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The base of the exponential function must be. Exponential function, its properties and graph - Knowledge Hypermarket

EXPONENTARY AND LOGARITHMIC FUNCTIONS VIII

§ 179 Basic properties of the exponential function

In this section we will study the basic properties of the exponential function

y = a x (1)

Let us remember that under A in formula (1) we mean any fixed positive number, different from 1.

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

In fact, with a positive A expression A x defined for any real number X .

Property 2. Exponential function takes only positive values.

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

A x > 0.

If X <. 0, то

A x =

Where - X already more than zero. That's why A - x > 0. But then

A x = > 0.

Finally, when X = 0

A 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 abscissa axis.

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

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

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

Let us give a rigorous proof of the 3rd property. Let A > 1 and X - an arbitrary positive number. Let's show that

A x > 1.

If the number X rational ( X = m / n ) , That A x = A m/ n = n √a m .

Since A > 1, then A 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 a number x :

X"< х < х" .

But then, by definition of a degree with an irrational exponent

A x" < A x < A x"" .

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

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

A x > 1.

If the number X was negative, then we would have

A x =

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

A x = < 1.

Thus, when A > 1 and arbitrary negative x

A x < 1.

The case when 0< A < 1, легко сводится к уже рассмотренному случаю. Учащимся предлагается убедиться в этом самостоятельно.

Property 4. If x = 0, then regardless of a A 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 A curve at = A x (see Fig. 246 and 247) intersects the axis at at a point with ordinate 1.

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

This property also allows for a simple geometric interpretation.

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

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

Let A > 1 and X 2 > X 1. Let's show that

A x 2 > A x 1

Since X 2 > X 1 ., then X 2 = X 1 + d , Where d - some positive number. That's why

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

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

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

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

In other words, if

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

b = c .

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

Property 6. If a > 1, then with an unlimited increase in the argument X (X -> ∞ ) function values at = A x also grow indefinitely (at -> ∞ ). When the argument decreases without limit X (X -> -∞ ) the values of this function tend to zero while remaining positive (at->0; at > 0).

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

If 0 <A < 1, then with an unlimited increase in the argument x (x -> ∞), the values of the function y = a x tend to zero, while remaining positive (at->0; at > 0). When the argument x decreases without limit (X -> -∞ ) the values of this function grow unlimitedly (at -> ∞ ).

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

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

All we have to do is establish the range of variation of the exponential function y = a x (A > 0, A =/= 1).

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

A x 0 = at 0 .

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

Proving this fact is beyond the scope of our program. Its geometric interpretation is that for any positive value at 0 function graph y = a x will definitely intersect with a straight line at = at 0 and, moreover, only at one point (Fig. 248).

From this we can draw the following conclusion, which we formulate as property 7.

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

Exercises

1368. Find the domains of definition of the following functions:

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

1370. Based on what property of the exponential function can it be stated that

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

1371. Which number is greater:

A) 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 A - a given positive number?

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

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

a) the most higher value; b) the smallest value?

Focus:

Definition. Function species is called exponential function .

Comment. Exclusion from base values a numbers 0; 1 and negative values a is explained by the following circumstances:

The analytical expression itself a x in these cases, it retains its meaning and can be used in solving problems. For example, for the expression x y dot x = 1; y = 1 is within the range of acceptable values.

Construct graphs of functions: and.



Graph of an Exponential Function
y= a x, a > 1	y= a x , 0< a < 1

Properties of the Exponential Function

Properties of the Exponential Function	y= a x, a > 1	y= a x , 0< a < 1
Function Domain
2. Function range
3. Intervals of comparison with unit	at x> 0, a x > 1	at x > 0, 0< a x < 1
	at x < 0, 0< a x < 1	at x < 0, a x > 1
4. Even, odd.	The function is neither even nor odd (a function of general form).
5.Monotony.	monotonically increases by R	decreases monotonically by R
6. Extremes.	The exponential function has no extrema.
7.Asymptote	O-axis x is a horizontal asymptote.
8. For any real values x And y;

When the table is filled out, tasks are solved in parallel with the filling.

Task No. 1. (To find the domain of definition of a function).

What argument values are valid for functions:

Task No. 2. (To find the range of values of a function).

The figure shows the graph of the function. Specify the domain of definition and range of values of the function:

Task No. 3. (To indicate the intervals of comparison with one).

Compare each of the following powers with one:

Task No. 4. (To study the function for monotonicity).

Compare by size real numbers m And n If:

Task No. 5. (To study the function for monotonicity).

Draw a conclusion regarding the basis a, If:

y(x) = 10 x ; f(x) = 6 x ; z(x) - 4 x

How are the graphs of exponential functions relative to each other for x > 0, x = 0, x< 0?

The following function graphs are plotted in one coordinate plane:

y(x) = (0,1) x ; f(x) = (0.5) x ; z(x) = (0.8) x .

How are the graphs of exponential functions relative to each other for x > 0, x = 0, x< 0?

Number one of the most important constants in mathematics. By definition, it equal to the limit of the sequence with unlimited increasing n . Designation e entered Leonard Euler in 1736. He calculated the first 23 digits of this number in decimal notation, and the number itself was named in honor of Napier “the non-Pier number.”

Number e plays a special role in mathematical analysis. Exponential function with base e, called exponent and is designated y = e x.

First signs numbers e easy to remember: two, comma, seven, year of birth of Leo Tolstoy - two times, forty-five, ninety, forty-five.

Homework:

Kolmogorov paragraph 35; No. 445-447; 451; 453.

Repeat the algorithm for constructing graphs of functions containing a variable under the modulus sign.

Lesson No.2

Topic: Exponential function, its properties and graph.

Target: Check the quality of mastering the concept of “exponential function”; to develop the skills and abilities to recognize an exponential function, to use its properties and graphs, to teach students to use analytical and graphical forms of writing an exponential function; 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

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 quite difficult work: We would have to take the cube root of 3 and the cube root of 9 and compare them. But we know that it increases, this in 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)

- 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 greater 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.

What special point exists in the graph of an exponential function?

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?

Why are exponential functions with for different reasons have only one intersection point?

Exponential functions are strictly monotonic throughout their entire 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) . Let us remember that 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

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 minimum value, but it has the greatest value at the point . Points b) , V) , G) Solve the notebooks yourself, we will check them orally.