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 whether a function is even or odd
1)
;
2)
;
3)
.
Solution.
1) The function is defined when 
. We'll find 
.
Those. 
. This means that this function is even.
2) The function is defined when 
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 function of general form.
Function 
is called increasing (decreasing) on a certain interval if in this interval each larger value of the argument corresponds to a larger (smaller) value of the function.
Functions increasing (decreasing) over a certain interval are called monotonic.
If the function 
differentiable on the interval 
and has a positive (negative) derivative 
, then the function 
increases (decreases) over this interval.
Example 6.3. Find intervals of monotonicity of functions
1)
;
3)
.
Solution.
1) This function is defined on the entire number line. Let's find the derivative.
The derivative is equal to zero if 
And 
. The domain of definition is the number axis, divided by dots 
,
at 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 increases over this interval.
2) This function is defined if 
or
.
We determine the sign of the quadratic trinomial in each interval.
Thus, the domain of definition 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 over the interval 
.
Dot 
called the maximum (minimum) point of the function 
, if there is such a neighborhood of the point 
that's for everyone 
from this neighborhood the inequality holds 
.
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 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 a function 
equal to zero 
, and the second derivative exists and is different from zero. If 
, That 
– maximum point, if 
, That 
– 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 
– 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 
– minimum points.
When passing through a point 
the derivative changes sign from “+” to “–”, so 
– maximum point.
,
.
2) The function is defined and continuous in the interval 
. Let's find the derivative 
.
Having solved the equation 
, we'll find 
And 
– critical points. If the denominator 
, i.e. 
, then the derivative does not exist. So, 
– third critical point. Let us determine the sign of the derivative in intervals.
Therefore, the function has a minimum at the point 
, maximum in points 
And 
.
3) A function is defined and continuous if 
, i.e. at 
.
Let's find the derivative 
.
Let's find critical points: 
Neighborhoods of points 
do not belong to the domain of definition, therefore they are not extrema. So, let's examine the critical points 
And 
.
4) The function is defined and continuous on the interval 
. Let's use rule 2. Find the derivative 
.
Let's find critical points:
Let's find the second derivative 
and determine its sign at the points 
At points 
function has a minimum.
At points 
the function has a maximum.
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.
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 the function . Substitute the following values x (\displaystyle x) into it:
- f (1) = 2 (1) 2 + 1 = 2 + 1 = 3 (\displaystyle f(1)=2(1)^(2)+1=2+1=3) (1 , 3) (\ displaystyle (1,3)) .
- f (2) = 2 (2) 2 + 1 = 2 (4) + 1 = 8 + 1 = 9 (\displaystyle f(2)=2(2)^(2)+1=2(4)+1 =8+1=9) . We got a point with coordinates (2, 9) (\displaystyle (2,9)).
- f (− 1) = 2 (− 1) 2 + 1 = 2 + 1 = 3 (\displaystyle f(-1)=2(-1)^(2)+1=2+1=3) . We got a point with coordinates (− 1, 3) (\displaystyle (-1,3)) .
- f (− 2) = 2 (− 2) 2 + 1 = 2 (4) + 1 = 8 + 1 = 9 (\displaystyle f(-2)=2(-2)^(2)+1=2( 4)+1=8+1=9) . We got a point with coordinates (− 2, 9) (\displaystyle (-2,9)) .
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.
- You can check the symmetry of the graph using individual points. If the value of y (\displaystyle y) x (\displaystyle x) matches the value of y (\displaystyle y) that matches the value of − x (\displaystyle -x) , the function is even. In our example with the function f (x) = 2 x 2 + 1 (\displaystyle f(x)=2x^(2)+1) we got the following coordinates of the points:
- (1.3) and (-1.3)
- (2.9) and (-2.9)
- Note that for x=1 and x=-1 the dependent variable is y=3, and for x=2 and x=-2 the dependent variable is y=9. Thus the function is even. In fact, to accurately determine the form of the function, you need to consider more than two points, but the described method is a good approximation.
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.
- If you substitute several positive and corresponding negative values of x (\displaystyle x) into the function, the values of y (\displaystyle y) will differ in sign. For example, given a function f (x) = x 3 + x (\displaystyle f(x)=x^(3)+x) . Substitute several values of x (\displaystyle x) into it:
- f (1) = 1 3 + 1 = 1 + 1 = 2 (\displaystyle f(1)=1^(3)+1=1+1=2) . We got a point with coordinates (1,2).
- f (− 1) = (− 1) 3 + (− 1) = − 1 − 1 = − 2 (\displaystyle f(-1)=(-1)^(3)+(-1)=-1- 1=-2)
- f (2) = 2 3 + 2 = 8 + 2 = 10 (\displaystyle f(2)=2^(3)+2=8+2=10)
- f (− 2) = (− 2) 3 + (− 2) = − 8 − 2 = − 10 (\displaystyle f(-2)=(-2)^(3)+(-2)=-8- 2=-10) . We received a point with coordinates (-2,-10).
- Thus, f(x) = -f(-x), that is, the function is odd.
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:
- f (1) = 1 2 + 2 (1) + 1 = 1 + 2 + 1 = 4 (\displaystyle f(1)=1^(2)+2(1)+1=1+2+1=4 ) . We got a point with coordinates (1,4).
- f (− 1) = (− 1) 2 + 2 (− 1) + (− 1) = 1 − 2 − 1 = − 2 (\displaystyle f(-1)=(-1)^(2)+2 (-1)+(-1)=1-2-1=-2) . We got a point with coordinates (-1,-2).
- f (2) = 2 2 + 2 (2) + 2 = 4 + 4 + 2 = 10 (\displaystyle f(2)=2^(2)+2(2)+2=4+4+2=10 ) . We got a point with coordinates (2,10).
- f (− 2) = (− 2) 2 + 2 (− 2) + (− 2) = 4 − 4 − 2 = − 2 (\displaystyle f(-2)=(-2)^(2)+2 (-2)+(-2)=4-4-2=-2) . We got a point with coordinates (2,-2).
- 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.
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 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.
How to insert mathematical formulas on a website?
If you ever need to add one or two mathematical formulas to a web page, then the easiest way to do this is as described in the article: mathematical formulas are easily inserted onto the site in the form of pictures that are automatically generated by Wolfram Alpha. In addition to simplicity, this universal method will help improve the visibility of the site in search engines. It has been working for a long time (and, I think, will work forever), but is already morally outdated.
If you regularly use mathematical formulas on your site, then I recommend you use MathJax - a special JavaScript library that displays mathematical notation in web browsers using MathML, LaTeX or ASCIIMathML markup.
There are two ways to start using MathJax: (1) using a simple code, you can quickly connect a MathJax script to your website, which will be automatically loaded from a remote server at the right time (list of servers); (2) download the MathJax script from a remote server to your server and connect it to all pages of your site. The second method - more complex and time-consuming - will speed up the loading of your site's pages, and if the parent MathJax server becomes temporarily unavailable for some reason, this will not affect your own site in any way. Despite these advantages, I chose the first method as it is simpler, faster and does not require technical skills. Follow my example, and in just 5 minutes you will be able to use all the features of MathJax on your site.
You can connect the MathJax library script from a remote server using two code options taken from the main MathJax website or on the documentation page:
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.
Any fractal is constructed according to a certain rule, which is consistently applied an unlimited number of times. Each such time is called an iteration.
The iterative algorithm for constructing a Menger sponge is quite simple: the original cube with side 1 is divided by planes parallel to its faces into 27 equal cubes. One central cube and 6 cubes adjacent to it along the faces are removed from it. The result is a set consisting of the remaining 20 smaller cubes. Doing the same with each of these cubes, we get a set consisting of 400 smaller cubes. Continuing this process endlessly, we get a Menger sponge.