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.
3. Study of the function for monotonicity.
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 
.
4. Study of the function at the extremum.
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.
Function is one of the most important mathematical concepts. Function - variable dependency at from variable x, if each value X matches a single value at. Variable X called the independent variable or argument. Variable at 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 takes (variable y), form the range of values of the function.
Function graph call 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 are plotted along the abscissa axis x, and the values of the variable are plotted along the ordinate axis y. 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) Function domain and function range.
The domain of a function is the set of all valid valid argument values x(variable x), for which the function y = f(x) determined.
The range of a function is the set of all real values y, which the function accepts.
In elementary mathematics, functions are studied only on the set of real numbers.
2) Function zeros.
Values X, at which y=0, 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 are such intervals of values x, on which the function values y 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) Even (odd) 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 is true f(-x) = - f(x). 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 the point a belongs to the domain of definition, then the 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.
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)
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).
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). 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.
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?
The advantage of a graphic task is its clarity. You can immediately see how the function behaves, where it increases and where it decreases. From the graph you can immediately find out some important characteristics of the function.
In general, analytical and graphical methods of defining a function 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 it’s easy to make a sign.
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?
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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
The function is even function, when f(-x)=f(x) for any x from the domain of definition. Such a function will be symmetrical about the Oy axis.
The function is odd function, when f(-x)=-f(x) for any x from the domain of definition. Such a function will be symmetric about the origin O (0;0) .
The function is not even, neither odd and is called general function, when it does not have 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 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
Bounded from below It is customary to call a function y=f(x), x \in X 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 .
Bounded from above a function y=f(x), x \in X is called when 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 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 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.
Increasing and decreasing function
It is customary to speak of a function that increases on the interval under consideration as increasing function then, 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 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
.png)
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
.png)
Extrema of the function
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.
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 more - the largest.
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).
Graph of an even function
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.
Graph of an odd function
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 the odd function y=x^3 is symmetrical about the origin.