function y = f(x) is called increasing (respectively decreasing) on the set X if on this set the values of the function increase (respectively decrease) as the argument increases.

In other words, the function y = f(x) increases on the set X if from x1∈X, x2∈X and x1It decreases on this set, if from x1∈X, x2∈X and x1 f(x2).

A function y = f(x) is called non-strictly increasing (respectively, non-strictly decreasing) on X if x1∈X, x2∈X and x1=f(x2)).

Functions that increase and decrease on X are called monotonic on X, and functions that are not strictly increasing or decreasing on X are called nonstrictly monotonic on X.

The following assertions are used to prove the monotonicity of functions:

1. If the function f is increasing on the set X, then for any number C the function f+C is also increasing on X.

2. If the function f is increasing on the set X and C > 0, then the function Cf is also increasing on X.

3. If the function f is increasing on the set X, then the function - f is decreasing on this set.

4. If the function f is increasing on the set X and retains its sign on the set X, then the function 1f is decreasing on this set.

5. If the functions f and g increase on the set X, then their sum f + g also increases on this set.

6. If the functions f and g are increasing and non-negative on the set X, then their product fg is also increasing on X.

7. If the function f is increasing and non-negative on the set X and n is a natural number, then the function fn is also increasing on X.

8. If both functions f(x) and g(x) are increasing or both decreasing, then the function h(x) = f(g(x)) is an increasing function. If one of the functions is increasing. And the other is decreasing, then h(x) = f(g(x)) is a decreasing function.

Let us formulate theorems about equations.

Theorem 1.

If the function f(x) is monotonic on the interval X, then the equation f(x) = C has at most one root on the interval X.

Theorem 2.

If the function f(x) is monotonic on the interval X, then the equation f(g(x)) = f(h(x)) is equivalent on the interval X to the equation g(x) = h(x).

Theorem 3.

If the function f(x) increases on the interval X, and g(x) decreases on the interval X, then the equation g(x) = f(x) has at most one root on the interval X.

Theorem 4.

If the function f(x) increases on the interval X, then the equation f(f(x)) = x is equivalent on the interval X to the equation f(x) = x.

1. Find all values of a for which the equation has exactly three roots

4-x-alog3(x2-2x+3)+2-x2+2xlog13(2x-a+2)=0.

Solution. Let us transform this equation into the form

2x2-2xlog3(x2-2x+3)= 22x-a-1log3(2x-a+2).

If we put u = x2-2x, v=2x-a-1, then we arrive at the equation

2ulog3(u+3)= 2vlog3(v+3).

The function f (t) = 2tlog3(t+3) increases monotonically for t >-2, so from the last equation we can go to the equivalent u = v, x2-2x = 2x-a-1⇔(x-1)2=2x -a.

This equation, as can be seen from the figure, has exactly three roots in the following cases:

1. The vertex of the graph of the function y \u003d 2x-a is located at the vertex of the parabola y \u003d (x-1) 2, which corresponds to a \u003d 1;

2. The left ray of the graph y \u003d 2x-a touches the parabola, and the right one intersects it at two points; this is possible with a=12;

3. The right beam touches, and the left one intersects the parabola, which takes place at a=32.

Let's explain the second case. Left beam equation y = 2a-2x, its slope is -2. Therefore, the slope of the tangent to the parabola is

2(x -1) = -2 ⇒ x = 0 and the touch point has coordinates (0; 1). From the condition that this point belongs to the ray, we find a=12.

The third case can be treated in a similar way, or symmetry considerations can be invoked.

Answer: 0.5; 1;1.5.

Other equations can be considered in more detail. (Appendix No. 4).

Chapter 4. Using the properties of convexity.

Let the function f(x) be defined on the interval X, it is called strictly convex down (up) on X, if for any u and v from X, u!=v and 0

Geometrically, this means that any point of the chord BC (that is, a segment with ends at the points B(u;f(u)) and C(v;f(v)) other than points B and C lies above (below) the point And the graph of the function f(x), corresponding to the same value of the argument (Appendix No. 5).

Functions that are strictly convex up and down are called strictly convex.

The following assertions are true.

Theorem 1.

Let the function f(x) be strictly downward convex on the interval X, u ,v ∈X, u

Theorem 1 implies the following assertion.

Theorem 2.

If the function f(x) is strictly convex on the interval X, then the functions u = u(x), v = v(x), u1=u1(x), v1 = v1(x) are such that, for all x, the ODZ equations f(u)+f(v) = f(u1) + f(v1) (1) their values u(x), v(x), u1(x), v1(x) are contained in X and the condition u +v = u1 +v1, then the equation f(u)+f(v) = f(u1) + f(v1) (2) on the ODZ is equivalent to the set of equations u (x) = u1(x), u(x) = v1(x) (3).

1. 41-sin4x+41-cos4x=412.

Solution. If we put fx= 41-x2, u=cos2x, v=sin2x, u1=v1=12, then this equation will be written in the form (1). Since f "x \u003d -x24 (1-x2) 3, f "" x \u003d -2 + x244 (1-x2) 7, then the function fx is strictly upward convex on the segment -1; 1. Obviously, the remaining conditions are satisfied Theorem 2 and, therefore, the equation is equivalent to the equation cos2x = 0.5, x = PI4 + PIk2, where k∈Z.

Answer: x = PI4 + PIk2, where k∈Z.

Theorem 3.

Let the function fx be strictly convex on the interval X and u,v, λv+(1-λ)u∈X. Then the equality f (λv+(1-λ)u) = λf(v)+(1-λ)f(u) (4) is true if and only if either u=v or λ=0, or λ=1.

Examples: sin2xcos3x+cos2xsin3x∙1+sin2xcos3x+cos2xsin3x= sin2xcos3x1+cos3x+cos2xsin3x1+sin3x.

The equation has the form (4) if fx=x1+x= x+x2, u=sin3x, v= cos3x, λ=sin2x.

Obviously, the function fx is strictly downward convex on R. Therefore, by Theorem 3, the original equation is equivalent to the set of equations sinx=0, sin2x=1, cos3x=sin3x.

Hence we get that its solutions will be PIk2, PI12+PIn3, where k,n∈Z.

Answer: PIk2, PI12+PIn3, where k,n∈Z.

The use of convexity properties is also used to solve more complex equations. (Appendix No. 6).

Chapter 5. Using the Even or Odd Properties of Functions.

The function fx is called even if for any value x taken from the domain of the function, the value - x also belongs to the domain of definition and the equality f-x= fx is satisfied. The function fx is called odd if for any value x taken from the domain of the function, the value - x also belongs to the domain of definition and the equality f-x=- fx is satisfied.

It follows from the definition that the domains of even and odd functions are symmetric with respect to zero (a necessary condition).

For any two symmetric values of the argument from the domain of definition, the even function takes equal numerical values, and the odd one takes equal in absolute value, but of the opposite sign.

Theorem 1.

The sum, difference, product, and quotient of two even functions are even functions.

Theorem 2.

The product and quotient of two odd functions are even functions.

Let we have the equation F(x)=0, where F(x) is an even or odd function.

To solve the equation F(x) = 0, where F(x) is an even or odd function, it is enough to find positive (or negative) roots symmetrical to those obtained, and for an odd function the root will be x = 0 if this value is included in the domain of definition F(x). For an even function, the value x = 0 is checked by direct substitution into the equation.

We have even functions on both sides of the equation. Therefore, it suffices to find solutions for x>=0. Since x=0 is not a root of the equation, consider two intervals: (0;2, 2;infinity.

a) On the interval (0;2 we have:

8x= 2x+2-x+2, 23x=24, x= 43.

b) On the interval 2;infinity we have:

8x= 2x+2+x-2.23x=22x, x=0.

But since x \u003d 0 is not the root of the equation, then for x> 0 this equation has a root x \u003d 43. Then x \u003d - 43 is also the root of the equation.

Answer: 43; - 43.

The author believes that the work can be used by teachers and students of general education types in optional classes, in preparation for mathematical olympiads, passing the exam, entrance exams to technical schools.

Portal for the student. Self-training

Topic: "Exponential function. Functional-graphical methods for solving equations, inequalities, systems"

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