Knowledge basic elementary functions, their properties and graphs no less important than knowing the multiplication tables. They are like the foundation, everything is based on them, everything is built from them and everything comes down to them.

In this article we will list all the main elementary functions, provide their graphs and give without conclusion or proof properties of basic elementary functions according to the scheme:

• behavior of a function at the boundaries of the domain of definition, vertical asymptotes (if necessary, see the article classification of discontinuity points of a function);
• even and odd;
• intervals of convexity (convexity upward) and concavity (convexity downward), inflection points (if necessary, see the article convexity of a function, direction of convexity, inflection points, conditions of convexity and inflection);
• inclined and horizontal asymptotes;
• singular points of functions;
• special properties of some functions (for example, the smallest positive period of trigonometric functions).

If you are interested in or, then you can go to these sections of the theory.

Basic elementary functions are: constant function (constant), nth root, power function, exponential, logarithmic function, trigonometric and inverse trigonometric functions.

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Permanent function.

A constant function is defined on the set of all real numbers by the formula , where C is some real number. A constant function associates each real value of the independent variable x with the same value of the dependent variable y - the value C. A constant function is also called a constant.

The graph of a constant function is a straight line parallel to the x-axis and passing through the point with coordinates (0,C). For example, let's show graphs of constant functions y=5, y=-2 and, which in the figure below correspond to the black, red and blue lines, respectively.

Properties of a constant function.

• Domain: the entire set of real numbers.
• The constant function is even.
• Range of values: set consisting of singular WITH .
• A constant function is non-increasing and non-decreasing (that’s why it’s constant).
• It makes no sense to talk about convexity and concavity of a constant.
• There are no asymptotes.
• The function passes through the point (0,C) of the coordinate plane.

Root of the nth degree.

Let's consider the basic elementary function, which is given by the formula , where n – natural number, greater than one.

Root of the nth degree, n is an even number.

Let's start with the nth root function for even values ​​of the root exponent n.

As an example, here is a picture with images of function graphs and , they correspond to black, red and blue lines.

The graphs of even-degree root functions have a similar appearance for other values ​​of the exponent.

Properties of the nth root function for even n.

The nth root, n is an odd number.

The nth root function with an odd root exponent n is defined on the entire set of real numbers. For example, here are the function graphs and , they correspond to black, red and blue curves.

For other odd values ​​of the root exponent, the function graphs will have a similar appearance.

Properties of the nth root function for odd n.

Power function.

Power function is given by a formula of the form .

Let's consider the form of graphs of a power function and the properties of a power function depending on the value of the exponent.

Let's start with a power function with an integer exponent a. In this case, the type of graphs of power functions and the properties of the functions depend on the evenness or oddness of the exponent, as well as on its sign. Therefore, we will first consider power functions for odd positive values ​​of the exponent a, then for even positive exponents, then for odd negative exponents, and finally, for even negative a.

The properties of power functions with fractional and irrational exponents (as well as the type of graphs of such power functions) depend on the value of the exponent a. We will consider them, firstly, for a from zero to one, secondly, for a greater than one, thirdly, for a from minus one to zero, fourthly, for a less than minus one.

At the end of this section, for completeness, we will describe a power function with zero exponent.

Power function with odd positive exponent.

Let's consider a power function with an odd positive exponent, that is, with a = 1,3,5,....

The figure below shows graphs of power functions - black line, - blue line, - red line, - green line. For a=1 we have linear function y=x.

Properties of a power function with an odd positive exponent.

Power function with even positive exponent.

Let's consider a power function with an even positive exponent, that is, for a = 2,4,6,....

As an example, we give graphs of power functions – black line, – blue line, – red line. For a=2 we have quadratic function, whose graph is quadratic parabola.

Properties of a power function with an even positive exponent.

Power function with odd negative exponent.

Look at the graphs of the power function for odd negative values exponent, that is, for a = -1, -3, -5,... .

The figure shows graphs of power functions as examples - black line, - blue line, - red line, - green line. For a=-1 we have inverse proportionality, whose graph is hyperbola.

Properties of a power function with an odd negative exponent.

Power function with even negative exponent.

Let's move on to the power function for a=-2,-4,-6,….

The figure shows graphs of power functions – black line, – blue line, – red line.

Properties of a power function with an even negative exponent.

A power function with a rational or irrational exponent whose value is greater than zero and less than one.

Pay attention! If a is a positive fraction with an odd denominator, then some authors consider the domain of definition of the power function to be the interval. It is stipulated that the exponent a is an irreducible fraction. Now the authors of many textbooks on algebra and the beginnings of analysis DO NOT DEFINE power functions with an exponent in the form of a fraction with an odd denominator for negative values ​​of the argument. We will adhere to precisely this view, that is, we will consider the set to be the domains of definition of power functions with fractional positive exponents. We recommend that students find out your teacher's opinion on this subtle point in order to avoid disagreements.

Let us consider a power function with a rational or irrational exponent a, and .

Let us present graphs of power functions for a=11/12 (black line), a=5/7 (red line), (blue line), a=2/5 (green line).

A power function with a non-integer rational or irrational exponent greater than one.

Let us consider a power function with a non-integer rational or irrational exponent a, and .

Let us present graphs of power functions given by the formulas (black, red, blue and green lines respectively).

>

For other values ​​of the exponent a, the graphs of the function will have a similar appearance.

Properties of the power function at .

A power function with a real exponent that is greater than minus one and less than zero.

Pay attention! If a is a negative fraction with an odd denominator, then some authors consider the domain of definition of a power function to be the interval . It is stipulated that the exponent a is an irreducible fraction. Now the authors of many textbooks on algebra and the beginnings of analysis DO NOT DEFINE power functions with an exponent in the form of a fraction with an odd denominator for negative values ​​of the argument. We will adhere to precisely this view, that is, we will consider the domains of definition of power functions with fractional fractional negative exponents to be a set, respectively. We recommend that students find out your teacher's opinion on this subtle point in order to avoid disagreements.

Let's move on to the power function, kgod.

To have a good idea of ​​the form of graphs of power functions for , we give examples of graphs of functions (black, red, blue and green curves, respectively).

Properties of a power function with exponent a, .

A power function with a non-integer real exponent that is less than minus one.

Let us give examples of graphs of power functions for , they are depicted by black, red, blue and green lines, respectively.

Properties of a power function with a non-integer negative exponent less than minus one.

When a = 0 and we have a function - this is a straight line from which the point (0;1) is excluded (it was agreed not to attach any significance to the expression 0 0).

Exponential function.

One of the main elementary functions is the exponential function.

The graph of the exponential function, where and takes different forms depending on the value of the base a. Let's figure this out.

First, consider the case when the base of the exponential function takes a value from zero to one, that is, .

As an example, we present graphs of the exponential function for a = 1/2 – blue line, a = 5/6 – red line. The graphs of the exponential function have a similar appearance for other values ​​of the base from the interval.

Properties of an exponential function with a base less than one.

Let us move on to the case when the base of the exponential function is greater than one, that is, .

As an illustration, we present graphs of exponential functions - blue line and - red line. For other values ​​of the base greater than one, the graphs of the exponential function will have a similar appearance.

Properties of an exponential function with a base greater than one.

Logarithmic function.

Next main elementary function is a logarithmic function, where , . The logarithmic function is defined only for positive values ​​of the argument, that is, for .

The graph of a logarithmic function takes different forms depending on the value of the base a.

Exponential function is a generalization of the product of n numbers equal to a:
y (n) = a n = a·a·a···a,
to the set of real numbers x:
y (x) = ax.
Here a is a fixed real number, which is called basis of the exponential function.
An exponential function with base a is also called exponent to base a.

The generalization is carried out as follows.
For natural x = 1, 2, 3,... , the exponential function is the product of x factors:
.
Moreover, it has properties (1.5-8) (), which follow from the rules for multiplying numbers. For zero and negative values ​​of integers, the exponential function is determined using formulas (1.9-10). For fractional values ​​x = m/n rational numbers, , it is determined by formula (1.11). For reals, the exponential function is defined as sequence limit:
,
where is an arbitrary sequence of rational numbers converging to x: .
With this definition, the exponential function is defined for all , and satisfies properties (1.5-8), as for natural x.

A rigorous mathematical formulation of the definition of an exponential function and the proof of its properties is given on the page “Definition and proof of the properties of an exponential function”.

Properties of the Exponential Function

The exponential function y = a x has the following properties on the set of real numbers ():
(1.1) defined and continuous, for , for all ;
(1.2) for a ≠ 1 has many meanings;
(1.3) strictly increases at , strictly decreases at ,
is constant at ;
(1.4) at ;
at ;
(1.5) ;
(1.6) ;
(1.7) ;
(1.8) ;
(1.9) ;
(1.10) ;
(1.11) , .

Other useful formulas.
.
Formula for converting to an exponential function with a different exponent base:

When b = e, we obtain the expression of the exponential function through the exponential:

Private values

, , , , .

The figure shows graphs of the exponential function
y (x) = ax
for four values degree bases: a = 2 , a = 8 , a = 1/2 and a = 1/8 . It can be seen that for a > 1 the exponential function increases monotonically. The larger the base of the degree a, the stronger the growth. At 0 < a < 1 the exponential function decreases monotonically. The smaller the exponent a, the stronger the decrease.

Ascending, descending

The exponential function for is strictly monotonic and therefore has no extrema. Its main properties are presented in the table.

	y = a x , a > 1	y = ax, 0 < a < 1
Domain of definition	- ∞ < x < + ∞	- ∞ < x < + ∞
Range of values	0 < y < + ∞	0 < y < + ∞
Monotone	monotonically increases	monotonically decreases
Zeros, y = 0	No	No
Intercept points with the ordinate axis, x = 0	y = 1	y = 1
	+ ∞	0
	0	+ ∞

Inverse function

The inverse of an exponential function with base a is the logarithm to base a.

If , then
.
If , then
.

Differentiation of an exponential function

To differentiate an exponential function, its base must be reduced to the number e, apply the table of derivatives and the differentiation rule complex function.

To do this you need to use the property of logarithms
and the formula from the derivatives table:
.

Let an exponential function be given:
.
We bring it to the base e:

Let's apply the rule of differentiation of complex functions. To do this, introduce the variable

Then

From the table of derivatives we have (replace the variable x with z):
.
Since is a constant, the derivative of z with respect to x is equal to
.
According to the rule of differentiation of a complex function:
.

Derivative of an exponential function

.
Derivative of nth order:
.
Deriving formulas > > >

An example of differentiating an exponential function

Find the derivative of a function
y = 3 5 x

Solution

Let's express the base of the exponential function through the number e.
3 = e ln 3
Then
.
Enter a variable
.
Then

From the table of derivatives we find:
.
Since 5ln 3 is a constant, then the derivative of z with respect to x is equal to:
.
According to the rule of differentiation of a complex function, we have:
.

Answer

Integral

Expressions using complex numbers

Consider the function complex number z:
f (z) = a z
where z = x + iy; i 2 = - 1 .
Let us express the complex constant a in terms of modulus r and argument φ:
a = r e i φ
Then


.
The argument φ is not uniquely defined. IN general view
φ = φ 0 + 2 πn,
where n is an integer. Therefore the function f (z) is also not clear. Its main significance is often considered
.

Series expansion

.

Used literature:
I.N. Bronstein, K.A. Semendyaev, Handbook of mathematics for engineers and college students, “Lan”, 2009.

Let's find the value of the expression for various rational values ​​of the variable x=2; 0; -3; -

Note that no matter what number we substitute for the variable x, we can always find the value of this expression. This means that we are considering an exponential function (E is equal to three to the power of x), defined on the set of rational numbers: .

Let's build a graph of this function by compiling a table of its values.

Let's draw a smooth line passing through these points (Figure 1)

Using the graph of this function, let’s consider its properties:

3.Increases throughout the entire area of ​​definition.

1. range of values ​​from zero to plus infinity.

8. The function is convex downwards.

If we construct graphs of functions in one coordinate system; y=(y is equal to two to the power of x, y is equal to five to the power of x, y is equal to seven to the power of x), then you can see that they have the same properties as y=(y is equal to three to the power of x) (Fig. .2), that is, all functions of the form y = (a is equal to a to the x power, for a greater than one) will have such properties

Let's plot the function:

1. Compiling a table of its values.

Let us mark the obtained points on the coordinate plane.

Let's draw a smooth line passing through these points (Figure 3).

Using the graph of this function, we indicate its properties:

1. The domain of definition is the set of all real numbers.

2. Is neither even nor odd.

3.Decreases throughout the entire domain of definition.

4. Has neither the largest nor the smallest values.

5.Limited below, but not limited above.

6.Continuous throughout the entire domain of definition.

7. range of values ​​from zero to plus infinity.

8. The function is convex downwards.

Similarly, if we construct graphs of functions in one coordinate system; y = (y is equal to one-half to the power of x, y is equal to one-fifth to the power of x, y is equal to one-seventh to the power of x), then you can notice that they have the same properties as y = (y is equal to one-third to the power x (Fig. 4), that is, all functions of the form y = (the y is equal to one divided by a to the x power, with a greater than zero but less than one) will have such properties.

Let's construct graphs of functions in one coordinate system

This means that the graphs of the functions y=y= will also be symmetrical (y is equal to a to the x power and y is equal to one divided by a to the x power) for the same value of a.

Let us summarize what has been said by defining the exponential function and indicating its main properties:

Definition: A function of the form y=, where (the y is equal to a to the power x, where a is positive and different from one), is called an exponential function.

It is necessary to remember the differences between the exponential function y= and the power function y=, a=2,3,4,…. both audibly and visually. The exponential function X is a power, and for a power function X is the basis.

Example 1: Solve the equation (three to the power x equals nine)

(Y is equal to three to the power of X and Y is equal to nine) Fig. 7

Note that they have one common point M (2;9) (em with coordinates two; nine), which means that the abscissa of the point will be the root of this equation. That is, the equation has a single root x = 2.

Example 2: Solve the equation

In one coordinate system, we will construct two graphs of the function y= (the y is equal to five to the power of x and the y is equal to one twenty-fifth) Fig. 8. The graphs intersect at one point T (-2; (te with coordinates minus two; one twenty-fifth). This means that the root of the equation is x = -2 (the number minus two).

Example 3: Solve the inequality

In one coordinate system we will construct two graphs of the function y=

(Y is equal to three to the power of X and Y is equal to twenty-seven).

Fig.9 The graph of the function is located above the graph of the function y=at

x Therefore, the solution to the inequality is the interval (from minus infinity to three)

Example 4: Solve the inequality

In one coordinate system, we will construct two graphs of the function y= (the y is equal to one fourth to the power of x and the y is equal to sixteen). (Fig. 10). The graphs intersect at one point K (-2;16). This means that the solution to the inequality is the interval (-2; (from minus two to plus infinity), since the graph of the function y= is located below the graph of the function at x

Our reasoning allows us to verify the validity of the following theorems:

Theme 1: If true if and only if m=n.

Theorem 2: If is true if and only if, inequality is true if and only if (Fig. *)

Theorem 4: If true if and only if (Fig.**), the inequality is true if and only if. Theorem 3: If true if and only if m=n.

Example 5: Graph the function y=

Let's modify the function by applying the property of degree y=

Let us construct an additional coordinate system and in new system coordinates, we will construct a graph of the function y = (the y is equal to two to the x power) Fig. 11.

Example 6: Solve the equation

In one coordinate system we will construct two graphs of the function y=

(Y is equal to seven to the power of X and Y is equal to eight minus X) Fig. 12.

The graphs intersect at one point E (1; (e with coordinates one; seven). This means that the root of the equation is x = 1 (x equal to one).

Example 7: Solve the inequality

In one coordinate system we will construct two graphs of the function y=

(Y is equal to one-fourth to the power of X and Y is equal to X plus five). The graph of the function y=is located below the graph of the function y=x+5 when the solution to the inequality is the interval x (from minus one to plus infinity).

Lesson No.2

Topic: Exponential function, its properties and graph.

Target: Check the quality of mastering the concept of “exponential function”; to develop skills in recognizing the exponential function, using its properties and graphs, teaching students to use analytical and graphic forms recording exponential functions; provide a working environment in the classroom.

Equipment: board, posters

Lesson form: class lesson

Lesson type: practical lesson

Lesson type: lesson in teaching skills and abilities

Lesson Plan

1. Organizational moment

2. Independent work and check homework

3. Problem solving

4. Summing up

5. Homework

Lesson progress.

1. Organizational moment :

Hello. Open your notebooks, write down today’s date and the topic of the lesson “Exponential Function”. Today we will continue to study the exponential function, its properties and graph.

2. Independent work and checking homework .

Target: check the quality of mastery of the concept of “exponential function” and check the completion of the theoretical part of the homework

Method: test task, frontal survey

As homework, you were given numbers from the problem book and a paragraph from the textbook. We won’t check your execution of numbers from the textbook now, but you will hand in your notebooks at the end of the lesson. Now the theory will be tested in the form of a small test. The task is the same for everyone: you are given a list of functions, you must find out which of them are indicative (underline them). And next to the exponential function you need to write whether it is increasing or decreasing.

Option 1 Answer B) D) - exponential, decreasing	Option 2 Answer D) - exponential, decreasing D) - exponential, increasing
Option 3 Answer A) - exponential, increasing B) - exponential, decreasing	Option 4 Answer A) - exponential, decreasing IN) - exponential, increasing

Now let’s remember together which function is called exponential?

A function of the form , where and , is called an exponential function.

What is the scope of this function?

All real numbers.

What is the range of the exponential function?

All positive real numbers.

Decreases if the base of the power is greater than zero but less than one.

In what case does an exponential function decrease in its domain of definition?

Increasing if the base of the power is greater than one.

3. Problem solving

Target: to develop skills in recognizing an exponential function, using its properties and graphs, teach students to use analytical and graphical forms of writing an exponential function

Method: demonstration by the teacher of solving typical problems, oral work, work at the blackboard, work in a notebook, conversation between the teacher and students.

The properties of the exponential function can be used when comparing 2 or more numbers. For example: No. 000. Compare the values ​​and if a) ..gif" width="37" height="20 src=">, then this is a rather complicated job: we would have to take the cube root of 3 and 9, and compare them. But we know that it increases, this in its own way turn means that as the argument increases, the value of the function increases, that is, we just need to compare the values ​​of the argument and , it is obvious that (can be demonstrated on a poster showing an increasing exponential function). And always, when solving such examples, you first determine the base of the exponential function, compare it with 1, determine monotonicity and proceed to compare the arguments. In the case of a decreasing function: when the argument increases, the value of the function decreases, therefore, we change the sign of inequality when moving from inequality of arguments to inequality of functions. Next, we solve orally: b)

-

IN)

-

G)

-

- No. 000. Compare the numbers: a) and

Therefore, the function increases, then

Why ?

Increasing function and

Therefore, the function is decreasing, then

Both functions increase throughout their entire domain of definition, since they are exponential with a base of power greater than one.

What is the meaning behind it?

We build graphs:

Which function increases faster when striving https://pandia.ru/text/80/379/images/image062_0.gif" width="20 height=25" height="25">

Which function decreases faster when striving https://pandia.ru/text/80/379/images/image062_0.gif" width="20 height=25" height="25">

On the interval which of the functions has higher value at a specific point?

D), https://pandia.ru/text/80/379/images/image068_0.gif" width="69" height="57 src=">. First, let's find out the scope of definition of these functions. Do they coincide?

Yes, the domain of these functions is all real numbers.

Name the scope of each of these functions.

The ranges of these functions coincide: all positive real numbers.

Determine the type of monotonicity of each function.

All three functions decrease throughout their entire domain of definition, since they are exponential with a base of powers less than one and greater than zero.

Which singular point does the graph of an exponential function exist?

What is the meaning behind it?

Whatever the basis of the degree of an exponential function, if the exponent contains 0, then the value of this function is 1.

We build graphs:

Let's analyze the graphs. How many points of intersection do the graphs of functions have?

Which function decreases faster when trying https://pandia.ru/text/80/379/images/image070.gif" width="41 height=57" height="57">

Which function increases faster when striving https://pandia.ru/text/80/379/images/image070.gif" width="41 height=57" height="57">

On the interval, which of the functions has greater value at a specific point?

On the interval, which of the functions has greater value at a specific point?

Why do exponential functions with different bases have only one intersection point?

Exponential functions are strictly monotonic throughout their domain of definition, so they can intersect only at one point.

The next task will focus on using this property. No. 000. Find the largest and smallest value given function on a given interval a) . Recall that a strictly monotonic function takes its minimum and maximum values ​​at the ends of a given segment. And if the function is increasing, then its highest value will be at the right end of the segment, and the smallest at the left end of the segment (demonstration on the poster, using the example of an exponential function). If the function is decreasing, then its largest value will be at the left end of the segment, and the smallest at the right end of the segment (demonstration on the poster, using the example of an exponential function). The function is increasing, because, therefore, the smallest value of the function will be at the point https://pandia.ru/text/80/379/images/image075_0.gif" width="145" height="29">. Points b ) , V) d) solve the notebooks yourself, we will check them orally.

Students solve the task in their notebooks

Decreasing function

Decreasing function greatest value of the function on the segment the smallest value of a function on a segment

Increasing function the smallest value of a function on a segment greatest value of the function on the segment

- No. 000. Find the largest and smallest value of the given function on the given interval a) . This task is almost the same as the previous one. But what is given here is not a segment, but a ray. We know that the function is increasing, and it has neither the largest nor the smallest value on the entire number line https://pandia.ru/text/80/379/images/image063_0.gif" width="68" height ="20">, and tends to at , i.e. on the ray the function at tends to 0, but does not have its smallest value, but it has the largest value at the point . Points b) , V) , G) Solve the notebooks yourself, we will check them orally.

Majority decision mathematical problems is somehow related to the transformation of numerical, algebraic or functional expressions. The above applies especially to the decision. In the versions of the Unified State Examination in mathematics, this type of problem includes, in particular, task C3. Learning to solve C3 tasks is important not only for the purpose of successful passing the Unified State Exam, but also for the reason that this skill will be useful when studying a mathematics course in high school.

When completing tasks C3, you have to decide various types equations and inequalities. Among them are rational, irrational, exponential, logarithmic, trigonometric, containing modules (absolute values), as well as combined ones. This article discusses the main types of exponential equations and inequalities, as well as various methods their decisions. Read about solving other types of equations and inequalities in the “” section in articles devoted to methods for solving C3 problems from Unified State Exam options in mathematics.

Before we begin to analyze specific exponential equations and inequalities, as a math tutor, I suggest you brush up on some theoretical material that we will need.

Exponential function

What is an exponential function?

Function of the form y = a x, Where a> 0 and a≠ 1 is called exponential function.

Basic properties of exponential function y = a x:

Graph of an Exponential Function

The graph of the exponential function is exponent:

Graphs of exponential functions (exponents)

Solving exponential equations

Indicative are called equations in which the unknown variable is found only in exponents of some powers.

To solve exponential equations you need to know and be able to use the following simple theorem:

Theorem 1. Exponential equation a f(x) = a g(x) (Where a > 0, a≠ 1) is equivalent to the equation f(x) = g(x).

In addition, it is useful to remember the basic formulas and operations with degrees:

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Example 1. Solve the equation:

Solution: We use the above formulas and substitution:

The equation then becomes:

Discriminant of the received quadratic equation positive:

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This means that this equation has two roots. We find them:

Moving on to reverse substitution, we get:

The second equation has no roots, since the exponential function is strictly positive over the entire domain of definition. Let's solve the second one:

Taking into account what was said in Theorem 1, we move on to the equivalent equation: x= 3. This will be the answer to the task.

Answer: x = 3.

Example 2. Solve the equation:

Solution: The equation has no restrictions on the range of permissible values, since the radical expression makes sense for any value x(exponential function y = 9 4 -x positive and not equal to zero).

We solve the equation by equivalent transformations using the rules of multiplication and division of powers:

The last transition was carried out in accordance with Theorem 1.

Answer:x= 6.

Example 3. Solve the equation:

Solution: both sides of the original equation can be divided by 0.2 x. This transition will be equivalent, since this expression is greater than zero for any value x(the exponential function is strictly positive in its domain of definition). Then the equation takes the form:

Answer: x = 0.

Example 4. Solve the equation:

Solution: we simplify the equation to an elementary one by means of equivalent transformations using the rules of division and multiplication of powers given at the beginning of the article:

Dividing both sides of the equation by 4 x, as in the previous example, is an equivalent transformation, since this expression is not equal to zero for any values x.

Answer: x = 0.

Example 5. Solve the equation:

Solution: function y = 3x, standing on the left side of the equation, is increasing. Function y = —x The -2/3 on the right side of the equation is decreasing. This means that if the graphs of these functions intersect, then at most one point. In this case, it is easy to guess that the graphs intersect at the point x= -1. There will be no other roots.

Answer: x = -1.

Example 6. Solve the equation:

Solution: we simplify the equation by means of equivalent transformations, keeping in mind everywhere that the exponential function is strictly greater than zero for any value x and using the rules for calculating the product and quotient of powers given at the beginning of the article:

Answer: x = 2.

Solving exponential inequalities

Indicative are called inequalities in which the unknown variable is contained only in exponents of some powers.

To solve exponential inequalities knowledge of the following theorem is required:

Theorem 2. If a> 1, then the inequality a f(x) > a g(x) is equivalent to an inequality of the same meaning: f(x) > g(x). If 0< a < 1, то exponential inequality a f(x) > a g(x) is equivalent to an inequality with the opposite meaning: f(x) < g(x).

Example 7. Solve the inequality:

Solution: Let's present the original inequality in the form:

Let's divide both sides of this inequality by 3 2 x, in this case (due to the positivity of the function y= 3 2x) the inequality sign will not change:

Let's use the substitution:

Then the inequality will take the form:

So, the solution to the inequality is the interval:

moving to the reverse substitution, we get:

Due to the positivity of the exponential function, the left inequality is satisfied automatically. Using the well-known property of the logarithm, we move on to the equivalent inequality:

Since the base of the degree is a number greater than one, equivalent (by Theorem 2) is the transition to the following inequality:

So, we finally get answer:

Example 8. Solve the inequality:

Solution: Using the properties of multiplication and division of powers, we rewrite the inequality in the form:

Let's introduce a new variable:

Taking this substitution into account, the inequality takes the form:

Multiplying the numerator and denominator of the fraction by 7, we obtain the following equivalent inequality:

So, the following values ​​of the variable satisfy the inequality t:

Then, moving to the reverse substitution, we get:

Since the base of the degree here is greater than one, the transition to the inequality will be equivalent (by Theorem 2):

Finally we get answer:

Example 9. Solve the inequality:

Solution:

We divide both sides of the inequality by the expression:

It is always greater than zero (due to the positivity of the exponential function), so the inequality sign does not need to be changed. We get:

t located in the interval:

Moving on to the reverse substitution, we find that the original inequality splits into two cases:

The first inequality has no solutions due to the positivity of the exponential function. Let's solve the second one:

Example 10. Solve the inequality:

Solution:

Parabola branches y = 2x+2-x 2 are directed downwards, therefore it is limited from above by the value that it reaches at its vertex:

Parabola branches y = x 2 -2x The +2 in the indicator are directed upward, which means it is limited from below by the value that it reaches at its vertex:

At the same time, the function also turns out to be bounded from below y = 3 x 2 -2x+2, which is on the right side of the equation. It reaches its smallest value at the same point as the parabola in the exponent, and this value is 3 1 = 3. So, the original inequality can only be true if the function on the left and the function on the right take on the value , equal to 3 (the intersection of the ranges of values ​​of these functions is only this number). This condition is satisfied at a single point x = 1.

Answer: x= 1.

In order to learn to decide exponential equations and inequalities it is necessary to constantly train in solving them. Various things can help you with this difficult task. methodological manuals, problem books on elementary mathematics, collections of competitive problems, mathematics classes at school, as well as individual lessons with a professional tutor. I sincerely wish you success in your preparation and excellent results in the exam.

Sergey Valerievich

P.S. Dear guests! Please do not write requests to solve your equations in the comments. Unfortunately, I have absolutely no time for this. Such messages will be deleted. Please read the article. Perhaps in it you will find answers to questions that did not allow you to solve your task on your own.