Outline of the lesson ringing function graph wrapped functions. Mutually wrapped up functions. Note by recording
We also stuck to the factory, if, according to the given function f and the given value of the argument, it is necessary to calculate the value of the function in the whole point. Ale inodi is communicated to the zealous employees: to know the function of the deyak in the meaning of the argument, in which function is given the meaning of y.
The function, yak, accepts its own meaning in one point of its domain of value, is called the reverse function. For example, the line function will be reverse function... And the quadratic function or the function of the sinus of nothing is ochіkuvati are werewolf functions. So as one and the same meaning of the function can be adopted with different arguments.
ringing function
Okay, so f є deyaka is quite a turnover function. The skin number in the area has the value of y0, it seems like only one number in the area of the value x0, and f (x0) = y0.
Since now the skin value x0 is put at the similarity of the value y0, then we can accept a new function. For example, for the linear function f (x) = k * x + b, the function g (x) = (x - b) / k will be revolving.
Yaksho deyaka function g in the skin point NS the region of the value of the turnover function f takes the value of such that f (y) = x, then it seems that the function g- є ringing function to f.
As soon as we will have a graph of the reverse function f, then in order to have a graph of the wrapped function, we can admit to such firm: the graph of the function f and the opposite function g will be symmetrical in the back.
If the function g is a reverse function before the function f, then the function g will be a reverse function. And the function f will be viable up to the function g. Name it seems that two functions f and g are mutually ringing one to one.
On the offensive little one, there are graphs of functions f and g, one to one.
In vivedemo, I will come to the theorem: if the function f grows (or decreases) for any advance A, then it is reverse. The function of g, which is effective in the area of significance of the function f, also has an increasing (or depending on changes) function. A theorem is given to be called the theorem about the wrapped function.
Meta lesson:
coverage:
- formvati knowledge z news by those up to software material;
- to gain the power of the turnover function and the knowledge of the function, the turnover of the given;
developing:
- development of self-control tips, subject matter;
- opanuvati for the understanding the funniest function and learn the methods of knowing the wrapped function;
Vikhovna: formulate communication competence.
ustatkuvannya: computer, projector, screen, SMART Board interactive board, distribution material ( robot self) For robots in a group.
Go to lesson.
1. Organizational moment.
meta – training of scholars to robotics at the lessons:
Viznachennya vіdsutnіkh,
Setting up scientists for a robot, organizing respect;
By those and the lesson.
2. Actualization of basic knowledge of scientists. Frontal experience.
meta - Establish the correctness and comprehension of the theoretical material, the repetition of the passed material.<Приложение 1 >
For scholars, a graph of functions is demonstrated on interactive education. The teacher is formulated zavdannya - look at the graph of the function and overhaul the power of the function. Learning to overhaul the power of the function at the basis of the scheme of further development. The right-handed teacher uses a marker on the graph of functions to write down the names of the authorities.
Powerful functions:
When the teacher is finished, the teacher will come to learn more about one power function - werewolves. For the comprehension of the introduction of new material, the teacher will teach the lads to learn about the main meals, on which scientists are responsible for the date of the end of the lesson. Nutrition recorded on special dosages and in the visual distribution material є at the skin examiner (moon before the lesson)
- Is the function called a reverse function?
- Whether the function is reversible?
- Is the function called a zorotnoy given?
- Yak knitted the area of \ u200b \ u200bvalue and how much is the value of the function and the rotary function?
- How is the function given analytically, how is the function given by the formula?
- How is the function set graphically, how can the graphical function be set?
3. Explanation of the new material.
meta - formulate knowledge of a new topic, depending on the program material; to gain the power of the turnover function and the knowledge of the function, the turnover of the given; razvivati objektnu mov.
The teacher should conduct a cycle of material up to the material of the paragraph. At the Interactive School, the teacher should carry out the adjustment of graphs in two functions, in some areas the meaning and many meanings are the same, but one of the functions is monotonous, but it is not, we ourselves give the function of the scientist to understand the turnover.
Let the teacher use the formulas for the value of the circulating function and carry out the proof of the theorems about the circulating function, the vikorist graph of the monotonous function on the interactive information.
Value 1: The function y = f (x), x X is called circulating, If you have your own meaning, you will receive only in one point without X.
Theorem: If the function y = f (x) is monotone on no X, then it is reverse.
Delivered:
- what a function y = f (x) growth on NS i nay x 1 ≠ x 2- two points without NS.
- For value high x 1<
x 2.
Todi z that, scho x 1< x 2 viplyaє, scho f (x 1) < f (x 2). - With such a rank, different meanings, the argument is given a different meaning of the function, so that the function is reversed.
(In the course of the brought theorems, the teacher should use a marker to rob all the necessary explanations on the chair)
Before the time you formulate the value of the revolving function, the teacher should ask the scholars for the value, as from the proponated functions of the werewolf? On the interactive screen, graphs of functions and recordings of similarly set functions are shown:
B) 
G) y = 2x + 5
D) y = -x 2 + 7
The teacher will introduce the value of the ringing function.
Business value 2: Come on reverse function y = f (x) marked on without NSі E (f) = Y... Supportive for skin y s Y then є single value NS, for what f (x) = y. Todi otrimaєmo function, yak is marked on Y, a NS- area of value
Qiu function to mean x = f -1 (y)і called ringing according to the relation to the function y = f (x).
Scientists will learn about the development of visnovoks about the connection between the area of design and the meaningless meaning of the functions.
To see the food about the ways of knowing the function of the zootnye given, the teacher screwed in two scholars. The guys in advance took away the teacher’s independent development of the analytical and graphic way of knowing the function of the oral data. The teacher has played the role of a consultant in the preparation of students before the lesson.
Povidomlennya the first scholar.
Respect: monotony of function, є sufficient mindful of the wrapped function. ale wono not є necessary mind.
A scholar in the butt of new situations, if the function is not monotonous, but reversed, if the function is not monotonous and not reversed, if monotonous and reversed
Let us learn to know the science in the way of knowing the enveloped function, given analytically.
knowledge algorithm
- Perekonatysya, scho the function is monotonous.
- Visloviti zminnu x through y.
- Redesignating zminnі. Change x = f -1 (y) write y = f -1 (x)
Let us put two butts on the well-known func-
Butt 1: Demonstrate, for the function y = 5x-3 іnu vivid function, і know її analytical viraz.
Decision. The line function y = 5x-3 is assigned to R, the growth is to R, and the region is to R and the region to the value є R. So, the ringing function is to R. It is important to know the analytic viraz, the virgin is equal to y = 5x-3 to x; function. Vaughn is assigned і growth to R.
Butt 2: Show that for the function y = x 2, x≤0 is a healthy function, and to know an analytical viraz.
The function is uninterrupted, monotonous in its area of significance, and it is also reversible. Having analyzed the areas of significance and many of the significance of the function, you should try to get a general idea of the analogical rotation for the oral function.
Another student will appear on account of graphic ways of knowing the wrapped function. In the course of his explanation, he taught the vicarist mobility of the interactive board.
To correct the graph of the function y = f -1 (x), which is equivalent to the function y = f (x), the graph of the function y = f (x) must be reconfigured symmetrically to the straight line y = x.
For an hour of explanation on the interactive presentation
Stay in the same coordinate system the graph of the function and the graph of the vocal function. Write down the analytic view of the wrapped function.
4. Initial registration of new material.
meta - Establish the correctness and assimilation of the understanding of the material, determine the clearing of the original understanding of the material, carry out the correction.
Learn to bet. Їm moonlit the leaves iz zavdannymi, in which stench and vikonuyut robot in pairs. The hour for the visit of the robot is interrupted (5-7 min). One pair of scientists work on computers, the projector is visible for a whole hour, and the lads cannot see it, as they work on computers.
At the end of the hour (to transfer, when a large number of scientists have entered into the robot), the robot will appear on the interactive screen (the projector will turn on again), and the robot will appear in the course of the change in the correctness of the guy. If necessary, the teacher will carry out a correction, which will explain the robot.
Self-operation of the robot in pairs<Dodatok 2 >
5. Take a lesson. Food, like boules, set before the cob of lectures. Deafening estimates for the lesson.
Homework §10. No. 10.6 (a, c) 10.8-10.9 (b) 10.12 (b)
Algebra and cobs analysis. 10 class In 2 parts for the educational setting (professional level) / A.G. Mordkovich, L.O.Denishcheva, T.A. Koreshkova and I.; ed ed. A.G. Mordkovich, Moscow: Mnemosyna, 2007
What kind of fun is it? How can I know the function, the sound of the given?
Viznachennya.
Let the function y = f (x) be assigned to the set D, and E is the valueless value. Sale function up to function y = f (x) - the whole function x = g (y), which is assigned to the set E and skin y∈E, put the same value as x∈D, but f (x) = y.
Thus, the area of the value of the function y = f (x) is the area of the value of the rotary to non-functional function, and the area of the value of y = f (x) is the area of the value of the wrapped function.
To know the function that is given to the function y = f (x), :
1) The formula of the function is substitute y for submission x, substitute for x - y:
2) 3 of the disclaimer of visibility y through x:
Know the function, the rotary function y = 2x-6.
Functions y = 2x-6 і y = 0.5x + 3 є are reversed.
Graphs of direct and rotary functions symmetric with respect to straight lines y = x(Bisectors I and III coordinate parts).
y = 2x-6 і y = 0.5x + 3 -. Graph of line functions є. We take two points to induce straightforwardness.
Unambiguously, the presence of y through x is possible in the same way, if the equal x = f (y) has the same solution. The price can be changed in the same way, as the skin-specific function y = f (x) is accepted in one point of the value domain (this function is called circulating).
Theorem (necessary and sufficient mind and turnover function)
While the function y = f (x) is assigned and continuous for a numerical interval, then for the reversibility of the function it is necessary and sufficient, but f (x) was strictly monotonic.
Moreover, if y = f (x) grows for an interval, then і is turned up to its function also grows for a whole interval; if y = f (x) is decreasing, then it is a rotary decay function.
As far as the turnover rate is concerned, it is not possible to see the prominence, the de function is only growing, or only ubuvay, and for all the time to know the function, the turn of the given.
Classic butt -. On the way)