The dependence of a variable y on a variable x, in which each value of x corresponds to a single value of y is called a function. For designation use the notation y=f(x). Each function has a number of basic properties, such as monotonicity, parity, periodicity and others.
Take a closer look at the parity property.
A function y=f(x) is called even if it satisfies the following two conditions:
2. The value of the function at point x, belonging to the domain of definition of the function, must be equal to the value of the function at point -x. That is, for any point x, the following equality must be satisfied from the domain of definition of the function: f(x) = f(-x).
If you plot a graph of an even function, it will be symmetrical about the Oy axis.
For example, the function y=x^2 is even. Let's check it out. The domain of definition is the entire numerical axis, which means it is symmetrical about point O.
Let's take an arbitrary x=3. f(x)=3^2=9.
f(-x)=(-3)^2=9. Therefore f(x) = f(-x). Thus, both conditions are met, which means the function is even. Below is a graph of the function y=x^2.
The figure shows that the graph is symmetrical about the Oy axis.
A function y=f(x) is called odd if it satisfies the following two conditions:
1. The domain of definition of a given function must be symmetrical with respect to point O. That is, if some point a belongs to the domain of definition of the function, then the corresponding point -a must also belong to the domain of definition of the given function.
2. For any point x, the following equality must be satisfied from the domain of definition of the function: f(x) = -f(x).
The graph of an odd function is symmetrical with respect to point O - the origin of coordinates. For example, the function y=x^3 is odd. Let's check it out. The domain of definition is the entire numerical axis, which means it is symmetrical about point O.
Let's take an arbitrary x=2. f(x)=2^3=8.
f(-x)=(-2)^3=-8. Therefore f(x) = -f(x). Thus, both conditions are met, which means the function is odd. Below is a graph of the function y=x^3.
The figure clearly shows that even function y=x^3 is symmetrical about the origin.
Which were familiar to you to one degree or another. It was also noted there that the stock of function properties will be gradually replenished. Two new properties will be discussed in this section.
Definition 1.
The function y = f(x), x є X, is called even if for any value x from the set X the equality f (-x) = f (x) holds.
Definition 2.
The function y = f(x), x є X, is called odd if for any value x from the set X the equality f (-x) = -f (x) holds.
Prove that y = x 4 is an even function.
Solution. We have: f(x) = x 4, f(-x) = (-x) 4. But(-x) 4 = x 4. This means that for any x the equality f(-x) = f(x) holds, i.e. the function is even.
Similarly, it can be proven that the functions y - x 2, y = x 6, y - x 8 are even.
Prove that y = x 3 ~ an odd function.
Solution. We have: f(x) = x 3, f(-x) = (-x) 3. But (-x) 3 = -x 3. This means that for any x the equality f (-x) = -f (x) holds, i.e. the function is odd.
Similarly, it can be proven that the functions y = x, y = x 5, y = x 7 are odd.
You and I have already been convinced more than once that new terms in mathematics most often have an “earthly” origin, i.e. they can be explained somehow. This is the case with both even and odd functions. See: y - x 3, y = x 5, y = x 7 - odd functions, while y = x 2, y = x 4, y = x 6 are even functions. And in general, for any function of the form y = x" (below we will specifically study these functions), where n is a natural number, we can conclude: if n is an odd number, then the function y = x" is odd; if n is an even number, then the function y = xn is even.
There are also functions that are neither even nor odd. Such, for example, is the function y = 2x + 3. Indeed, f(1) = 5, and f (-1) = 1. As you can see, here, therefore, neither the identity f(-x) = f ( x), nor the identity f(-x) = -f(x).
So, a function can be even, odd, or neither.
Studying the question of whether given function even or odd is usually called the study of a function for parity.
In definitions 1 and 2 we're talking about about the values of the function at points x and -x. This assumes that the function is defined at both point x and point -x. This means that point -x belongs to the domain of definition of the function simultaneously with point x. If a numerical set X, together with each of its elements x, also contains the opposite element -x, then X is called a symmetric set. Let's say, (-2, 2), [-5, 5], (-oo, +oo) are symmetric sets, while 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 .
An example of a limited function: y=\sin x is limited on the entire number axis, since \left | \sin x \right | \neq 1.
It is customary to speak of a function that increases on the interval under consideration as increasing function then when higher value x will correspond 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 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}) .
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
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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
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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.
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.
Calculation steps:
Function study.
1) D(y) – Definition domain: the set of all those values of the variable x. for which the algebraic expressions f(x) and g(x) make sense.
If a 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) Properties of the function: even/odd, periodicity:
Odd And even functions are called whose graphs are symmetric with respect to changes 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 (symmetrical relative to the center of coordinates).
Even function- a function that does not change its value when the sign of the independent variable changes (symmetrical about the ordinate).
Neither even nor odd function (function general view) - 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 general functions).
Odd functions
Odd power where is an arbitrary integer.
Even functions
Even power where is an arbitrary integer.
Periodic function- a function that repeats its values at some regular argument interval, that is, it does not change its value when adding some fixed non-zero number to the argument ( period functions) over the entire domain of definition.
3) Zeros (roots) of a function are the points where it becomes zero.
Finding the intersection point 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 at which the graph intersects the axis are called function zeros. To find the zeros of a function you need to solve the equation, that is, find those meanings of "x", at which the function becomes zero.
4) Intervals of constancy of signs, signs in them.
Intervals where the function f(x) maintains sign.
The interval of constancy of sign is the interval at every point of which the function is positive or negative.
ABOVE the x-axis.
BELOW the axle.
5) Continuity (points of discontinuity, nature of the 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.
If the limit of the function exists, but the function is not defined at this point, or the limit does not coincide with the value of the function at this point:
,
then the point is called removable break point functions (in complex analysis, a removable singular point).
If we “correct” the function at the point of removable discontinuity and put 
, then we get a function that is continuous at a given point. This operation on a function is called extending the function to continuous or redefinition of the function by continuity, which justifies the name of the point as a point removable rupture.
Discontinuity points of the first and second kind
If a 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 associated with the existence of numerical functions unilateral limits:
if both one-sided limits exist and are finite, then such a point is called discontinuity 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 point of discontinuity of the second kind.
Asymptote - straight, which has the property that the distance from a point on the curve to this straight tends to zero as the point moves away 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 asymptote - straight species, subject to the existence limit
.
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 using the horizontal asymptote formula, 
.
6) Finding intervals of monotonicity. Find intervals of monotonicity of a function f(x)(that is, intervals of increasing and decreasing). 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 intervals where this inequality holds, the function f(x)increases. Where the reverse inequality holds f(x)0, function f(x) is decreasing.
Finding a local extremum. Having found the intervals of monotonicity, we can immediately determine the local extremum points where an increase is replaced by a decrease, local maxima are located, and where a decrease is replaced by an increase, local minima are located. 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)
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1. Find the derivative of the function: f(x). 2. Find the points at which the derivative is zero: f(x)=0x 1, x 2 ,... 3. Determine the affiliation of points X 1 ,X 2 , … segment [ a; b]: let x 1a;b, A x 2a;b . |
Converting graphs.
Verbal description of the function.
Graphic method.
The graphical method of specifying a function is the most visual and is often used in technology. IN mathematical analysis The graphical method of specifying functions is used as an illustration.
Function graph f is the set of all points (x;y) of the coordinate plane, where y=f(x), and x “runs through” the entire domain of definition of this function.
A subset of the coordinate plane is a graph of a function if it has no more than one common point with any straight line parallel to the Oy axis.
Example. Are the figures shown below graphs of functions?
Advantage graphic task is its visibility. You can immediately see how the function behaves, where it increases and where it decreases. From the graph you can immediately recognize some important characteristics functions.
In general, analytical and graphic ways function assignments go hand in hand. Working with the formula helps to build a graph. And the graph often suggests solutions that you wouldn’t even notice in the formula.
Almost any student knows the three ways to define a function that we just looked at.
Let's try to answer the question: "Are there other ways to define a function?"
There is such a way.
The function can be quite unambiguously specified in words.
For example, the function y=2x can be specified by the following verbal description: each real value of the argument x is associated with its double value. The rule is established, the function is specified.
Moreover, you can verbally specify a function that is extremely difficult, if not impossible, to define using a formula.
For example: each value of the natural argument x is associated with the sum of the digits that make up the value of x. For example, if x=3, then y=3. If x=257, then y=2+5+7=14. And so on. It is problematic to write this down in a formula. But the sign is easy to make.
The method of verbal description is a rather rarely used method. But sometimes it does.
If there is a law of one-to-one correspondence between x and y, then there is a function. What law, in what form it is expressed - a formula, a tablet, a graph, words - does not change the essence of the matter.
Let us consider functions whose domains of definition are symmetrical with respect to the origin, i.e. for anyone X from the domain of definition number (- X) also belongs to the domain of definition. Among these functions are even and odd.
Definition. The function f is called even, if for any X from its domain of definition
Example. Consider the function 
It is even. Let's check it out.
For anyone X equalities are satisfied
Thus, both conditions are met, which means the function is even. Below is a graph of this function.
Definition. The function f is called odd, if for any X from its domain of definition 
Example. Consider the function 
It is odd. Let's check it out.
The domain of definition is the entire numerical axis, which means it is symmetrical about the point (0;0).
For anyone X equalities are satisfied
Thus, both conditions are met, which means the function is odd. Below is a graph of this function.
The graphs shown in the first and third figures are symmetrical about the ordinate axis, and the graphs shown in the second and fourth figures are symmetrical about the origin.
Which of the functions whose graphs are shown in the figures are even and which are odd?
