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Goals:
- formulate the concept of even and odd functions, teach the ability to determine and use these properties when studying functions and constructing graphs;
- develop students’ creative activity, logical thinking, ability to compare and generalize;
- cultivate hard work and mathematical culture; develop communication skills .
Equipment: multimedia installation, interactive board, Handout.
Forms of work: frontal and group with elements of search and research activities.
Information sources:
1. Algebra 9th class A.G. Mordkovich. Textbook.
2. Algebra 9th grade A.G. Mordkovich. Problem book.
3. Algebra 9th grade. Tasks for student learning and development. Belenkova E.Yu. Lebedintseva E.A.
DURING THE CLASSES
1. Organizational moment
Setting goals and objectives for the lesson.
2. Checking homework
No. 10.17 (9th grade problem book. A.G. Mordkovich).
A) at = f(X), f(X) = 
b) f (–2) = –3; f (0) = –1; f(5) = 69;
c) 1. D( f) = [– 2; + ∞)
2. E( f) = [– 3; + ∞)
3. f(X) = 0 at X ~ 0,4
4. f(X) >0 at X > 0,4 ; f(X)
< 0 при – 2 <
X <
0,4.
5. The function increases with X € [– 2; + ∞)
6. The function is limited from below.
7. at naim = – 3, at naib doesn't exist
8. The function is continuous.
(Have you used a function exploration algorithm?) Slide.
2. Let’s check the table you were asked from the slide.
| Fill the table | |||||
Domain |
Function zeros |
Intervals of sign constancy |
Coordinates of the points of intersection of the graph with Oy | ||
 |
x = –5, |
x € (–5;3) U |
x € (–∞;–5) U |
||
 |
x ∞ –5, |
x € (–5;3) U |
x € (–∞;–5) U |
||
x ≠ –5, |
x € (–∞; –5) U |
x € (–5; 2) |
|||
3. Updating knowledge
– Functions are given.
– Specify the scope of definition for each function.
– Compare the value of each function for each pair of argument values: 1 and – 1; 2 and – 2.
– For which of these functions in the domain of definition the equalities hold f(– X)
= f(X), f(– X) = – f(X)? (enter the obtained data into the table) Slide
| f(1) and f(– 1) | f(2) and f(– 2) | graphics | f(– X) = –f(X) | f(– X) = f(X) | ||
| 1. f(X) = | ||||||
| 2. f(X) = X 3 | ||||||
| 3. f(X) = | X | | ||||||
| 4.f(X) = 2X – 3 | ||||||
| 5. f(X) = | X ≠ 0 |
|||||
| 6. f(X)= | X > –1 | and not defined |
4. New material
– Carrying out this work, guys, we have identified one more property of the function, unfamiliar to you, but no less important than the others - this is the evenness and oddness of the function. Write down the topic of the lesson: “Even and odd functions”, our task is to learn to determine the evenness and oddness of a function, to find out the significance of this property in the study of functions and plotting graphs.
So, let's find the definitions in the textbook and read (p. 110) . Slide
Def. 1 Function at = f (X), defined on the set X is called even, if for any value XЄ X is executed equality f(–x)= f(x). Give examples.
Def. 2 Function y = f(x), defined on the set X is called odd, if for any value XЄ X the equality f(–х)= –f(х) holds. Give examples.
Where did we meet the terms “even” and “odd”?
Which of these functions will be even, do you think? Why? Which ones are odd? Why?
For any function of the form at= x n, Where n– an integer, it can be argued that the function is odd when n– odd and the function is even when n– even.
– View functions at= and at = 2X– 3 are neither even nor odd, because equalities are not satisfied f(– X) = – f(X), f(–
X) = f(X)
The study of whether a function is even or odd is called the study of a function's parity. Slide
In definitions 1 and 2 we were talking about the values of the function at x and – x, thereby it is assumed that the function is also defined at the value X, and at – X.
Def 3. If a numerical set, together with each of its elements x, also contains the opposite element –x, then the set X called a symmetric set.
Examples:
(–2;2), [–5;5]; (∞;∞) are symmetric sets, and , [–5;4] are asymmetric.
– Do even functions have a domain of definition that is a symmetric set? The odd ones?
– If D( f) is an asymmetric set, then what is the function?
– Thus, if the function at = f(X) – even or odd, then its domain of definition is D( f) is a symmetric set. Is the converse statement true: if the domain of definition of a function is a symmetric set, then is it even or odd?
– This means that the presence of a symmetric set of the domain of definition is a necessary condition, but not sufficient.
– So how do you examine a function for parity? Let's try to create an algorithm.
Slide
Algorithm for studying a function for parity
1. Determine whether the domain of definition of the function is symmetrical. If not, then the function is neither even nor odd. If yes, then go to step 2 of the algorithm.
2. Write an expression for f(–X).
3. Compare f(–X).And f(X):
- If f(–X).= f(X), then the function is even;
- If f(–X).= – f(X), then the function is odd;
- If f(–X) ≠ f(X) And f(–X) ≠ –f(X), then the function is neither even nor odd.
Examples:
Examine function a) for parity at= x 5 +; b) at= ; V) at= .
Solution.
a) h(x) = x 5 +,
1) D(h) = (–∞; 0) U (0; +∞), symmetric set.
2) h (– x) = (–x) 5 + – x5 –= – (x 5 +),
3) h(– x) = – h (x) => function h(x) = x 5 + odd.
b) y =,
at = f(X), D(f) = (–∞; –9)? (–9; +∞), an asymmetric set, which means the function is neither even nor odd.
V) f(X) = , y = f (x),
1) D( f) = (–∞; 3] ≠ ; b) (∞; –2), (–4; 4]?
Option 2
1. Is the given set symmetric: a) [–2;2]; b) (∞; 0], (0; 7) ?
A); b) y = x (5 – x 2).
a) y = x 2 (2x – x 3), b) y =
Graph the Function at = f(X), If at = f(X) is an even function.
Graph the Function at = f(X), If at = f(X) is an odd function.
Peer review on the slide.
6. Homework: No. 11.11, 11.21, 11.22;
Proof of the geometric meaning of the parity property.
***(Assignment of the Unified State Examination option).
1. The odd function y = f(x) is defined on the entire number line. For any non-negative value of the variable x, the value of this function coincides with the value of the function g( X) = X(X + 1)(X + 3)(X– 7). Find the value of the function h( X) = at X = 3.
7. Summing up
Converting graphs.
Verbal description of the function.
Graphic method.
The graphical method of specifying a function is the most visual and is often used in technology. IN mathematical analysis The graphical method of specifying functions is used as an illustration.
The graph of a function f is the set of all points (x;y) of the coordinate plane, where y=f(x), and x “runs through” the entire domain of definition of this function.
A subset of the coordinate plane is a graph of a function if it has at most one common point from any straight line parallel to the Oy axis.
Example. Are the figures shown below graphs of functions?
The advantage of a graphic task is its clarity. You can immediately see how the function behaves, where it increases and where it decreases. From the graph you can immediately recognize some important characteristics functions.
In general, analytical and graphical methods of defining a function go hand in hand. Working with the formula helps to build a graph. And the graph often suggests solutions that you wouldn’t even notice in the formula.
Almost any student knows the three ways to define a function that we just looked at.
Let's try to answer the question: "Are there other ways to define a function?"
There is such a way.
The function can be quite unambiguously specified in words.
For example, the function y=2x can be specified by the following verbal description: each real value of the argument x is associated with its double value. The rule is established, the function is specified.
Moreover, you can verbally specify a function that is extremely difficult, if not impossible, to define using a formula.
For example: each value of the natural argument x is associated with the sum of the digits that make up the value of x. For example, if x=3, then y=3. If x=257, then y=2+5+7=14. And so on. It is problematic to write this down in a formula. But the sign is easy to make.
The method of verbal description is a rather rarely used method. But sometimes it does.
If there is a law of one-to-one correspondence between x and y, then there is a function. What law, in what form it is expressed - a formula, a tablet, a graph, words - does not change the essence of the matter.
Let us consider functions whose domains of definition are symmetrical with respect to the origin, i.e. for anyone X from the domain of definition number (- X) also belongs to the domain of definition. Among such functions, even and odd are distinguished.
Definition. A function f is called even if for any X from its domain of definition
Example. Consider the function 
It is even. Let's check it out.
For anyone X equalities are satisfied
Thus, both conditions are met, which means the function is even. Below is a graph of this function.
Definition. A function f is called odd if for any X from its domain of definition 
Example. Consider the function 
It is odd. Let's check it out.
The domain of definition is the entire numerical axis, which means it is symmetrical about the point (0;0).
For anyone X equalities are satisfied
Thus, both conditions are met, which means the function is odd. Below is a graph of this function.
The graphs shown in the first and third figures are symmetrical about the ordinate axis, and the graphs shown in the second and fourth figures are symmetrical about the origin.
Which of the functions whose graphs are shown in the figures are even and which are odd?
How to insert mathematical formulas on a website?
If you ever need to add one or two mathematical formulas to a web page, then the easiest way to do this is as described in the article: mathematical formulas are easily inserted onto the site in the form of pictures that are automatically generated by Wolfram Alpha. Besides simplicity, this universal method will help improve website visibility search engines. It has been working for a long time (and, I think, will work forever), but is already morally outdated.
If you regularly use mathematical formulas on your site, then I recommend you use MathJax - a special JavaScript library that displays mathematical notation in web browsers using MathML, LaTeX or ASCIIMathML markup.
There are two ways to start using MathJax: (1) using a simple code, you can quickly connect a MathJax script to your website, which will be automatically loaded from a remote server at the right time (list of servers); (2) download the MathJax script from a remote server to your server and connect it to all pages of your site. The second method - more complex and time-consuming - will speed up the loading of your site's pages, and if the parent MathJax server becomes temporarily unavailable for some reason, this will not affect your own site in any way. Despite these advantages, I chose the first method as it is simpler, faster and does not require technical skills. Follow my example, and in just 5 minutes you will be able to use all the features of MathJax on your site.
You can connect the MathJax library script from a remote server using two code options taken from the main MathJax website or on the documentation page:
One of these code options needs to be copied and pasted into the code of your web page, preferably between tags and or immediately after the tag. According to the first option, MathJax loads faster and slows down the page less. But the second option automatically monitors and loads the latest versions of MathJax. If you insert the first code, it will need to be updated periodically. If you insert the second code, the pages will load more slowly, but you will not need to constantly monitor MathJax updates.
The easiest way to connect MathJax is in Blogger or WordPress: in the site control panel, add a widget designed to insert third-party JavaScript code, copy the first or second version of the download code presented above into it, and place the widget closer to the beginning of the template (by the way, this is not at all necessary , since the MathJax script is loaded asynchronously). That's all. Now learn the markup syntax of MathML, LaTeX, and ASCIIMathML, and you are ready to insert mathematical formulas into your site's web pages.
Any fractal is constructed according to a certain rule, which is applied sequentially an unlimited number of times. Each such time is called an iteration.
The iterative algorithm for constructing a Menger sponge is quite simple: the original cube with side 1 is divided by planes parallel to its faces into 27 equal cubes. One central cube and 6 cubes adjacent to it along the faces are removed from it. The result is a set consisting of the remaining 20 smaller cubes. Doing the same with each of these cubes, we get a set consisting of 400 smaller cubes. Continuing this process endlessly, we get a Menger sponge.
Function is one of the most important mathematical concepts. A function is the dependence of the variable y on the variable x, if each value of x corresponds to a single value of y. The variable x is called the independent variable or argument. The variable y is called the dependent variable. All values of the independent variable (variable x) form the domain of definition of the function. All values that the dependent variable (variable y) takes form the range of the function.
The graph of a function is the set of all points of the coordinate plane, the abscissas of which are equal to the values of the argument, and the ordinates are equal to the corresponding values of the function, that is, the values of the variable x are plotted along the abscissa axis, and the values of the variable y are plotted along the ordinate axis. To graph a function, you need to know the properties of the function. The main properties of the function will be discussed below!
To build a graph of a function, we recommend using our program - Graphing functions online. If you have any questions while studying the material on this page, you can always ask them on our forum. Also on the forum they will help you solve problems in mathematics, chemistry, geometry, probability theory and many other subjects!
Basic properties of functions.
1) The domain of definition of the function and the range of values of the function.
The domain of a function is the set of all valid real values of the argument x (variable x) for which the function y = f(x) is defined.
The range of a function is the set of all real y values that the function accepts.
IN elementary mathematics functions are studied only on the set of real numbers.
2) Zeros of the function.
Values of x for which y=0 are called function zeros. These are the abscissas of the points of intersection of the function graph with the Ox axis.
3) Intervals of constant sign of a function.
Intervals of constant sign of a function - such intervals of values x on which the values of the function y are either only positive or only negative are called intervals of constant sign of the function.
4) Monotonicity of the function.
An increasing function (in a certain interval) is a function in which a larger value of the argument from this interval corresponds to a larger value of the function.
A decreasing function (in a certain interval) is a function in which a larger value of the argument from this interval corresponds to a smaller value of the function.
5) Evenness (oddness) of the function.
An even function is a function whose domain of definition is symmetrical with respect to the origin and for any x f(-x) = f(x). The graph of an even function is symmetrical about the ordinate.
An odd function is a function whose domain of definition is symmetrical with respect to the origin and for any x from the domain of definition the equality f(-x) = - f(x) is true. The graph of an odd function is symmetrical about the origin.
Even function
1) The domain of definition is symmetrical with respect to the point (0; 0), that is, if point a belongs to the domain of definition, then point -a also belongs to the domain of definition.
2) For any value x f(-x)=f(x)
3) The graph of an even function is symmetrical about the Oy axis.
An odd function has the following properties:
1) The domain of definition is symmetrical about the point (0; 0).
2) for any value x belonging to the domain of definition, the equality f(-x)=-f(x) is satisfied
3) The graph of an odd function is symmetrical with respect to the origin (0; 0).
Not every function is even or odd. Functions general view are neither even nor odd.
6) Limited and unlimited functions.
A function is called bounded if there is a positive number M such that |f(x)| ≤ M for all values of x. If such a number does not exist, then the function is unlimited.
7) Periodicity of the function.
A function f(x) is periodic if there is a non-zero number T such that for any x from the domain of definition of the function the following holds: f(x+T) = f(x). This smallest number is called the period of the function. All trigonometric functions are periodic. (Trigonometric formulas).
A function f is called periodic if there is a number such that for any x from the domain of definition the equality f(x)=f(x-T)=f(x+T) holds. T is the period of the function.
Every periodic function has an infinite number of periods. In practice, the smallest positive period is usually considered.
The values of a periodic function are repeated after an interval equal to the period. This is used when constructing graphs.
