Even and odd function how. Even and odd functions. Methods for specifying a function

. To do this, use graph paper or a graphing calculator. Select any number of numeric values for the independent variable x (\displaystyle x) and plug them into the function to calculate the values for the dependent variable y (\displaystyle y) . Plot 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 x (\displaystyle x) and corresponding negative numeric values into the function. For example, given a function f (x) = 2 x 2 + 1 (\displaystyle f(x)=2x^(2)+1) . Substitute the following values x (\displaystyle x) into it:

Check whether the graph of the function is symmetrical about the Y axis. By symmetry we mean 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) is the same as 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 whether 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 of y (\displaystyle y) (for a positive value of x (\displaystyle x) ) corresponds to a negative value of (\displaystyle y) (\displaystyle y) (for a negative value of x (\displaystyle x) ), and vice versa. Odd functions have symmetry about the origin.

Check if the graph of the function has any symmetry. The last type of function is a function whose graph has no symmetry, that is, there is no mirror image both relative to the ordinate axis and relative to the origin. For example, given the function .

Substitute several positive and corresponding negative values of x (\displaystyle x) into the function:
According to the results obtained, there is no symmetry. The values of y (\displaystyle y) for opposite values of x (\displaystyle x) are not the same 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 as follows: f (x) = (x + 1) 2 (\displaystyle f(x)=(x+1)^(2)) . When written in this form, the function appears even because there is an even exponent. But this example proves that the type of 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 obtained exponents.

Even function.

A function whose sign does not change when the sign changes is called even. x.

x equality holds f(–x) = f(x). Sign x does not affect the sign y.

The graph of an even function is symmetrical about the coordinate axis (Fig. 1).

Examples of an even function:

y=cos x

y = x 2

y = –x 2

y = x 4

y = x 6

y = x 2 + x

Explanation:
Let's take the function y = x 2 or y = –x 2 .
For any value x the function is positive. Sign x does not affect the sign y. The graph is symmetrical about the coordinate axis. This is an even function.

Odd function.

A function whose sign changes when the sign changes is called odd. x.

In other words, for any value x equality holds f(–x) = –f(x).

The graph of an odd function is symmetrical with respect to the origin (Fig. 2).

Examples of odd function:

y= sin x

y = x 3

y = –x 3

Explanation:

Let's take the function y = – x 3 .
All meanings at it will have a minus sign. That is a sign x influences 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 functions are even or odd. There are functions that do not obey 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. Functions that describe these processes are called periodic functions. That is, these are functions in whose graphs there are elements that repeat at certain numerical intervals.

In July 2020, NASA launches an expedition to Mars. The spacecraft will deliver to Mars an electronic medium with the names of all registered expedition participants.

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One of these code options needs to be copied and pasted into the code of your web page, preferably between tags and or immediately after the tag. According to the first option, MathJax loads faster and slows down the page less. But the second option automatically monitors and loads the latest versions of MathJax. If you insert the first code, it will need to be updated periodically. If you insert the second code, the pages will load more slowly, but you will not need to constantly monitor MathJax updates.

The easiest way to connect MathJax is in Blogger or WordPress: in the site control panel, add a widget designed to insert third-party JavaScript code, copy the first or second version of the download code presented above into it, and place the widget closer to the beginning of the template (by the way, this is not at all necessary , since the MathJax script is loaded asynchronously). That's all. Now learn the markup syntax of MathML, LaTeX, and ASCIIMathML, and you are ready to insert mathematical formulas into your site's web pages.

Another New Year's Eve... frosty weather and snowflakes on the window glass... All this prompted me to write again about... fractals, and what Wolfram Alpha knows about it. There is an interesting article on this subject, which contains examples of two-dimensional fractal structures. Here we will look at more complex examples of three-dimensional fractals.

A fractal can be visually represented (described) as a geometric figure or body (meaning that both are a set, in this case, a set of points), the details of which have the same shape as the original figure itself. That is, this is a self-similar structure, examining the details of which when magnified, we will see the same shape as without magnification. Whereas in the case of an ordinary geometric figure (not a fractal), upon magnification we will see details that have a simpler shape than the original figure itself. For example, at a high enough magnification, part of an ellipse looks like a straight line segment. This does not happen with fractals: with any increase in them, we will again see the same complex shape, which will be repeated again and again with each increase.

Benoit Mandelbrot, the founder of the science of fractals, wrote in his article Fractals and Art in the Name of Science: “Fractals are geometric shapes that are as complex in their details as in their overall form. That is, if part of the fractal will be enlarged to the size of the whole, it will appear as a whole, either exactly, or perhaps with a slight deformation."

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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

A function is an even function when f(-x)=f(x) for any x in the domain. Such a function will be symmetrical about the Oy axis.

A function is an odd function when f(-x)=-f(x) for any x in the domain. Such a function will be symmetric about the origin O (0;0) .

A function is neither even nor odd and is called a general function when it has no 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 a 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

A function y=f(x), x \in X is usually called bounded below 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 .

A function y=f(x), x \in X is called bounded above if 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 from below: y=\sqrt(1-x^(2)), x \in [-1;1] since y=\sqrt(1+x^(2)) \neq 1 for any x \ in [-1;1] .

A function y=f(x), x \in X is usually called bounded when there is a number K > 0 for which the inequality \left | f(x)\right | \neq K for any x \in X .

An example of a bounded function: y=\sin x is bounded on the entire number line, since \left | \sin x \right | \neq 1 .

Increasing and decreasing function

It is customary to speak of a function that increases over the interval under consideration as an increasing function 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 a decreasing function when a larger value of x corresponds to a smaller value of the function y(x) . It follows that, taking from the interval under consideration two arbitrary values of the argument x_(1) and x_(2) , and x_(1) > x_(2) , the result will be y(x_(1))< y(x_{2}) .

The roots of a function are usually called 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

The 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 then the inequality f(x ) > f(x_(0)) . y_(min) - designation of the function at the min point.

The 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 then the inequality f(x )< 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

When the derivative changes sign from plus to minus, then x_(0) will be the minimum point;

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:

The derivative f"(x) is sought;

Stationary and critical points of the function are found and those belonging to the segment are selected;

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 the larger one will be the largest.

Function is one of the most important mathematical concepts. A function is the dependence of the variable y on the variable x, if each value of x corresponds to a single value of y. The variable x is called the independent variable or argument. The variable y is 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 (variable y) takes form the range of the function.

The graph of a function is 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 x are plotted along the abscissa axis, and the values of the variable y are plotted along the ordinate axis. 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) The domain of definition of the function and the range of values of the function.

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

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

2) Zeros of the function.

Values of x for which y=0 are 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 - such intervals of values x on which the values of the function y are 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 in which a larger value of 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) Evenness (oddness) of the 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 f(-x) = - f(x) is true. 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 point a belongs to the domain of definition, then 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.

An 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) is satisfied
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 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).

A 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) holds. 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.