Tg even or odd function. Graph of even and odd functions

Graphs of even and odd functions have the following features:

If a function is even, then its graph is symmetrical about the ordinate. If a function is odd, then its graph is symmetrical about the origin.

Example. Construct a graph of the function \(y=\left|x \right|\).

Solution. Consider the function: \(f\left(x \right)=\left|x \right|\) and substitute the opposite \(-x \) instead of \(x \). As a result of simple transformations we get: $$f\left(-x \right)=\left|-x \right|=\left|x \right|=f\left(x \right)$$ In other words, if replace the argument with the opposite sign, the function will not change.

This means that this function is even, and its graph will be symmetrical with respect to the ordinate axis (vertical axis). The graph of this function is shown in the figure on the left. This means that when constructing a graph, you can only draw half, and the second part (to the left of the vertical axis, draw symmetrically to the right part). By determining the symmetry of a function before starting to plot its graph, you can greatly simplify the process of constructing or studying the function. If it is difficult to perform a check in general form, you can do it simpler: substitute in the equation same values different signs. For example -5 and 5. If the function values ​​turn out to be the same, then we can hope that the function will be even. From a mathematical point of view, this approach is not entirely correct, but from a practical point of view it is convenient. To increase the reliability of the result, you can substitute several pairs of such opposite values.

Example. Construct a graph of the function \(y=x\left|x \right|\).

Solution. Let's check the same as in the previous example: $$f\left(-x \right)=x\left|-x \right|=-x\left|x \right|=-f\left(x \right) $$ This means that the original function is odd (the sign of the function has changed to the opposite).

Conclusion: the function is symmetrical about the origin. You can build only one half, and draw the second symmetrically. This kind of symmetry is more difficult to draw. This means that you are looking at the chart from the other side of the sheet, and even upside down. Or you can do this: take the drawn part and rotate it around the origin 180 degrees counterclockwise.

Example. Construct a graph of the function \(y=x^3+x^2\).

Solution. Let's perform the same check for sign change as in the previous two examples. $$f\left(-x \right)=\left(-x \right)^3+\left(-x \right)^2=-x^2+x^2$$ As a result, we get that: $$f\left(-x \right)\not=f\left(x \right),f\left(-x \right)\not=-f\left(x \right)$$ And this means, that the function is neither even nor odd.

Conclusion: the function is not symmetrical either with respect to the origin or the center of the coordinate system. This happened because it is the sum of two functions: even and odd. The same situation will happen if you subtract two different functions. But multiplication or division will lead to a different result. For example, the product of an even and an odd function produces an odd function. Or the quotient of two odd numbers leads to an even function.

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

Let's determine 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 of definition, the corresponding values ​​of y (function) turn out to be equal.

Let's give a more strict 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 point) also lies in this scope,

• f(-x) = f(x).

From the above definition follows the condition necessary for the domain of definition of such a function, 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 above, therefore, the conclusion follows: the even function has a form symmetrical with respect to the ordinate axis (Oy).

How to determine the parity of a function in practice?

Let it be specified using the formula h(x)=11^x+11^(-x). Following the algorithm that follows directly from the definition, we first examine 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 opposite value (-x) for the argument (x).
We get:
h(-x) = 11^(-x) + 11^x.
Since addition satisfies the commutative (commutative) law, it is obvious that h(-x) = h(x) and the given functional dependence is even.

Let's check the parity of the function h(x)=11^x-11^(-x). Following the same algorithm, we get that h(-x) = 11^(-x) -11^x. Taking out the minus, in the end we have
h(-x)=-(11^x-11^(-x))=- h(x). Therefore, 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 adding similar functions, they get an even one;
• 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 multiplying 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 you square an odd function, you get an even one.

The parity of a function can be used to solve 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 solutions for non-negative values ​​of the variable. The resulting roots of the equation must be combined with the opposite numbers. One of them is subject to verification.

This is also successfully used to solve non-standard problems with a parameter.

For example, is there any value of the parameter a for which the equation 2x^6-x^4-ax^2=1 will 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 opposite number. The conclusion is obvious: the roots of an equation that are different from zero are included in the set of its solutions in “pairs”.

It is clear that the number itself is not 0, 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 we substitute it into the equation, we get 2=2. Thus, in addition to “paired” ones, 0 is also a root, which proves their odd number.

Function is one of the most important mathematical concepts. Function - variable dependency at from 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 definition of the function. All values ​​that the dependent variable takes (variable y), form the range of values ​​of the function.

Function graph 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 ordinate axis y. To graph a function, you need to know the properties of the function. The main properties of the function will be discussed below!

To build a graph of a function, 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 they will help you solve problems in mathematics, chemistry, geometry, probability theory and many other subjects!

Basic properties of functions.

1) Function domain and function range.

The domain of a function is the set of all valid valid argument values x(variable x), for which the function y = f(x) determined.
The range of a function is the set of all real values y, which 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, called function zeros. These are the abscissas of the points of intersection of the function graph with the Ox axis.

3) Intervals of constant sign of a function.

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

4) Monotonicity of the function.

An increasing function (in a certain interval) is a function for which higher value the argument from this interval corresponds to a larger value of the function.

A decreasing function (in a certain interval) is a function in which a larger value of the argument from this interval corresponds to a smaller value of the function.

5) Even (odd) function.

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

An odd function is a function whose domain of definition is symmetrical with respect to the origin and for any X from the domain of definition the equality is true 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 about the point (0; 0).
2) for any value x, belonging 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 is such a positive number M such that |f(x)| ≤ M for all values ​​of x. If such a number does not exist, then the function is unlimited.

7) Periodicity of the function.

A function f(x) is periodic if there is a non-zero number T such that for any x from the domain of definition of the function the following holds: 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 is 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 a periodic function are repeated after an interval equal to the period. This is used when constructing graphs.

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Methods for specifying a function

Let the function be given by the formula: y=2x^(2)-3. By assigning any values ​​to the independent variable x, you can calculate, using this formula, the corresponding values ​​of the dependent variable y. For example, if x=-0.5, then, using the formula, we find that the corresponding value of y is y=2 \cdot (-0.5)^(2)-3=-2.5.

Taking any value taken by the argument x in the formula y=2x^(2)-3, you can calculate only one value of the function that corresponds to it. The function can be represented as a table:

x	−2	−1	0	1	2	3
y	−4	−3	−2	−1	0	1

Using this table, you can see that for the argument value −1 the function value −3 will correspond; and the value x=2 will correspond to y=0, etc. It is also important to know that each argument value in the table corresponds to only one function value.

More functions can be specified using graphs. Using a graph, it is established which value of the function correlates with a certain value x. Most often, this will be an approximate value of the function.

Even and odd function

The function is even function, when f(-x)=f(x) for any x from the domain of definition. Such a function will be symmetrical about the Oy axis.

The function is odd function, when f(-x)=-f(x) for any x from the domain of definition. Such a function will be symmetric about the origin O (0;0) .

The function is not even, neither odd and is called general function, when it does not have symmetry about the axis or origin.

Let us examine the following function for parity:

f(x)=3x^(3)-7x^(7)

D(f)=(-\infty ; +\infty) with a symmetric domain of definition relative to the origin. f(-x)= 3 \cdot (-x)^(3)-7 \cdot (-x)^(7)= -3x^(3)+7x^(7)= -(3x^(3)-7x^(7))= -f(x).

This means that the function f(x)=3x^(3)-7x^(7) is odd.

Periodic function

The function y=f(x) , in the domain of which the equality f(x+T)=f(x-T)=f(x) holds for any x, is called periodic function with period T \neq 0 .

Repeating the graph of a function on any segment of the x-axis that has length T.

The intervals where the function is positive, that is, f(x) > 0, are segments of the abscissa axis that correspond to the points of the function graph lying above the abscissa axis.

f(x) > 0 on (x_(1); x_(2)) \cup (x_(3); +\infty)

Intervals where the function is negative, that is, f(x)< 0 - отрезки оси абсцисс, которые отвечают точкам графика функции, лежащих ниже оси абсцисс.

f(x)< 0 на (-\infty; x_(1)) \cup (x_(2); x_(3))

Limited function

Bounded from below It is customary to call a function y=f(x), x \in X when there is a number A for which the inequality f(x) \geq A holds for any x \in X .

An example of a function bounded from below: y=\sqrt(1+x^(2)) since y=\sqrt(1+x^(2)) \geq 1 for any x .

Bounded from above a function y=f(x), x \in X is called when there is a number B for which the inequality f(x) \neq B holds for any x \in X .

An example of a function bounded below: y=\sqrt(1-x^(2)), x \in [-1;1] since y=\sqrt(1+x^(2)) \neq 1 for any x \in [-1;1] .

Limited It is customary to call a function y=f(x), x \in X when there is a number K > 0 for which the inequality \left | f(x)\right | \neq K for any x \in X .

Example limited function: y=\sin x is limited on the entire number axis, since \left | \sin x \right | \neq 1.

Increasing and decreasing function

It is customary to speak of a function that increases on the interval under consideration as increasing function then, when a larger value of x corresponds to a larger value of the function y=f(x) . It follows that taking two arbitrary values ​​of the argument x_(1) and x_(2) from the interval under consideration, with x_(1) > x_(2) , the result will be y(x_(1)) > y(x_(2)).

A function that decreases on the interval under consideration is called decreasing function when a larger value of x corresponds to a smaller value of the function y(x) . It follows that, taking two arbitrary values ​​of the argument x_(1) and x_(2) from the interval under consideration, with x_(1) > x_(2) , the result will be y(x_(1))< y(x_{2}) .

Function Roots It is customary to call the points at which the function F=y(x) intersects the abscissa axis (they are obtained by solving the equation y(x)=0).

a) If for x > 0 an even function increases, then it decreases for x< 0

b) When an even function decreases at x > 0, then it increases at x< 0

c) When an odd function increases at x > 0, then it also increases at x< 0

d) When an odd function decreases for x > 0, then it will also decrease for x< 0

Extrema of the function

Minimum point of the function y=f(x) is usually called a point x=x_(0) whose neighborhood will have other points (except for the point x=x_(0)), and for them the inequality f(x) > f will then be satisfied (x_(0)) . y_(min) - designation of the function at the min point.

Maximum point of the function y=f(x) is usually called a point x=x_(0) whose neighborhood will have other points (except for the point x=x_(0)), and for them the inequality f(x) will then be satisfied< f(x^{0}) . y_{max} - обозначение функции в точке max.

Prerequisite

According to Fermat's theorem: f"(x)=0 when the function f(x) that is differentiable at the point x_(0) will have an extremum at this point.

Sufficient condition

1. When the derivative changes sign from plus to minus, then x_(0) will be the minimum point;
2. x_(0) - will be a maximum point only when the derivative changes sign from minus to plus when passing through the stationary point x_(0) .

The largest and smallest value of a function on an interval

Calculation steps:

1. The derivative f"(x) is sought;
2. Stationary and critical points of the function are found and those belonging to the segment are selected;
3. The values ​​of the function f(x) are found at stationary and critical points and ends of the segment. The smaller of the results obtained will be the smallest value of the function, and more - the largest.

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Attention! Slide previews are for informational purposes only and may not represent all the features of the presentation. If you are interested in this work, please download the full version.

Goals:

• form the concept of parity and oddness of a function, teach the ability to determine and use these properties when function research, plotting;
• develop students' creative activity, logical thinking, ability to compare, generalize;
• cultivate hard work and 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 9th class A.G. Mordkovich. Textbook.
2. Algebra 9th grade A.G. Mordkovich. Problem book.
3. Algebra 9th grade. Tasks for student learning and development. Belenkova E.Yu. Lebedintseva E.A.

PROGRESS OF THE LESSON

1. Organizational moment

Setting goals and objectives for the lesson.

2. Checking homework

No. 10.17 (9th grade problem book. 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 at X ~ 0,4
4. f(X) >0 at X > 0,4 ; f(X) < 0 при – 2 < X < 0,4.
5. The function increases when X € [– 2; + ∞)
6. The function is limited from below.
7. at naim = – 3, at naib doesn't exist
8. The function is continuous.

(Have you used a function exploration algorithm?) Slide.

2. Let’s check the table you were asked from the slide.

Fill out the table
	Domain of definition	Function zeros	Intervals of sign constancy		Coordinates of the points of intersection of the graph with Oy
		x = –5, x = 2	x € (–5;3) U U(2;∞)	x € (–∞;–5) U U (–3;2)
	x ∞ –5, x ≠ 2		x € (–5;3) U U(2;∞)	x € (–∞;–5) U U (–3;2)
	x ≠ –5, x ≠ 2		x € (–∞; –5) U U(2;∞)	x € (–5; 2)

3. Updating knowledge

– Functions are given.
– Specify the scope 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 these functions in the domain of definition the equalities hold f(– X) = f(X), f(– X) = – f(X)? (enter the obtained data into the table) Slide

		f(1) and f(– 1)	f(2) and f(– 2)	graphics	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

– Carrying out this work, guys, we have identified 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 to determine the evenness and oddness of a function, to find out the significance of this property in the study of functions and plotting graphs.
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 is executed 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) holds. Give examples.

Where did we meet the terms “even” and “odd”?
Which of these functions will be even, do you think? Why? Which ones are odd? Why?
For any function of the form at= x n, Where n– an integer, it can be argued that the function is odd when n– odd and the function is even when n– even.
– View functions at= and at = 2X– 3 are neither even nor odd, because equalities are not satisfied f(– X) = – f(X), f(– X) = f(X)

The study of whether a function is even or odd is called the study of a function for parity. Slide

In definitions 1 and 2 we were talking about the values ​​of the function at x and – x, thereby it is assumed that the function is also defined at the value X, and at – X.

Def 3. If a numerical set, together with each of its elements x, also contains the opposite element –x, then the set X called a symmetric set.

Examples:

(–2;2), [–5;5]; (∞;∞) are symmetric sets, and , [–5;4] are asymmetric.

– U even functions is the 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) – even or odd, then its domain of definition is D( f) is a symmetric set. Is the converse statement true: if the domain of definition of a function is a symmetric set, then is it even or odd?
– This means that the presence of a symmetric set of the domain of definition is a necessary condition, but not sufficient.
– So how do you examine a function for parity? Let's try to create an algorithm.

Slide

Algorithm for studying a function for parity

1. Determine whether the domain of definition 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:

Examine function a) for parity at= x 5 +; b) at= ; V) at= .

Solution.

a) h(x) = x 5 +,

1) D(h) = (–∞; 0) U (0; +∞), symmetric set.

2) h (– x) = (–x) 5 + – x5 –= – (x 5 +),

3) h(– x) = – h (x) => function h(x)= x 5 + odd.

b) y =,

at = f(X), D(f) = (–∞; –9)? (–9; +∞), an asymmetric set, which means the function is neither even nor odd.

V) f(X) = , y = f (x),

1) D( f) = (–∞; 3] ≠ ; b) (∞; –2), (–4; 4]?