The sum of even and odd functions. Even and Odd Functions

Evenness and oddness of a function are one of its main properties, and evenness occupies an impressive part of the school course in mathematics. It largely determines the nature of the behavior of the function and greatly facilitates the construction of the corresponding graph.

Let us define the parity of the function. Generally speaking, the function under study is considered even if for opposite values ​​of the independent variable (x) located in its domain, the corresponding values ​​of y (function) are equal.

Let us give a more rigorous definition. Consider some function f (x), which is defined in the domain D. It will be even if for any point x located in the domain of definition:

• -x (opposite dot) also lies in the given scope,

• f(-x) = f(x).

From the above definition, the condition necessary for the domain of definition of such a function follows, namely, symmetry with respect to the point O, which is the origin of coordinates, since if some point b is contained in the domain of definition of an even function, then the corresponding point - b also lies in this domain. From the foregoing, therefore, the conclusion follows: an even function has a form that is symmetrical with respect to the ordinate axis (Oy).

How to determine the parity of a function in practice?

Let it be given using the formula h(x)=11^x+11^(-x). Following the algorithm that follows directly from the definition, we first of all study its domain of definition. Obviously, it is defined for all values ​​of the argument, that is, the first condition is satisfied.

The next step is to substitute the argument (x) with its opposite value (-x).
We get:
h(-x) = 11^(-x) + 11^x.
Since addition satisfies the commutative (displacement) law, it is obvious that h(-x) = h(x) and the given functional dependence is even.

Let's check the evenness of the function h(x)=11^x-11^(-x). Following the same algorithm, we get h(-x) = 11^(-x) -11^x. Taking out the minus, as a result, we have
h(-x)=-(11^x-11^(-x))=- h(x). Hence h(x) is odd.

By the way, it should be recalled that there are functions that cannot be classified according to these criteria, they are called neither even nor odd.

Even functions have a number of interesting properties:

• as a result of the addition of similar functions, an even one is obtained;
• as a result of subtracting such functions, an even one is obtained;
• even, also even;
• as a result of multiplying two such functions, an even one is obtained;
• as a result of multiplication of odd and even functions, an odd one is obtained;
• as a result of dividing the odd and even functions, an odd one is obtained;
• the derivative of such a function is odd;
• If we square an odd function, we get an even one.

The parity of a function can be used in solving equations.

To solve an equation like g(x) = 0, where the left side of the equation is an even function, it will be enough to find its solutions for non-negative values ​​of the variable. The obtained roots of the equation must be combined with opposite numbers. One of them is subject to verification.

The same is successfully used to solve non-standard problems with a parameter.

For example, is there any value for the parameter a that would make the equation 2x^6-x^4-ax^2=1 have three roots?

If we take into account that the variable enters the equation in even powers, then it is clear that replacing x by - x given equation won't change. It follows that if a certain number is its root, then so is the opposite number. The conclusion is obvious: the roots of the equation, other than zero, are included in the set of its solutions in “pairs”.

It is clear that the number 0 itself is not, that is, the number of roots of such an equation can only be even and, naturally, for any value of the parameter it cannot have three roots.

But the number of roots of the equation 2^x+ 2^(-x)=ax^4+2x^2+2 can be odd, and for any value of the parameter. Indeed, it is easy to check that the set of roots given equation contains solutions in "pairs". Let's check if 0 is a root. When substituting it into the equation, we get 2=2. Thus, in addition to "paired" 0 is also a root, which proves their odd number.

A function is called even (odd) if for any and the equality

.

The graph of an even function is symmetrical about the axis
.

The graph of an odd function is symmetrical about the origin.

Example 6.2. Examine for even or odd functions

1)
; 2)
; 3)
.

Decision.

1) The function is defined with
. Let's find
.

Those.
. So this function is even.

2) The function is defined for

Those.
. Thus, this function is odd.

3) the function is defined for , i.e. for

,
. Therefore, the function is neither even nor odd. Let's call it a general function.

3. Investigation of a function for monotonicity.

Function
is called increasing (decreasing) on ​​some interval if in this interval each larger value of the argument corresponds to a larger (smaller) value of the function.

Functions increasing (decreasing) on ​​some interval are called monotonic.

If the function
differentiable on the interval
and has a positive (negative) derivative
, then the function
increases (decreases) in this interval.

Example 6.3. Find intervals of monotonicity of functions

1)
; 3)
.

Decision.

1) This function is defined on the entire number axis. Let's find the derivative.

The derivative is zero if
and
. Domain of definition - numerical axis, divided by points
,
for intervals. Let us determine the sign of the derivative in each interval.

In the interval
the derivative is negative, the function decreases on this interval.

In the interval
the derivative is positive, therefore, the function is increasing on this interval.

2) This function is defined if
or

.

We determine the sign of the square trinomial in each interval.

Thus, the scope of the function

Let's find the derivative
,
, if
, i.e.
, but
. Let us determine the sign of the derivative in the intervals
.

In the interval
the derivative is negative, therefore, the function decreases on the interval
. In the interval
the derivative is positive, the function increases on the interval
.

4. Investigation of a function for an extremum.

Dot
is called the maximum (minimum) point of the function
, if there is such a neighborhood of the point that for everyone
this neighborhood satisfies the inequality

.

The maximum and minimum points of a function are called extremum points.

If the function
at the point has an extremum, then the derivative of the function at this point is equal to zero or does not exist (a necessary condition for the existence of an extremum).

The points at which the derivative is equal to zero or does not exist are called critical.

5. Sufficient conditions for the existence of an extremum.

Rule 1. If during the transition (from left to right) through the critical point derivative
changes sign from "+" to "-", then at the point function
has a maximum; if from "-" to "+", then the minimum; if
does not change sign, then there is no extremum.

Rule 2. Let at the point
first derivative of the function
zero
, and the second derivative exists and is nonzero. If a
, then is the maximum point, if
, then is the minimum point of the function.

Example 6.4 . Explore the maximum and minimum functions:

1)
; 2)
; 3)
;

4)
.

Decision.

1) The function is defined and continuous on the interval
.

Let's find the derivative
and solve the equation
, i.e.
.from here
are critical points.

Let us determine the sign of the derivative in the intervals ,
.

When passing through points
and
the derivative changes sign from “–” to “+”, therefore, according to rule 1
are the minimum points.

When passing through a point
derivative changes sign from "+" to "-", so
is the maximum point.

,
.

2) The function is defined and continuous in the interval
. Let's find the derivative
.

By solving the equation
, find
and
are critical points. If the denominator
, i.e.
, then the derivative does not exist. So,
is the third critical point. Let us determine the sign of the derivative in intervals.

Therefore, the function has a minimum at the point
, maximum at points
and
.

3) A function is defined and continuous if
, i.e. at
.

Let's find the derivative

.

Let's find the critical points:

Neighborhoods of points
do not belong to the domain of definition, so they are not extremum t. So let's explore the critical points
and
.

4) The function is defined and continuous on the interval
. We use rule 2. Find the derivative
.

Let's find the critical points:

Let's find the second derivative
and determine its sign at the points

At points
function has a minimum.

At points
function has a maximum.

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 when function research, plotting;
• 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

– Performing this work, guys, we have revealed one more property of the function, unfamiliar to you, but no less important than the rest - this is the even and odd 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 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

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. AT mathematical analysis the graphical way of setting 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?

advantage graphic task is its visibility. 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 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 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. Etc. 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?

Function is one of the most important mathematical concepts. Function - variable dependency at from a variable x, if each value X matches a single value at. variable X called the independent variable or argument. variable at called the dependent variable. All values ​​of the independent variable (variable x) form the domain of the function. All values ​​that the dependent variable takes (variable y), form the range of the function.

Function Graph they call 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 are plotted along the abscissa axis x, and the values ​​of the variable are plotted along the y-axis y. To plot a function, you need to know the properties of the function. The main properties of the function will be discussed below!

To plot a function graph, 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 you will be helped to solve problems in mathematics, chemistry, geometry, probability theory and many other subjects!

Basic properties of functions.

1) Function scope and function range.

The scope of a function is the set of all valid valid values ​​of the argument x(variable x) for which the function y = f(x) defined.
The range of a function is the set of all real values y that the function accepts.

In elementary mathematics, functions are studied only on the set of real numbers.

2) Function zeros.

Values X, at which y=0, is called function zeros. These are the abscissas of the points of intersection of the graph of the function with the x-axis.

3) Intervals of sign constancy of a function.

The intervals of sign constancy of a function are such intervals of values x, on which the values ​​of the function y either only positive or only negative are called intervals of sign constancy of the function.

4) Monotonicity of the function.

Increasing function (in some interval) - a function in which a larger value of the argument from this interval corresponds to a larger value of the function.

Decreasing function (in some interval) - a function in which a larger value of the argument from this interval corresponds to a smaller value of the function.

5) Even (odd) functions.

An even function is a function whose domain of definition is symmetric with respect to the origin and for any X f(-x) = f(x). The graph of an even function is symmetrical about the y-axis.

An odd function is a function whose domain of definition is symmetric with respect to the origin and for any X from the domain of definition the equality f(-x) = - f(x). 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 the point a belongs to the domain of definition, then the 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.

odd function has the following properties:
1) The domain of definition is symmetrical with respect to the point (0; 0).
2) for any value x, which belongs to the domain of definition, the equality f(-x)=-f(x)
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 exists a positive number M such that |f(x)| ≤ M for all values ​​of x . If there is no such number, then the function is unbounded.

7) Periodicity of the function.

A function f(x) is periodic if there exists a non-zero number T such that for any x from the domain of the function, f(x+T) = f(x). Such smallest number is called the period of the function. All trigonometric functions are periodic. (Trigonometric formulas).

Function f is called periodic if there exists a number such that for any x from the domain of definition the equality f(x)=f(x-T)=f(x+T). 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 the periodic function are repeated after an interval equal to the period. This is used when plotting graphs.