Which of the functions is exemplary. An exponential function, its properties and graph - Knowledge Hypermarket

EXPONENTIAL AND LOGARITHMIC FUNCTIONS VIII

§ 179 Basic properties of the exponential function

In this section, we will study the main properties of the exponential function

y = a x (1)

Recall that under but in formula (1) we mean any fixed positive number other than 1.

Property 1. The domain of the exponential function is the set of all real numbers.

Indeed, for a positive but expression but x defined for any real number X .

Property 2. Exponential function takes only positive values.

Indeed, if X > 0, then, as was proved in § 176,

but x > 0.

If X <. 0, то

but x =

where - X already greater than zero. That's why but - x > 0. But then

but x = > 0.

Finally, at X = 0

but x = 1.

The 2nd property of the exponential function has a simple graphical interpretation. It lies in the fact that the graph of this function (see Fig. 246 and 247) is located entirely above the x-axis.

Property 3. If but >1, then at X > 0 but x > 1, and at X < 0 but x < 1. If but < 1, тoh, on the contrary, X > 0 but x < 1, and at X < 0 but x > 1.

This property of the exponential function also allows for a simple geometric interpretation. At but > 1 (fig. 246) curves y = a x located above the line at = 1 at X > 0 and below the straight line at = 1 at X < 0.

If but < 1 (рис. 247), то, наоборот, кривые y = a x located below the line at = 1 at X > 0 and above this straight line at X < 0.

Let us give a rigorous proof of the 3rd property. Let be but > 1 and X is an arbitrary positive number. Let us show that

but x > 1.

If number X rational ( X = m / n ) , then but x = but m / n = n √a m .

Insofar as but > 1, then but m > 1, but the root of a number greater than one is obviously also greater than 1.

If X irrational, then there are positive rational numbers X" And X" , which serve as decimal approximations of the number x :

X"< х < х" .

But then, by definition of degree c irrational indicator

but x" < but x < but x"" .

As shown above, the number but x" more than one. Therefore, the number but x , more than but x" , must also be greater than 1,

So, we have shown that a >1 and arbitrary positive X

but x > 1.

If the number X was negative, then we would have

but x =

where number is X would be positive. That's why but - x > 1. Therefore,

but x = < 1.

Thus, at but > 1 and arbitrary negative x

but x < 1.

Case when 0< but < 1, легко сводится к уже рассмотренному случаю. Учащимся предлагается убедиться в этом самостоятельно.

Property 4. If x = 0, then regardless of a but x =1.

This follows from the definition of degree zero; the zero power of any number other than zero is equal to 1. Graphically, this property is expressed in the fact that for any but curve at = but x (see fig. 246 and 247) crosses the axis at at the point with ordinate 1.

Property 5. At but >1 exponential function = but x is monotonically increasing, and for a < 1 - monotonically decreasing.

This property also allows for a simple geometric interpretation.

At but > 1 (Fig. 246) curve at = but x with growth X rises higher and higher, and but < 1 (рис. 247) - опускается все ниже и ниже.

Let us give a rigorous proof of the 5th property.

Let be but > 1 and X 2 > X one . Let us show that

but x 2 > but x 1

Insofar as X 2 > X 1 ., then X 2 = X 1 + d , where d is some positive number. That's why

but x 2 - but x 1 = but x 1 + d - but x 1 = but x 1 (but d - 1)

According to the 2nd property of the exponential function but x 1 > 0. Since d > 0, then by the 3rd property of the exponential function but d > 1. Both factors in the product but x 1 (but d - 1) are positive, therefore this product itself is positive. Means, but x 2 - but x 1 > 0, or but x 2 > but x 1 , which was to be proved.

So, at a > 1 function at = but x is monotonically increasing. Similarly, it is proved that but < 1 функция at = but x is monotonically decreasing.

Consequence. If two powers of the same positive number other than 1 are equal, then their exponents are also equal.

In other words, if

but b = but c (but > 0 and but =/= 1),

b = c .

Indeed, if the numbers b And from were not equal, then due to the monotonicity of the function at = but x most of them would correspond to but >1 is greater, and at but < 1 меньшее значение этой функции. Таким образом, было бы или but b > but c , or but b < but c . Both of these contradict the condition but b = but c . It remains to be recognized that b = c .

Property 6. If a > 1, then with an unlimited increase in the argument X (X -> ∞ ) function values at = but x also grow indefinitely (at -> ∞ ). With an unlimited decrease in the argument X (X -> -∞ ) the values of this function tend to zero, while remaining positive (at->0; at > 0).

Taking into account the above-proved monotonicity of the function at = but x , we can say that in the case under consideration, the function at = but x increases monotonically from 0 to ∞ .

If 0 <but < 1, then with an unlimited increase in the argument x (x -> ∞), the values of the function y \u003d a x tend to zero, while remaining positive (at->0; at > 0). With an unlimited decrease in the argument x (X -> -∞ ) the values of this function grow indefinitely (at -> ∞ ).

Due to the monotonicity of the function y = ax we can say that in this case the function at = but x decreases monotonically from ∞ to 0.

The 6th property of the exponential function is clearly reflected in figures 246 and 247. We will not strictly prove it.

We only need to establish the range of the exponential function y = ax (but > 0, but =/= 1).

Above we proved that the function y = ax takes only positive values and either increases monotonically from 0 to ∞ (at but > 1), or decreases monotonically from ∞ to 0 (at 0< but <. 1). Однако остался невыясненным следующий вопрос: не претерпевает ли функция y = ax when you change any jumps? Does it take any positive values? This question is answered positively. If but > 0 and but =/= 1, then whatever the positive number at 0 must be found X 0 , such that

but x 0 = at 0 .

(Due to the monotonicity of the function y = ax specified value X 0 would be the only one, of course.)

The proof of this fact is beyond the scope of our program. Its geometric interpretation is that for any positive value at 0 function graph y = ax must intersect with the line at = at 0 and, moreover, only at one point (Fig. 248).

From this we can draw the following conclusion, which we formulate in the form of property 7.

Property 7. The area of change of the exponential function y \u003d a x (but > 0, but =/= 1)is the set of all positive numbers.

Exercises

1368. Find the domains of the following functions:

1369. Which of the given numbers is greater than 1 and which is less than 1:

1370. On the basis of what property of the exponential function can one assert that

a) (5/7) 2.6 > (5/7) 2.5; b) (4/3) 1.3 > (4/3) 1.2

1371. Which number is greater:

but) π - √3 or (1 / π ) - √3; c) (2 / 3) 1 + √6 or (2 / 3) √2 + √5 ;

b) ( π / 4) 1 + √3 or ( π / 4) 2; d) (√3 ) √2 - √5 or (√3) √3 - 2 ?

1372. Are the inequalities equivalent:

1373. What can be said about numbers X And at , if a x = and y , where but is a given positive number?

1374. 1) Is it possible among all values of a function at = 2x highlight:

2) Is it possible among all function values at = 2 | x| highlight:

but) highest value; b) the smallest value?

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) = x.
Here a is fixed real number, which is called the base of the exponential function.
An exponential function with base a is also called exponential 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 the properties (1.5-8) (), which follow from the rules for multiplying numbers. At zero and negative values integers , the exponential function is determined by the formulas (1.9-10). For fractional values x = m/n rational numbers, , it is determined by formula (1.11). For real , 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 the properties (1.5-8), as well as for natural x .

A rigorous mathematical formulation of the definition of an exponential function and a 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) is defined and continuous, for , for all ;
(1.2) when 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
.
The formula for converting to an exponential function with a different power base:

For b = e , we get the expression of the exponential function in terms of the exponent:

Private values

, , , , .

The figure shows graphs of the exponential function
y (x) = x
for four values degree bases:a= 2 , a = 8 , a = 1/2 and a = 1/8 . It can be seen that for a > 1 exponential function is monotonically increasing. The larger the base of the degree a, the stronger the growth. At 0 < a < 1 exponential function is monotonically decreasing. How less indicator degree a , the stronger the decrease.

Ascending, descending

The exponential function at is strictly monotonic, so it has no extrema. Its main properties are presented in the table.

	y = a x , a > 1	y = x, 0 < a < 1
Domain	- ∞ < x < + ∞	- ∞ < x < + ∞
Range of values	0 < y < + ∞	0 < y < + ∞
Monotone	increases monotonically	decreases monotonically
Zeros, y= 0	No	No
Points of intersection with the y-axis, x = 0	y= 1	y= 1
	+ ∞	0
	0	+ ∞

Inverse function

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

If , then
.
If , then
.

Differentiation of the exponential function

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

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

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

We apply the rule of differentiation of a complex function. To do this, we introduce a 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
.
According to the rule of differentiation of a complex function:
.

Derivative of exponential function

.
Derivative of the nth order:
.
Derivation of formulas > > >

An example of differentiating an exponential function

Find the derivative of a function
y= 35 x

Solution

We express the base of the exponential function in terms of the number e.
3 = e log 3
Then
.
We introduce a variable
.
Then

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

Answer

Integral

Expressions in terms of complex numbers

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

.
The argument φ is not uniquely defined. IN general view
φ = φ 0 + 2 pn,
where n is an integer. Therefore, the function f (z) is also ambiguous. Often considered its main importance
.

Expansion in series

References:
I.N. Bronstein, K.A. Semendyaev, Handbook of Mathematics for Engineers and Students of Higher Educational Institutions, Lan, 2009.

The solution of most mathematical problems is somehow connected with the transformation of numerical, algebraic or functional expressions. This applies especially to the solution. In the USE variants in mathematics, this type of task includes, in particular, task C3. Learning how to solve C3 tasks is important not only for the purpose successful delivery Unified State Examination, but also for the reason that this skill is useful when studying a mathematics course in higher education.

Performing tasks C3, you have to decide different kinds 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 under the heading "" in articles devoted to methods for solving C3 problems from USE options mathematics.

Before proceeding to the analysis of specific exponential equations and inequalities, as a math tutor, I suggest you brush up on some theoretical material which we will need.

Exponential function

What is an exponential function?

View function y = a x, where a> 0 and a≠ 1, called exponential function.

Main exponential function properties y = a x:

Graph of an exponential function

The graph of the exponential function is exhibitor:

Graphs of exponential functions (exponents)

Solution of exponential equations

indicative called equations in which the unknown variable is found only in exponents of any powers.

For solutions 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 actions with degrees:

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Example 1 Solve the equation:

Solution: use the above formulas and substitution:

The equation then becomes:

Received discriminant quadratic equation positive:

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

Going back to 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 pass 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 area of admissible 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 on its domain). Then the equation takes the form:

Answer: x = 0.

Example 4 Solve the equation:

Solution: we simplify the equation to an elementary one by 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-2/3, standing on the right side of the equation, is decreasing. This means that if the graphs of these functions intersect, then at most at 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 equivalent transformations, bearing 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 partial powers given at the beginning of the article:

Answer: x = 2.

Solving exponential inequalities

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

For solutions 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 of the opposite meaning: f(x) < g(x).

Example 7 Solve the inequality:

Solution: represent the original inequality in the form:

Divide both parts of this inequality by 3 2 x, and (due to the positiveness of the function y= 3 2x) the inequality sign will not change:

Let's use a substitution:

Then the inequality takes the form:

So, the solution to the inequality is the interval:

passing to the reverse substitution, we get:

The left inequality, due to the positiveness of the exponential function, is fulfilled automatically. Taking advantage known property logarithm, we pass to the equivalent inequality:

Since the base of the degree is a number greater than one, equivalent (by Theorem 2) will be 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:

With this substitution, the inequality takes the form:

Multiply the numerator and denominator of the fraction by 7, we get the following equivalent inequality:

So the inequality is satisfied the following values variable t:

Then, going back to substitution, we get:

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

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 (because the exponential function is positive), so the inequality sign does not need to be changed. We get:

t , which are in the interval:

Passing 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, hence it is bounded from above by the value it reaches at its vertex:

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

At the same time, the function turns out to be bounded from below y = 3 x 2 -2x+2 on the right side of the equation. She reaches her the 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 the value 3 at one point (by the intersection the ranges of these functions is only this number). This condition is satisfied at a single point x = 1.

Answer: x= 1.

To learn how to solve exponential equations and inequalities, you need to constantly train in their solution. In this difficult matter, various teaching aids, problem books in elementary mathematics, collections of competitive problems, mathematics classes at school, as well as individual sessions with a professional tutor. I sincerely wish you success in your preparations and brilliant results on the exam.

Sergey Valerievich

P.S. Dear guests! Please do not write requests for solving your equations in the comments. Unfortunately, I don't have time for this at all. 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.

Find the value of the expression for various rational values of the variable x=2; 0; -3; -

Note, no matter what number we substitute instead of the variable x, you can always find the value of this expression. So, we are considering an exponential function (y equals three to the x power), defined on the set of rational numbers: .

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

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

Using the graph of this function, consider its properties:

3. Increases over the entire definition area.

range from zero to plus infinity.

8. The function is convex down.

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

Let's plot the function:

1. Compiling a table of its values.

We mark the obtained points on the coordinate plane.

Let's draw a smooth line passing through these points (Fig. 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 over the entire domain of definition.

4. Has neither the largest nor the smallest values.

5. Limited from below, but not limited from above.

6. Continuous over the entire domain of definition.

7. value range from zero to plus infinity.

8. The function is convex down.

Similarly, if in one coordinate system to build graphs of functions; y=(y equals one second to the x power, y equals one fifth to the x power, y equals one seventh to the x power), you can see that they have the same properties as y=(y equals one third to the power of x). x) (Fig. 4), that is, all functions of the form y \u003d (y is equal to one divided by a to the power of x, with a greater than zero but less than one) will have such properties

Let us 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 power of x and y equal to one divided by a to the power x) for the same value of a.

We summarize what has been said by giving a definition of an exponential function and indicating its main properties:

Definition: A function of the form y \u003d, where (y is equal to a to the power of 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 aurally and visually. The exponential function X is a degree, and power function X is the basis.

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

(y equals three to the power of x and y equals 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 given 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 \u003d (y is equal to five to the power of x and 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). Hence, the root of the equation is x \u003d -2 (number minus two).

Example 3: Solve the inequality

In one coordinate system, we construct two graphs of the function y \u003d

(y equals three to the power of x and y equals twenty-seven).

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

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 \u003d (y is equal to one fourth to the power of x and y is equal to sixteen). (Fig. 10). 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), because the graph of the function y \u003d is located below the graph of the function at x

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

Terem 1: If is true if and only if m=n.

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

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

Example 5: Plot the function y=

We modify the function by applying the degree property y=

Let's build additional system coordinates and in new system coordinates, we will plot the function y \u003d (y is equal to two to the power of x) Fig.11.

Example 6: Solve the equation

In one coordinate system, we construct two graphs of the function y \u003d

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

Graphs intersect at one point E (1; (e with coordinates one; seven). Hence, the root of the equation is x = 1 (x equal to one).

Example 7: Solve the inequality

In one coordinate system, we construct two graphs of the function y \u003d

(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 at, the solution to the inequality is the interval x (from minus one to plus infinity).