Even function.
Even A function whose sign does not change when the sign is changed is called x.
x equality f(–x) = f(x). Sign x does not affect sign y.
The graph of an even function is symmetrical about the coordinate axis (Fig. 1).
Even function examples:
y= cos x
y = x 2
y = –x 2
y = x 4
y = x 6
y = x 2 + x
Explanation:
Let's take a function y = x 2 or y = –x 2 .
For any value x the function is positive. Sign x does not affect sign y. The graph is symmetrical about the coordinate axis. This is an even function.
odd function.
odd is a function whose sign changes when the sign is changed x.
In other words, for any value x equality f(–x) = –f(x).
The graph of an odd function is symmetrical with respect to the origin (Fig. 2).
Examples of an odd function:
y= sin x
y = x 3
y = –x 3
Explanation:
Take the function y = - x 3 .
All values at it will have a minus sign. That is the sign x affects the sign y. If the independent variable is a positive number, then the function is positive; if the independent variable is a negative number, then the function is negative: f(–x) = –f(x).
The graph of the function is symmetrical about the origin. This is an odd function.
Properties of even and odd functions:
NOTE:
Not all features are even or odd. There are functions that are not subject to such gradation. For example, the root function at = √X does not apply to either even or odd functions (Fig. 3). When listing the properties of such functions, an appropriate description should be given: neither even nor odd.
Periodic functions.
As you know, periodicity is the repetition of certain processes at a certain interval. The functions describing these processes are called periodic functions. That is, these are functions in whose graphs there are elements that repeat at certain numerical intervals.
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 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). 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 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).
but) 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, |
х € (–5;3) U |
х € (–∞;–5) U |
||
 |
x ∞ -5, |
х € (–5;3) U |
х € (–∞;–5) U |
||
x ≠ -5, |
x € (–∞; –5) U |
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) ?
but); b) y \u003d x (5 - x 2).
a) y \u003d x 2 (2x - x 3), b) y \u003d
Plot the Function at = f(X), if at = f(X) is an even function.
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
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 with -x will not change the given equation. 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.
- (Math.) The 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 ... ...
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F(x) = x is an example of an odd function. f(x) = x2 is an example of an even function. f(x) = x3 ... Wikipedia
F(x) = x is an example of an odd function. f(x) = x2 is an example of an even function. f(x) = x3 ... Wikipedia
F(x) = x is an example of an odd function. f(x) = x2 is an example of an even function. f(x) = x3 ... Wikipedia
F(x) = x is an example of an odd function. f(x) = x2 is an example of an even function. f(x) = x3 ... Wikipedia
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