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 research.
1) D(y) - Domain of definition: the set of all those values of the variable x. under which the algebraic expressions f(x) and g(x) make sense.
If the function is given by a formula, then the domain of definition consists of all values of the independent variable for which the formula makes sense.
2) Function properties: even/odd, periodicity:
odd and even are called functions whose graphs are symmetric with respect to the change in the sign of the argument.
odd function- a function that changes the value to the opposite when the sign of the independent variable changes (symmetric about the center of coordinates).
Even function- a function that does not change its value when the sign of the independent variable changes (symmetric about the y-axis).
Neither even nor odd function (general function) is a function that does not have symmetry. This category includes functions that do not fall under the previous 2 categories.
Functions that do not belong to any of the categories above are called neither even nor odd(or generic functions).
Odd functions
An odd power where is an arbitrary integer.
Even functions
An even power where is an arbitrary integer.
Periodic function is a function that repeats its values at some regular interval of the argument, i.e., does not change its value when some fixed nonzero number is added to the argument ( period functions) over the entire domain of definition.
3) Zeros (roots) of a function are the points where it vanishes.
Finding the point of intersection of the graph with the axis Oy. To do this, you need to calculate the value f(0). Find also the points of intersection of the graph with the axis Ox, why find the roots of the equation f(x) = 0 (or make sure there are no roots).
The points where the graph intersects the axis are called function zeros. To find the zeros of the function, you need to solve the equation, that is, find those x values, for which the function vanishes.
4) Intervals of constancy of signs, signs in them.
Intervals where the function f(x) retains its sign.
The constancy interval is the interval at every point in which function is positive or negative.
ABOVE the x-axis.
BELOW axis.
5) Continuity (points of discontinuity, character of discontinuity, asymptotes).
continuous function- a function without "jumps", that is, one in which small changes in the argument lead to small changes in the value of the function.
Removable breakpoints
If the limit of the function exists, but the function is not defined at this point, or the limit does not match the value of the function at this point:
,
then the point is called break point functions (in complex analysis, a removable singular point).
If we "correct" the function at the point of a removable discontinuity and put 
, then we get a function that is continuous at this point. Such an operation on a function is called extending the function to continuous or extension of the function by continuity, which justifies the name of the point, as points disposable gap.
Discontinuity points of the first and second kind
If the function has a discontinuity at a given point (that is, the limit of the function at a given point is absent or does not coincide with the value of the function at a given point), then for numerical functions there are two possible options related to the existence of numerical functions unilateral limits:
if both one-sided limits exist and are finite, then such a point is called breaking point of the first kind. Removable discontinuity points are discontinuity points of the first kind;
if at least one of the one-sided limits does not exist or is not a finite value, then such a point is called breaking point of the second kind.
Asymptote - straight, which has the property that the distance from a point of the curve to this straight tends to zero as the point moves along the branch to infinity.
vertical
Vertical asymptote - limit line 
.
As a rule, when determining the vertical asymptote, they look for not one limit, but two one-sided ones (left and right). This is done in order to determine how the function behaves as it approaches the vertical asymptote from different directions. For example:
Horizontal
Horizontal asymptote - straight species, subject to the existence limit
.
oblique
Oblique asymptote - straight species, subject to the existence limits
Note: A function can have no more than two oblique (horizontal) asymptotes.
Note: if at least one of the two limits mentioned above does not exist (or is equal to ), then the oblique asymptote at (or ) does not exist.
if in item 2.), then , and the limit is found by the horizontal asymptote formula, 
.
6) Finding intervals of monotonicity. Find monotonicity intervals of a function f(x) (i.e. intervals of increase and decrease). This is done by examining the sign of the derivative f(x). To do this, find the derivative f(x) and solve the inequality f(x)0. On the intervals where this inequality is satisfied, the function f(x) increases. Where the reverse inequality holds f(x)0, function f(x) decreases.
Finding a local extremum. Having found the intervals of monotonicity, we can immediately determine the points of a local extremum where the increase is replaced by a decrease, there are local maxima, and where the decrease is replaced by an increase, local minima. Calculate the value of the function at these points. If a function has critical points that are not local extremum points, then it is useful to calculate the value of the function at these points as well.
Finding the largest and smallest values of the function y = f(x) on a segment(continuation)
|
1. Find the derivative of a function: f(x). 2. Find points where the derivative is zero: f(x)=0x 1, x 2 ,... 3. Determine the ownership of points X 1 ,X 2 , … segment [ a; b]: let x 1a;b, a x 2a;b . 4. Find function values at selected points and at the ends of the segment: f(x 1), f(x 2),..., f(x a),f(x b), 5. Selection of the largest and smallest values of the function from those found. Comment. If on the segment [ a; b] there are discontinuity points, then it is necessary to calculate one-sided limits in them, and then take their values into account in choosing the largest and smallest values of the function. |
7) Finding intervals of convexity and concavity. This is done by examining the sign of the second derivative f(x). Find the inflection points at the junctions of the convex and concavity intervals. Calculate the value of the function at the inflection points. If the function has other points of continuity (other than inflection points) at which the second derivative is equal to 0 or does not exist, then at these points it is also useful to calculate the value of the function. Finding f(x) , we solve the inequality f(x)0. On each of the solution intervals, the function will be downward convex. Solving the reverse inequality f(x)0, we find the intervals on which the function is convex upwards (that is, concave). We define inflection points as those points at which the function changes the direction of convexity (and is continuous).
Function inflection point- this is the point at which the function is continuous and when passing through which the function changes the direction of convexity.
Conditions of existence
Necessary condition for the existence of an inflection point: if the function is twice differentiable in some punctured neighborhood of the point , then either 
.
Hide Show
Ways to set a function
Let the function be given by the formula: y=2x^(2)-3 . By assigning any value to the independent variable x , you can use this formula to calculate the corresponding values of the dependent variable y . For example, if x=-0.5 , then using the formula, we get that the corresponding value of y is y=2 \cdot (-0.5)^(2)-3=-2.5 .
Given any value taken by the x argument in the formula y=2x^(2)-3 , only one function value can be calculated 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 figure out that for the value of the argument -1, the value of the function -3 will correspond; and the value x=2 will correspond to y=0, and so on. It is also important to know that each argument value in the table corresponds to only one function value.
More functions can be set using graphs. With the help of the graph, it is established which value of the function correlates with a certain value of 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. Such a function will be symmetrical about the Oy axis.
The function is odd function when f(-x)=-f(x) for any x in the domain. Such a function will be symmetrical about the origin O (0;0) .
The function is not even, nor odd and called general function when it does not have symmetry about the axis or origin.
We examine the following function for parity:
f(x)=3x^(3)-7x^(7)
D(f)=(-\infty ; +\infty) with a symmetrical domain of definition about the origin. f(-x)= 3 \cdot (-x)^(3)-7 \cdot (-x)^(7)= -3x^(3)+7x^(7)= -(3x^(3)-7x^(7))= -f(x).
Hence, the function f(x)=3x^(3)-7x^(7) is odd.
Periodic function
The function y=f(x) , in the domain of which f(x+T)=f(x-T)=f(x) is true for any x, is called periodic function with period T \neq 0 .
Repetition of the graph of the function on any segment of the abscissa axis, which has length T .
Intervals where the function is positive, that is, f (x) > 0 - segments of the abscissa axis, which correspond to the points of the graph of the function that lie above the abscissa axis.
f(x) > 0 on (x_(1); x_(2)) \cup (x_(3); +\infty)
Gaps where the function is negative, i.e. 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))
Function limitation
bounded from below it is customary to call a function y=f(x), x \in X when there exists a number A for which the inequality f(x) \geq A holds for any x \in X .
An example of a function bounded 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 if there exists 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 exists a number K > 0 for which the inequality \left | f(x) \right | \neq K for any x \in X .
Example of a bounded function: y=\sin x is bounded on the whole number line because \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 when a larger value of x will correspond to a larger value of the function y=f(x) . From here it turns out that taking from the considered interval two arbitrary values of the argument x_(1) and x_(2) , and x_(1) > x_(2) , it 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 will correspond to a smaller value of the function y(x) . From here it turns out that taking from the considered interval two arbitrary values of the argument x_(1) and x_(2) , and x_(1) > x_(2) , it will be y(x_(1))< y(x_{2}) .
Function roots it is customary to name the points at which the function F=y(x) intersects the abscissa axis (they are obtained as a result of solving the equation y(x)=0 ).
a) If an even function increases for x > 0, then it decreases for x< 0
b) When an even function decreases for x > 0, then it increases for x< 0
c) When an odd function increases for x > 0, then it also increases for x< 0
d) When an odd function decreases for x > 0, then it will also decrease for x< 0
Function extremes
Function minimum point y=f(x) it is customary to call such a point x=x_(0) , in which its 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 point min.
Function maximum point y=f(x) it is customary to call such a point x=x_(0) , in which its neighborhood will have other points (except for the point x=x_(0) ), and then the inequality f(x) will be satisfied for them< f(x^{0}) . y_{max} - обозначение функции в точке max.
Necessary condition
According to Fermat's theorem: f"(x)=0, then when the function f(x) , which is differentiable at the point x_(0) , an extremum will appear at this point.
Sufficient condition
- When the sign of the derivative changes 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 the function on the interval
Calculation steps:
- Looking for derivative f"(x) ;
- Stationary and critical points of the function are found and those belonging to the interval are chosen;
- The values of the function f(x) are found at the stationary and critical points and ends of the segment. The smallest of the results will be the smallest value of the function, and more - greatest.
- (Math.) A 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 ... ...
F(x) = x is an example of an odd function. f(x) = x2 is an example of an even function. f(x) = x3 ... Wikipedia
A function that satisfies the equality f (x) = f (x). See Even and Odd Functions... Great Soviet Encyclopedia
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
Special functions introduced by the French mathematician E. Mathieu in 1868 when solving problems on the oscillation of an elliptical membrane. M. f. are also used in the study of the propagation of electromagnetic waves in an elliptical cylinder ... Great Soviet Encyclopedia
The "sin" request is redirected here; see also other meanings. The "sec" request is redirected here; see also other meanings. "Sine" redirects here; see also other meanings ... Wikipedia
In July 2020, NASA launches an expedition to Mars. The spacecraft will deliver to Mars an electronic carrier with the names of all registered members of the expedition.
If this post solved your problem or you just liked it, share the link to it with your friends on social networks.
One of these code options needs to be copied and pasted into the code of your web page, preferably between the tags
and or right after the tag . According to the first option, MathJax loads faster and slows down the page less. But the second option automatically tracks and loads the latest versions of MathJax. If you insert the first code, then it will need to be updated periodically. If you paste the second code, then 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 load 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 MathML, LaTeX, and ASCIIMathML markup syntax and you're ready to embed math formulas into your web pages.
Another New Year's Eve... frosty weather and snowflakes on the window pane... All this prompted me to write again about... fractals, and what Wolfram Alpha knows about it. On this occasion, there is an interesting article in which there are examples of two-dimensional fractal structures. Here we will consider 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, it is a self-similar structure, considering 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), when zoomed in, we will see details that have a simpler shape than the original figure itself. For example, at a sufficiently high 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 with each increase will be repeated again and again.
Benoit Mandelbrot, the founder of the science of fractals, in his article Fractals and Art for Science wrote: "Fractals are geometric shapes that are as complex in their details as they are in their overall form. That is, if part of the fractal will be enlarged to the size of the whole, it will look like the whole, or exactly, or perhaps with a slight deformation.
