Even chart. Even and odd functions. Investigation of a function for monotonicity

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)
.

Solution.

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)
.

Solution.

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
, That is the maximum point, if
, That is the minimum point of the function.

Example 6.4 . Explore the maximum and minimum functions:

1)
; 2)
; 3)
;

4)
.

Solution.

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 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.

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. 1 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= ; V) 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.

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 \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

. 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.

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.

An increasing function (in a certain interval) is 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). This 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.

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. 1 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.

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= ; V) 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.

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 \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).

7. Summing up

3. Investigation of a function for monotonicity.

4. Investigation of a function for an extremum.

5. Sufficient conditions for the existence of an extremum.

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