- Substitute positive numeric values into the function x (\displaystyle x) and corresponding negative numeric values. For example, given the function f (x) = 2 x 2 + 1 (\displaystyle f(x)=2x^(2)+1). Substitute the following values into it x (\displaystyle x):
Check whether the graph of the function is symmetrical about the Y axis. Symmetry means a mirror image of the graph relative to the ordinate. 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 y (\displaystyle y)(with a positive value x (\displaystyle x)) corresponds to a negative value y (\displaystyle y)(with a negative value 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 ones into the function negative values x (\displaystyle x):
- According to the results obtained, there is no symmetry. Values y (\displaystyle y) For opposite meanings x (\displaystyle x) do not coincide 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 like this: 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.
Definition 1. The function is called even
(odd
), if together with each variable value 
meaning - X also belongs 
and the equality holds
Thus, a function can be even or odd only if its domain of definition is symmetrical about the origin of coordinates on the number line (number X And - X belong at the same time 
). For example, the function 
is neither even nor odd, since its domain of definition 
not symmetrical about the origin.
Function 
even, because 
symmetrical about the origin and.
Function 
odd, because 
And 
.
Function 
is not even and odd, since although 
and is symmetrical with respect to the origin, equalities (11.1) are not satisfied. For example,.
The graph of an even function is symmetrical about the axis Oh, because if the point 
also belongs to the schedule. The graph of an odd function is symmetrical about the origin, since if 
belongs to the graph, then the point 
also belongs to the schedule.
When proving whether a function is even or odd, the following statements are useful.
Theorem 1. a) The sum of two even (odd) functions is an even (odd) function.
b) The product of two even (odd) functions is an even function.
c) The product of an even and odd function is an odd function.
d) If f– even function on the set X, and the function g
defined on the set 
, then the function 
– even.
d) If f– odd function on the set X, and the function g
defined on the set 
and even (odd), then the function 
– even (odd).
Proof. Let us prove, for example, b) and d).
b) Let 
And 
– even functions. Then, therefore. The case of odd functions is treated similarly 
And 
.
d) Let f is an even function. Then.
The remaining statements of the theorem can be proved in a similar way. The theorem is proven.
Theorem 2. Any function 
, defined on the set X, symmetrical about the origin, can be represented as a sum of even and odd functions.
Proof. Function 
can be written in the form
.
Function 
– even, because 
, and the function 
– odd, because. Thus, 
, Where 
– even, and 
– odd functions. The theorem is proven.
Definition 2. Function 
called periodic
, if there is a number 
, such that for any 
numbers 
And 
also belong to the domain of definition 
and the equalities are satisfied
Such a number T called period
functions 
.
From Definition 1 it follows that if T– period of the function 
, then the number – T Same
is the period of the function
(since when replacing T on – T equality is maintained). Using the method of mathematical induction it can be shown that if T– period of the function f, then 
, is also a period. It follows that if a function has a period, then it has infinitely many periods.
Definition 3. The smallest of the positive periods of a function is called its main period.
Theorem 3. If T– main period of the function f, then the remaining periods are multiples of it.
Proof. Let us assume the opposite, that is, that there is a period 
functions f
(
>0), not multiple T. Then, dividing 
on T with the remainder, we get 
, Where 
. That's why
that is 
– period of the function f, and 
, and this contradicts the fact that T– main period of the function f. The statement of the theorem follows from the resulting contradiction. The theorem is proven.
It is well known that trigonometric functions are periodic. Main period 
And 
equals 
,
And 
. Let's find the period of the function 
. Let 
- the period of this function. Then
(because 
.
oror 
.
Meaning T, determined from the first equality, cannot be a period, since it depends on X, i.e. is a function of X, and not a constant number. The period is determined from the second equality: 
. There are infinitely many periods, with 
the smallest positive period is obtained at 
:
. This is the main period of the function 
.
An example of a more complex periodic function is the Dirichlet function
Note that if T is a rational number, then 
And 
are rational numbers for rational X and irrational when irrational X. That's why
for any rational number T. Therefore, any rational number T is the period of the Dirichlet function. It is clear that this function does not have a main period, since there are positive rational numbers, arbitrarily close to zero (for example, a rational number can be made a choice n arbitrarily close to zero).
Theorem 4. If the function f
defined on the set X and has a period T, and the function g
defined on the set 
, then a complex function 
also has a period T.
Proof. We have, therefore
that is, the statement of the theorem is proven.
For example, since cos
x
has a period 
, then the functions 
have a period 
.
Definition 4. Functions that are not periodic are called non-periodic .
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 each 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.
Removable Break Points
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 direct 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
Horizontal asymptote - straight species, subject to the existence limit
.
Inclined
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 formula horizontal asymptote, 
.
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 lowest values functions y = f(x) on the interval(continuation)
|
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 . 4. Find the values of the function at selected points and at the ends of the segment: f(x 1), f(x 2),..., f(x a),f(x b), 5. Selecting the largest and smallest function values from those found. Comment. If on the segment [ a; b] there are discontinuity points, then it is necessary to calculate one-sided limits at 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 inflection points at the junctions of the convex and concave intervals. Calculate the value of the function at the inflection points. If a function has other points of continuity (except for inflection points) at which the second derivative is 0 or does not exist, then it is also useful to calculate the value of the function at these points. Having found f(x), we solve the inequality f(x)0. On each of the solution intervals the function will be convex downward. Solving the inverse inequality f(x)0, we find the intervals on which the function is convex upward (that is, concave). We define inflection points as those points at which the function changes direction of convexity (and is continuous).
Inflection point of a function- 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
A necessary condition for the existence of an inflection point: if the function is twice differentiable in some punctured neighborhood of the point , then or 
.
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.
We have already seen 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 )
