How to investigate a function for parity. Even and odd functions

- ; an even function is called when for any two different values of its argument f (x) =f(x) , for example, y= |x|; odd - such a function when f (x) \u003d - f (x), for example, y \u003d x2n + 1, where n ... ... Economic and Mathematical Dictionary

even and odd functions- An even function is called when for any two different values of its argument f (x) =f(x) , for example, y= |x|; such a function is odd when f(x) = f(x), for example, y= x2n+1, where n is any natural number. Functions that are neither... Technical Translator's Handbook

PARITY- a quantum number that characterizes the symmetry of the wave function of a physical system or an elementary particle under some discrete transformations: if under such a transformation? does not change sign, then the parity is positive, if it changes, then the parity ... ... Big Encyclopedic Dictionary

LEVEL PARITY- parity of the state of physical. system (wave parity. functions) corresponding to a given energy level. Such a characterization of levels is possible for a system h c, between which el. magn. or poison. parity-preserving forces. Taking into account the weak interaction ... ... Physical Encyclopedia

Parity

Parity (mathematics)- Parity in number theory is the ability of an integer to be divided without a remainder by 2. The parity of a function in mathematical analysis determines whether the function changes sign when the sign of the argument changes: for an even / odd function. Parity in quantum mechanics ... ... Wikipedia

TRIGONOMETRIC FUNCTIONS- class of elementary functions: sine, cosine, tangent, cotangent, secant, cosecant. Designated accordingly: sin x, cos x, tg x, ctg x, sec x, cosec x. Trigonometric functions of a real argument. Let A be a point of a circle centered at ... ... Mathematical Encyclopedia

INTERNAL PARITY- (P), one of the characteristics of (quantum numbers) elements. tsy, which determines the behavior of its wave function y during spatial inversion (mirror reflection), i.e., when the coordinates x® x, y® y, z® z are changed. If, with such a reflection, y does not change sign, V. h. h tsy ... ... Physical Encyclopedia

Charge parity- Charge conjugation is the operation of replacing a particle with an antiparticle (for example, an electron with a positron). Charge parity Charge parity is a quantum number that determines the behavior of the wave function of a particle during the operation of replacing a particle with an antiparticle ... ... Wikipedia

Cyclic parity check- Algorithm for calculating the checksum (English Cyclic redundancy code, CRC cyclic redundancy code) is a method of digitally identifying a certain data sequence, which consists in calculating the control value of its cyclic ... ... Wikipedia

- (Math.) A function y \u003d f (x) is called even if it does not change when the independent variable only changes sign, that is, if f (x) \u003d f (x). If f (x) = f (x), then the function f (x) is called odd. For example, y \u003d cosx, y \u003d 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 that satisfies 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

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 the study of the propagation of 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. "Sine" redirects here; see also other meanings ... Wikipedia

The dependence of the variable y on the variable x, in which each value of x corresponds to a single value of y is called a function. The notation is y=f(x). Each function has a number of basic properties, such as monotonicity, parity, periodicity, and others.

Consider the parity property in more detail.

A function y=f(x) is called even if it satisfies the following two conditions:

2. The value of the function at the point x belonging to the scope of the function must be equal to the value of the function at the point -x. That is, for any point x, from the domain of the function, the following equality f (x) \u003d f (-x) must be true.

Graph of an even function

If you build a graph of an even function, it will be symmetrical about the y-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 that it is symmetrical about the point O.

Take an arbitrary x=3. f(x)=3^2=9.

f(-x)=(-3)^2=9. Therefore, f(x) = f(-x). Thus, both conditions are satisfied for us, which means that the function is even. Below is a graph of the function y=x^2.

The figure shows that the graph is symmetrical about the y-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 the given function must be symmetrical with respect to the point O. That is, if some point a belongs to the domain of the function, then the corresponding point -a must also belong to the domain of the given function.

2. For any point x, from the domain of the function, the following equality f (x) \u003d -f (x) must be satisfied.

The graph of an odd function is symmetrical with respect to the point O - the origin. 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 that it is symmetrical about the point O.

Take an arbitrary x=2. f(x)=2^3=8.

f(-x)=(-2)^3=-8. Therefore f(x) = -f(x). Thus, both conditions are satisfied for us, which means that 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 with respect to the origin.

Back forward

Attention! The slide preview is for informational purposes only and may not represent the full extent of the presentation. If you are interested in this work, please download the full version.

Goals:

to form the concept of even and odd functions, to teach the ability to determine and use these properties in the study of functions, plotting graphs;
to develop the creative activity of students, logical thinking, the ability to compare, generalize;
to cultivate diligence, 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 class 9 A.G. Mordkovich. Textbook.
2. Algebra Grade 9 A.G. Mordkovich. Task book.
3. Algebra grade 9. Tasks for learning and development of students. Belenkova E.Yu. Lebedintseva E.A.

DURING THE CLASSES

1. Organizational moment

Setting goals and objectives of the lesson.

2. Checking homework

No. 10.17 (Problem book 9th grade 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 for 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 hire = - 3, at naib doesn't exist
8. The function is continuous.

(Did you use the feature exploration algorithm?) Slide.

2. Let's check the table that you were asked on the slide.

Fill the table
	Domain	Function zeros	Constancy intervals		Coordinates of the points of intersection of the graph with Oy
	Domain	Function zeros
		x = -5, x = 2	х € (–5;3) U U(2;∞)	х € (–∞;–5) U U (–3;2)
	x ∞ -5, x ≠ 2		х € (–5;3) U U(2;∞)	х € (–∞;–5) U U (–3;2)
	x ≠ -5, x ≠ 2		x € (–∞; –5) U U(2;∞)	x € (–5; 2)

3. Knowledge update

– Functions are given.
– Specify the domain 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 the given functions in the domain of definition are the equalities f(– X) = f(X), f(– X) = – f(X)? (put the data in the table) Slide

		f(1) and f(– 1)	f(2) and f(– 2)	charts	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 have revealed 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 how to determine the even and odd functions, find out the significance of this property in the study of functions and plotting.
So, let's find the definitions in the textbook and read (p. 110) . Slide

Def. one Function at = f (X) defined on the set X is called even, if for any value XЄ X in progress 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) is fulfilled. Give examples.

Where did we meet the terms "even" and "odd"?
Which of these functions will be even, do you think? Why? Which are odd? Why?
For any function of the form at= x n, where n is an integer, it can be argued that the function is odd for n is odd and the function is even for n- even.
– View functions at= and at = 2X– 3 is neither even nor odd, because equalities are not met f(– X) = – f(X), f(– X) = f(X)

The study of the question of whether a function is even or odd is called the study of a function for parity. Slide

Definitions 1 and 2 dealt with the values of the function at x and - x, thus it is assumed that the function is also defined at the value X, and at - X.

ODA 3. If a number set together with each of its elements x contains the opposite element x, then the set X is called a symmetric set.

Examples:

(–2;2), [–5;5]; (∞;∞) are symmetric sets, and , [–5;4] are nonsymmetric.

- Do even functions have a domain of definition - 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) is even or odd, then its domain of definition is D( f) is a symmetric set. But is the converse statement true, if the domain of a function is a symmetric set, then it is even or odd?
- So the presence of a symmetric set of the domain of definition is a necessary condition, but not a sufficient one.
– So how can we investigate the function for parity? Let's try to write an algorithm.

Slide

Algorithm for examining a function for parity

1. Determine whether the domain 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:

Investigate the function for parity a) at= x 5 +; b) at= ; in) at= .

Solution.

a) h (x) \u003d x 5 +,

1) D(h) = (–∞; 0) U (0; +∞), symmetric set.

2) h (- x) \u003d (-x) 5 + - x5 - \u003d - (x 5 +),

3) h (- x) \u003d - h (x) \u003d\u003e function h(x)= x 5 + odd.

b) y =,

at = f(X), D(f) = (–∞; –9)? (–9; +∞), asymmetric set, so the function is neither even nor odd.

in) 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 \u003d x (5 - x 2). 2. Examine the function for parity:

a) y \u003d x 2 (2x - x 3), b) y \u003d

3. In fig. plotted at = f(X), for all X, satisfying the condition X? 0.
Plot the Function at = f(X), if at = f(X) is an even function.

3. In fig. plotted at = f(X), for all x satisfying x? 0.
Plot 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 USE option).

1. The odd function y \u003d f (x) is defined on the entire real 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

Chart conversion.

Verbal description of the function.

Graphic way.

The graphical way of specifying a function is the most illustrative and is often used in engineering. In mathematical analysis, the graphical way of specifying functions is used as an illustration.

Function Graph 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 the given function.

A subset of the coordinate plane is a graph of some function if it has at most one common point with any line parallel to the Oy axis.

Example. Are the figures below graphs of functions?

The advantage of a graphic task is its clarity. You can immediately see how the function behaves, where it increases, where it decreases. From the graph, you can immediately find out some important characteristics of the function.

In general, analytical and graphical ways 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 won’t notice in the formula.

Almost any student knows the three ways to define a function that we have just covered.

Let's try to answer the question: "Are there other ways to define a function?"

There is such a way.

A function can be quite unambiguously defined in words.

For example, the function y=2x can be defined by the following verbal description: each real value of the argument x is assigned its doubled value. The rule is set, the function is set.

Moreover, it is possible to specify a function verbally, which is extremely difficult, if not impossible, to specify by 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 difficult to write this down in a formula. But the table is easy to make.

The method of verbal description is a rather rarely used method. But sometimes it happens.

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 - by a formula, tablet, graph, words - does not change the essence of the matter.

Consider functions whose domains of definition are symmetrical with respect to the origin of coordinates, i.e. for anyone X out of scope 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 out of its domain

Example. Consider the function

She is even. Let's check it out.

For anyone X the equalities

Thus, both conditions are satisfied for us, which means that the function is even. Below is a graph of this function.

Definition. The function f is called odd, if for any X out of its domain

Example. Consider the function

She is odd. Let's check it out.

The domain of definition is the entire numerical axis, which means that it is symmetrical about the point (0; 0).

For anyone X the equalities

Thus, both conditions are satisfied for us, which means that the function is odd. Below is a graph of this function.

The graphs shown in the first and third figures are symmetrical about the y-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?

Portal for the student. Self-training

Graph of an even function

Graph of an odd function

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