- (math.) A function y = f (x) is called even if it does not change when the independent variable only changes sign, that is, if f (x) = f (x). If f (x) = f (x), then the function f (x) is called odd. For example, y = cosx, y = x2... ...
F(x) = x is an example of an odd function. f(x) = x2 is an example of an even function. f(x) = x3 ... Wikipedia
A function satisfying the equality f (x) = f (x). See Even and Odd Functions... Great Soviet Encyclopedia
F(x) = x is an example of an odd function. f(x) = x2 is an example of an even function. f(x) = x3 ... Wikipedia
F(x) = x is an example of an odd function. f(x) = x2 is an example of an even function. f(x) = x3 ... Wikipedia
F(x) = x is an example of an odd function. f(x) = x2 is an example of an even function. f(x) = x3 ... Wikipedia
F(x) = x is an example of an odd function. f(x) = x2 is an example of an even function. f(x) = x3 ... Wikipedia
Special functions introduced by the French mathematician E. Mathieu in 1868 when solving problems on the oscillation of an elliptical membrane. M. f. are also used in studying the distribution electromagnetic waves in an elliptical cylinder... Great Soviet Encyclopedia
The "sin" request is redirected here; see also other meanings. The "sec" request is redirected here; see also other meanings. The "Sine" request is redirected here; see also other meanings... Wikipedia
The dependence of a variable y on a variable x, in which each value of x corresponds to a single value of y, is called a function. For designation use the notation y=f(x). Each function has a number of basic properties, such as monotonicity, parity, periodicity and others.
Take a closer look at the parity property.
A function y=f(x) is called even if it satisfies the following two conditions:
2. The value of the function at point x, belonging to the domain of definition of the function, must be equal to the value of the function at point -x. That is, for any point x, the following equality must be satisfied from the domain of definition of the function: f(x) = f(-x).
Graph of an even function
If you plot a graph of an even function, it will be symmetrical about the Oy axis.
For example, the function y=x^2 is even. Let's check it out. The domain of definition is the entire numerical axis, which means it is symmetrical about point O.
Let's take an arbitrary x=3. f(x)=3^2=9.
f(-x)=(-3)^2=9. Therefore f(x) = f(-x). Thus, both conditions are met, which means the function is even. Below is a graph of the function y=x^2.
The figure shows that the graph is symmetrical about the Oy axis.
Graph of an odd function
A function y=f(x) is called odd if it satisfies the following two conditions:
1. The domain of definition of a given function must be symmetrical with respect to point O. That is, if some point a belongs to the domain of definition of the function, then the corresponding point -a must also belong to the domain of definition of the given function.
2. For any point x, the following equality must be satisfied from the domain of definition of the function: f(x) = -f(x).
The graph of an odd function is symmetrical with respect to point O - the origin of coordinates. For example, the function y=x^3 is odd. Let's check it out. The domain of definition is the entire numerical axis, which means it is symmetrical about point O.
Let's take an arbitrary x=2. f(x)=2^3=8.
f(-x)=(-2)^3=-8. Therefore f(x) = -f(x). Thus, both conditions are met, which means the function is odd. Below is a graph of the function y=x^3.
The figure clearly shows that the odd function y=x^3 is symmetrical about the origin.
Back Forward
Attention! Slide previews are for informational purposes only and may not represent all the features of the presentation. If you are interested in this work, please download the full version.
Goals:
- form the concept of parity and oddness of a function, teach the ability to determine and use these properties when function research, plotting;
- develop creative student activity, logical thinking, ability to compare, generalize;
- cultivate hard work and mathematical culture; develop communication skills .
Equipment: multimedia installation, interactive whiteboard, handouts.
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.
PROGRESS OF THE LESSON
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 when 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 out the table | |||||
Domain of definition |
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) | graphs | 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
– While doing this work, guys, we identified another 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 number set along 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.
Mutual check on slide.
6. Homework: №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
Even function.
Even is a function whose sign does not change when the sign changes x.
x equality holds f(–x) = f(x). Sign x does not affect the sign y.
The graph of an even function is symmetrical about the coordinate axis (Fig. 1).
Examples of an even function:
y=cos x
y = x 2
y = –x 2
y = x 4
y = x 6
y = x 2 + x
Explanation:
Let's take the function y = x 2 or y = –x 2 .
For any value x the function is positive. Sign x does not affect the sign y. The graph is symmetrical about the coordinate axis. This is an even function.
Odd function.
Odd is a function whose sign changes when the sign changes x.
In other words, for any value x equality holds f(–x) = –f(x).
The graph of an odd function is symmetrical with respect to the origin (Fig. 2).
Examples of odd function:
y= sin x
y = x 3
y = –x 3
Explanation:
Let's take the function y = – x 3 .
All meanings at it will have a minus sign. That is a sign x influences the sign y. If the independent variable is a positive number, then the function is positive, if the independent variable is negative number, then the function is negative: f(–x) = –f(x).
The graph of the function is symmetrical about the origin. This is an odd function.
Properties of even and odd functions:
NOTE:
Not all functions are even or odd. There are functions that do not obey such gradation. For example, the root function at = √X does not apply to either even or odd functions (Fig. 3). When listing the properties of such functions, an appropriate description should be given: neither even nor odd.
Periodic functions.
As you know, periodicity is the repetition of certain processes at a certain interval. The functions that describe these processes are called periodic functions . That is, these are functions in whose graphs there are elements that repeat at certain numerical intervals.
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.
Function graph f is the set of all points (x;y) 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 find out some important characteristics of the function.
In general, analytical and graphic ways function assignments 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 specify 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 it’s easy to make a sign.
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 these functions are even and odd.
Definition. The 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. The 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?
