Definition of even and odd. How to determine even and odd functions

Function zeros
The zero of the function is the value X, at which the function becomes 0, that is, f(x)=0.

Zeros are the points of intersection of the graph of the function with the axis Oh.

Function parity
A function is called even if for any X from the domain of definition, the equality f(-x) = f(x)

An even function is symmetrical about the axis OU

Odd function
A function is called odd if for any X from the domain of definition, the equality f(-x) = -f(x) is satisfied.

An odd function is symmetrical with respect to the origin.
A function that is neither even nor odd is called a general function.

Function Increment
The function f(x) is called increasing if the larger value of the argument corresponds to the larger value of the function, i.e. x 2 >x 1 → f(x 2)> f(x 1)

Decreasing function
The function f(x) is called decreasing if the larger value of the argument corresponds to the smaller value of the function, i.e. x 2 >x 1 → f(x 2)
The intervals on which the function either only decreases or only increases are called intervals of monotony. The function f(x) has 3 intervals of monotonicity:
(-∞ x 1), (x 1 , x 2), (x 3 ; +∞)

Find intervals of monotonicity using the service Intervals of increasing and decreasing functions

Local maximum
Dot x 0 is called a local maximum point if for any X from a neighborhood of a point x 0 the following inequality holds: f(x 0) > f(x)

Local minimum
Dot x 0 is called a local minimum point if for any X from a neighborhood of a point x 0 the following inequality holds: f(x 0)< f(x).

Local maximum points and local minimum points are called local extremum points.

x 1 , x 2 - local extremum points.

Function Periodicity
The function f(x) is called periodic, with period T, if for any X f(x+T) = f(x) .

Constancy intervals
Intervals on which the function is either only positive or only negative are called intervals of constant sign.

f(x)>0 for x∈(x 1 , x 2)∪(x 2 , +∞), f(x)<0 при x∈(-∞,x 1)∪(x 1 , x 2)

Function continuity
The function f(x) is called continuous at the point x 0 if the limit of the function as x → x 0 is equal to the value of the function at this point, i.e. .

break points
The points at which the continuity condition is violated are called points of discontinuity of the function.

x0- breaking point.

General scheme for plotting functions

1. Find the domain of the function D(y).
2. Find the intersection points of the graph of functions with the coordinate axes.
3. Investigate the function for even or odd.
4. Investigate the function for periodicity.
5. Find intervals of monotonicity and extremum points of the function.
6. Find intervals of convexity and inflection points of the function.
7. Find the asymptotes of the function.
8. Based on the results of the study, build a graph.

Example: Explore the function and build its graph: y = x 3 - 3x
8) Based on the results of the study, we will construct a graph of the function:

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
		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 satisfied. 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 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= .

Decision.

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

Definition 1. The function is called even (odd ) if together with each value of the variable
meaning - X also belongs
and the equality

Thus, a function can be even or odd only when its domain of definition is symmetrical with respect to the origin on the real line (numbers X and - X simultaneously belong
). For example, the function
is neither even nor odd, since its domain of definition
not symmetrical about the origin.

Function
even, because
symmetrical with respect to the origin of coordinates and.

Function
odd because
and
.

Function
is neither even nor odd, since although
and is symmetric with respect to the origin, equalities (11.1) are not satisfied. For example,.

The graph of an even function is symmetrical about the axis OU, since if the point

also belongs to the graph. The graph of an odd function is symmetrical about the origin, because if
belongs to the graph, then the point
also belongs to the graph.

When proving whether a function is even or odd, the following statements are useful.

Theorem 1. a) The sum of two even (odd) functions is an even (odd) function.

b) The product of two even (odd) functions is an even function.

c) The product of an even and an odd function is an odd function.

d) If f is an even function on the set X, and the function g defined on the set
, then the function
- even.

e) If f is an odd function on the set X, and the function g defined on the set
and even (odd), then the function
- even (odd).

Proof. Let us prove, for example, b) and d).

b) Let
and
are even functions. Then, therefore. The case of odd functions is considered similarly
and
.

d) Let f is an even function. Then.

The other assertions of the theorem are proved similarly. The theorem has been proven.

Theorem 2. Any function
, defined on the set X, which is symmetric with respect to the origin, can be represented as the sum of an even and an odd function.

Proof. Function
can be written in the form

.

Function
is even, because
, and the function
is odd because. Thus,
, where
- even, and
is an odd function. The theorem has been proven.

Definition 2. Function
called periodical if there is a number
, such that for any
numbers
and
also belong to the domain of definition
and the equalities

Such a number T called period functions
.

Definition 1 implies that if T– function period
, then the number T too is the period of the function
(because when replacing T on the - T equality is maintained). Using the method of mathematical induction, it can be shown that if T– function period f, then and
, is also a period. It follows that if a function has a period, then it has infinitely many periods.

Definition 3. The smallest of the positive periods of a function is called its main period.

Theorem 3. If T is the main period of the function f, then the remaining periods are multiples of it.

Proof. Assume the opposite, that is, that there is a period functions f (>0), not multiple T. Then, dividing on the T with the remainder, we get
, where
. So

i.e – function period f, and
, which contradicts the fact that T is the main period of the function f. The assertion of the theorem follows from the obtained contradiction. The theorem has been proven.

It is well known that trigonometric functions are periodic. Main period
and
equals
,
and
. Find the period of the function
. Let be
is the period of this function. Then

(as
.

ororor
.

Meaning T, determined from the first equality, cannot be a period, since it depends on X, i.e. is a function of X, not a constant number. The period is determined from the second equality:
. There are infinitely many periods
the smallest positive period is obtained when
:
. This is the main period of the function
.

An example of a more complex periodic function is the Dirichlet function

Note that if T is a rational number, then
and
are rational numbers under rational X and irrational when irrational X. So

for any rational number T. Therefore, any rational number T is the period of the Dirichlet function. It is clear that this function has no main period, since there are positive rational numbers arbitrarily close to zero (for example, a rational number can be made by choosing n arbitrarily close to zero).

Theorem 4. If function f set on the set X and has a period T, and the function g set on the set
, then the complex function
also has a period T.

Proof. We have therefore

that is, the assertion of the theorem is proved.

For example, since cos x has a period
, then the functions
have a period
.

Definition 4. Functions that are not periodic are called non-periodic .

. To do this, use graph paper or a graphical calculator. Select any number of numeric values ​​for the independent variable x (\displaystyle x) and plug them into the function to calculate the values ​​of the dependent variable y (\displaystyle y). Put the found coordinates of the points on the coordinate plane, and then connect these points to build a graph of the function.

• Substitute positive numeric values ​​into the function x (\displaystyle x) and corresponding negative numeric values. For example, given a function f (x) = 2 x 2 + 1 (\displaystyle f(x)=2x^(2)+1). Substitute the following values ​​into it x (\displaystyle x):

Check if the graph of the function is symmetrical about the y-axis. Symmetry refers to the mirror image of the graph about the y-axis. If the part of the graph to the right of the y-axis (positive values ​​of the independent variable) matches the part of the graph to the left of the y-axis (negative values ​​of the independent variable), the graph is symmetrical about the y-axis. If the function is symmetrical about the y-axis, the function is even.

Check if the graph of the function is symmetrical about the origin. The origin is the point with coordinates (0,0). Symmetry about the origin means that a positive value y (\displaystyle y)(with a positive value x (\displaystyle x)) corresponds to a negative value y (\displaystyle y)(with a negative value x (\displaystyle x)), and vice versa. Odd functions have symmetry with respect to the origin.

Check if the graph of the function has any symmetry. The last type of function is a function whose graph does not have symmetry, that is, there is no mirror image both relative to the y-axis and relative to the origin. For example, given a function.

• Substitute several positive and corresponding negative values ​​into the function x (\displaystyle x):
• According to the results obtained, there is no symmetry. Values y (\displaystyle y) for opposite values x (\displaystyle x) do not match and are not opposite. Thus, the function is neither even nor odd.
• Please note that the function f (x) = x 2 + 2 x + 1 (\displaystyle f(x)=x^(2)+2x+1) can be written like this: f (x) = (x + 1) 2 (\displaystyle f(x)=(x+1)^(2)). Written in this form, the function appears to be even because there is an even exponent. But this example proves that the form of a function cannot be quickly determined if the independent variable is enclosed in parentheses. In this case, you need to open the brackets and analyze the resulting exponents.

Which to one degree or another were familiar to you. It was also noted there that the stock of function properties will be gradually replenished. Two new properties will be discussed in this section.

Definition 1.

The function y \u003d f (x), x є X, is called even if for any value x from the set X the equality f (-x) \u003d f (x) is true.

Definition 2.

The function y \u003d f (x), x є X, is called odd if for any value x from the set X the equality f (-x) \u003d -f (x) is true.

Prove that y = x 4 is an even function.

Decision. We have: f (x) \u003d x 4, f (-x) \u003d (-x) 4. But (-x) 4 = x 4 . Hence, for any x, the equality f (-x) = f (x), i.e. the function is even.

Similarly, it can be proved that the functions y - x 2, y \u003d x 6, y - x 8 are even.

Prove that y = x 3 is an odd function.

Decision. We have: f (x) \u003d x 3, f (-x) \u003d (-x) 3. But (-x) 3 = -x 3 . Hence, for any x, the equality f (-x) \u003d -f (x), i.e. the function is odd.

Similarly, it can be proved that the functions y \u003d x, y \u003d x 5, y \u003d x 7 are odd.

You and I have repeatedly convinced ourselves that new terms in mathematics most often have an “earthly” origin, i.e. they can be explained in some way. This is the case for both even and odd functions. See: y - x 3, y \u003d x 5, y \u003d x 7 are odd functions, while y \u003d x 2, y \u003d x 4, y \u003d x 6 are even functions. And in general, for any function of the form y \u003d x "(below we will specifically study these functions), where n is a natural number, we can conclude: if n is an odd number, then the function y \u003d x" is odd; if n is an even number, then the function y = xn is even.

There are also functions that are neither even nor odd. Such, for example, is the function y \u003d 2x + 3. Indeed, f (1) \u003d 5, and f (-1) \u003d 1. As you can see, here Hence, neither the identity f (-x) \u003d f ( x), nor the identity f(-x) = -f(x).

So, a function can be even, odd, or neither.

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

Definitions 1 and 2 deal with the values ​​of the function at the points x and -x. This assumes that the function is defined both at the point x and at the point -x. This means that the point -x belongs to the domain of the function at the same time as the point x. If a numerical set X together with each of its elements x contains the opposite element -x, then X is called a symmetric set. Let's say (-2, 2), [-5, 5], (-oo, +oo) are symmetric sets, while )