Examples of even functions. Even and odd functions. Function period. Function extremes. The largest and smallest value of the function on the interval

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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
	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.

– U even functions the domain of definition 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) 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= .

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. odd function y = 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

Evenness and oddness of a function are one of its main properties, and evenness occupies an impressive part school course 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 quite enough to find its solution 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 of a 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.

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;
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.

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;

(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

		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

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.

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= .

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

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: