Function is a mathematical value showing the dependence of one element "at" from another "X".
In other words, dependence at is called a variable function X, if each value that can take X matches one or more defined values at. Variable X- this is function argument.
Value at always depends on the size X, hence the argument X is independent variable, and the function at - dependent variable.
Let's explain with an example:
Let T is the boiling point of water, and R- Atmosphere pressure. When observing, it was found that for each value that can take R, always corresponds to the same value T. In this way, T is the argument function R.
Functional dependency T from R allows, when observing the boiling point of water without a barometer, to determine the pressure according to special tables, for example:
It can be seen that there are values argumentT, which the boiling point cannot accept, for example, it cannot be less than "absolute zero" (- 273 ° C). That is, the impossible value T= - 300 °C, no value corresponds R. Therefore, the definition says: “every value that can take X…", and not for each value of x ...
Wherein R is argument functionT. So the dependency R from T allows, when monitoring pressure without a thermometer, to determine the boiling point of water according to a similar table:
If each value of the argument X corresponds to one function value at, then the function is called unambiguous; if two or more, then ambiguous(two-digit, three-digit). If it is not specified that a function is multivalued, it should be understood that it is single-valued.
For example:
Sum ( S) angles of the polygon are number function (n) sides. Argument n can only take integer values, but not less than 3 . Addiction S from n expressed through the formula:
S = π (n - 2).
The unit of measure in this example is the radian. Wherein n- this is argument function S and functional dependency n from S expressed by the formula:
n = S/ π + 2.
ArgumentS can only take values that are multiples π , (π , 2 π , 3 π etc.).
Let's explain one more example:
Side of a square X is a function of its area S (x = √ S). The argument can take any positive value.
Argument- it's always variable, a function is usually also a variable value depending on the argument, but the possibility of its constancy is not ruled out.
For example:
The distance of a moving point from a stationary one is a function of the travel time, it usually changes, but when the point moves around the circle, the distance from the center remains constant.
In this case, the duration of movement in a circle is not distance function from the center.
So when the function is constant value, then the argument and function cannot be interchanged.
The length of the segment on the coordinate axis is found by the formula:
The length of the segment on the coordinate plane is sought by the formula:
To find the length of a segment in a three-dimensional coordinate system, the following formula is used:
The coordinates of the middle of the segment (for the coordinate axis only the first formula is used, for the coordinate plane - the first two formulas, for the three-dimensional coordinate system - all three formulas) are calculated by the formulas:
Function is a correspondence of the form y= f(x) between variables, due to which each considered value of some variable x(argument or independent variable) corresponds to a certain value of another variable, y(dependent variable, sometimes this value is simply called the value of the function). Note that the function assumes that one value of the argument X there can only be one value of the dependent variable at. However, the same value at can be obtained with various X.
Function scope are all values of the independent variable (function argument, usually X) for which the function is defined, i.e. its meaning exists. The domain of definition is indicated D(y). By and large, you are already familiar with this concept. The scope of a function is otherwise called the domain of valid values, or ODZ, which you have been able to find for a long time.
Function range are all possible values of the dependent variable of this function. Denoted E(at).
Function rises in the interval where greater value argument corresponds to the larger value of the function. Function Decreasing on the interval on which the larger value of the argument corresponds to the smaller value of the function.
Function intervals are the intervals of the independent variable at which the dependent variable retains its positive or negative sign.
Function zeros are those values of the argument for which the value of the function is equal to zero. At these points, the graph of the function intersects the abscissa axis (OX axis). Very often, the need to find the zeros of a function means simply solving the equation. Also, often the need to find intervals of constant sign means the need to simply solve the inequality.
Function y = f(x) are called even X
This means that for any opposite values of the argument, the values of the even function are equal. Schedule even function always symmetrical about the y-axis of the y.
Function y = f(x) are called odd, if it is defined on a symmetric set and for any X from the domain of definition the equality is fulfilled:
This means that for any opposite values of the argument, the values of the odd function are also opposite. The graph of an odd function is always symmetrical about the origin.
The sum of the roots of an even and odd features(points of intersection of the x-axis OX) is always zero, because for every positive root X has a negative root X.
It is important to note that some function does not have to be even or odd. There are many functions that are neither even nor odd. Such functions are called functions general view , and none of the above equalities or properties hold for them.
Linear function is called a function that can be given by the formula:
The graph of a linear function is a straight line and in the general case looks like this (an example is given for the case when k> 0, in this case the function is increasing; for the case k < 0 функция будет убывающей, т.е. прямая будет наклонена в другую сторону - слева направо):
The graph of a parabola is given by a quadratic function:
A quadratic function, like any other function, intersects the OX axis at the points that are its roots: ( x one ; 0) and ( x 2; 0). If there are no roots, then the quadratic function does not intersect the OX axis, if there is one root, then at this point ( x 0; 0) the quadratic function only touches the OX axis, but does not intersect it. A quadratic function always intersects the OY axis at a point with coordinates: (0; c). Schedule quadratic function(parabola) may look like this (the figure shows examples that are far from exhausting all possible types parabola):
Wherein:
Parabola vertex coordinates can be calculated using the following formulas. X tops (p- in the figures above) of a parabola (or the point at which the square trinomial reaches its maximum or minimum value):
Y tops (q- in the figures above) of a parabola or the maximum if the branches of the parabola are directed downwards ( a < 0), либо минимальное, если ветви параболы направлены вверх (a> 0), value square trinomial:
power function
Here are some examples of graphs of power functions:
Inversely proportional dependence call the function given by the formula:
Depending on the sign of the number k an inversely proportional graph can have two principal options:
Asymptote is the line to which the line of the graph of the function approaches infinitely close, but does not intersect. The asymptotes for the inverse proportionality graphs shown in the figure above are the coordinate axes, to which the graph of the function approaches infinitely close, but does not intersect them.
exponential function with base a call the function given by the formula:
a the graph of an exponential function can have two fundamental options (we will also give examples, see below):
logarithmic function call the function given by the formula:
Depending on whether the number is greater or less than one a The graph of a logarithmic function can have two fundamental options:
Function Graph y = |x| as follows:
Function at = f(x) is called periodical, if there exists such a non-zero number T, what f(x + T) = f(x), for anyone X out of function scope f(x). If the function f(x) is periodic with period T, then the function:
where: A, k, b are constant numbers, and k not equal to zero, also periodic with a period T 1 , which is determined by the formula:
Most examples of periodic functions are trigonometric functions. Here are the graphs of the main trigonometric functions. The following figure shows part of the graph of the function y= sin x(the whole graph continues indefinitely to the left and right), the graph of the function y= sin x called sinusoid:
Function Graph y= cos x called cosine wave. This graph is shown in the following figure. Since the graph of the sine, it continues indefinitely along the OX axis to the left and to the right:
Function Graph y=tg x called tangentoid. This graph is shown in the following figure. Like the graphs of other periodic functions, this graph repeats indefinitely along the OX axis to the left and to the right.
And finally, the graph of the function y=ctg x called cotangentoid. This graph is shown in the following figure. Like the graphs of other periodic and trigonometric functions, this graph repeats indefinitely along the OX axis to the left and to the right.
Successful, diligent and responsible implementation of these three points will allow you to show on the VU excellent result, the maximum of what you are capable of.
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Function zeros
The zero of the function is the value X, at which the function becomes 0, that is, f(x)=0.
Zeros are the points of intersection of the graph of the function with the axis Oh.
Function parity
A function is called even if for any X from the domain of definition, the equality f(-x) = f(x) 
An even function is symmetrical about the axis OU
Odd function
A function is called odd if for any X from the domain of definition, the equality f(-x) = -f(x) is satisfied. 
An odd function is symmetrical with respect to the origin.
A function that is neither even nor odd is called a general function. 
Function Increment
The function f(x) is called increasing if the larger value of the argument corresponds to the larger value of the function, i.e. 
Decreasing function
The function f(x) is called decreasing if the larger value of the argument corresponds to the smaller value of the function, i.e. 
The intervals on which the function either only decreases or only increases are called intervals of monotony. The function f(x) has 3 intervals of monotonicity: 
Find intervals of monotonicity using the service Intervals of increasing and decreasing functions
Local maximum
Dot x 0 is called a local maximum point if for any X from a neighborhood of a point x 0 the following inequality holds: f(x 0) > f(x) 
Local minimum
Dot x 0 is called a local minimum point if for any X from a neighborhood of a point x 0 the following inequality holds: f(x 0)< f(x).
Local maximum points and local minimum points are called local extremum points. 
local extremum points.
Function Periodicity
The function f(x) is called periodic, with period T, if for any X f(x+T) = f(x) . 
Constancy intervals
Intervals on which the function is either only positive or only negative are called intervals of constant sign. 
Function continuity
The function f(x) is called continuous at the point x 0 if the limit of the function as x → x 0 is equal to the value of the function at this point, i.e. 
.
break points
The points at which the continuity condition is violated are called points of discontinuity of the function. 
x0- breaking point.
General scheme for plotting functions
1. Find the domain of the function D(y).
2. Find the intersection points of the graph of functions with the coordinate axes.
3. Investigate the function for even or odd.
4. Investigate the function for periodicity.
5. Find intervals of monotonicity and extremum points of the function.
6. Find intervals of convexity and inflection points of the function.
7. Find the asymptotes of the function.
8. Based on the results of the study, build a graph.
Example: Explore the function and build its graph: y = x 3 - 3x
1) The function is defined on the entire real axis, i.e. its domain of definition is D(y) = (-∞; +∞).
2) Find the points of intersection with the coordinate axes:
with the OX axis: solve the equation x 3 - 3x \u003d 0
with axis ОY: y(0) = 0 3 – 3*0 = 0
3) Find out if the function is even or odd:
y(-x) = (-x) 3 - 3(-x) = -x 3 + 3x = - (x 3 - 3x) = -y(x)
It follows that the function is odd.
4) The function is non-periodic.
5) Find the intervals of monotonicity and the extremum points of the function: y’ = 3x 2 - 3.
Critical points: 3x 2 - 3 = 0, x 2 =1, x= ±1.
y(-1) = (-1) 3 – 3(-1) = 2
y(1) = 1 3 – 3*1 = -2
6) Find the convexity intervals and inflection points of the function: y'' = 6x
Critical points: 6x = 0, x = 0.
y(0) = 0 3 – 3*0 = 0
7) The function is continuous, it has no asymptotes.
8) Based on the results of the study, we will construct a graph of the function.
To understand this topic, consider the function shown on the graph // Let's show how the function graph allows you to determine its properties.
The scope of the function is yavl. interval [ 3.5; 5.5].
The range of the function yavl. interval [ 1; 3].
1. At x = -3, x = - 1, x = 1.5, x = 4.5, the value of the function is zero.
The value of the argument, at which the value of the function is zero, is called the zero of the function.
//those. for this function the numbers -3;-1;1.5; 4.5 are zeros.
2. On the intervals [ 4.5; 3) and (1; 1.5) and (4.5; 5.5] the graph of the function f is located above the abscissa axis, and at intervals (-3; -1) and (1.5; 4.5) under the axis abscissa, this is explained as follows - on the intervals [ 4.5; 3) and (1; 1.5) and (4.5; 5.5] the function takes positive values, and on the intervals (-3; -1) and ( 1.5; 4.5) are negative.
Each of the indicated intervals (where the function takes values of the same sign) is called the interval of constant sign of the function f.//i.e. for example, if we take the interval (0; 3), then it is not a constant-sign interval of the given function.
In mathematics, when searching for intervals of sign constancy of a function, it is customary to indicate intervals maximum length. //Those. interval (2; 3) is constancy interval function f, but the answer should include the interval [ 4,5; 3) containing the interval (2; 3).
3. If you move along the x-axis from 4.5 to 2, you will notice that the graph of the function goes down, that is, the values of the function decrease. //In mathematics, it is customary to say that on the interval [ 4,5; 2] the function is decreasing.
As x increases from 2 to 0, the graph of the function goes up, i.e. function values increase. //In mathematics, it is customary to say that on the interval [ 2; 0] the function is increasing.
The function f is called if for any two values of the argument x1 and x2 from this interval such that x2 > x1, the inequality f (x2) > f (x1) is satisfied. // or The function is called increasing over some interval, if for any values of the argument from this interval, the larger value of the argument corresponds to the larger value of the function.//i.e. the more x, the more y.
The function f is called decreasing over some interval, if for any two values of the argument x1 and x2 from this interval such that x2 > x1, the inequality f(x2) decreasing on some interval is satisfied, if for any values of the argument from this interval a larger value of the argument corresponds to a smaller value of the function. //those. the more x, the less y.
If a function is increasing over the entire domain of definition, then it is called increasing.
If a function is decreasing over the entire domain of definition, then it is called waning.
Example 1 graph of increasing and decreasing functions, respectively.
Example 2
Define yavl. is the linear function f(x) = 3x + 5 increasing or decreasing?
Proof. Let's use the definitions. Let x1 and x2 be arbitrary values of the argument, and x1< x2., например х1=1, х2=7
FUNCTION FUNCTION (from the Latin functio - performance, implementation), 1) activity, duty, work; external manifestation of the properties of an object in a given system of relations (for example, the function of the sense organs, the function of money). 2) Function in sociology - the role that a certain social or process performs in relation to the whole (for example, the function of the state, family, etc. in society). 3) Function in mathematics - a correspondence between variables, due to which each value of one value x (independent variable, argument) corresponds to a certain value of another value y (dependent variable, function). Functions can be defined, for example, by a formula, a graph, a table, a rule.
Modern Encyclopedia. 2000 .
Synonyms:- (lat. functio - execution) duty, range of activities. “A function is an existence that we think in action” (Goethe). The science of the functions of the organs of living beings - physiology; special science of functions nervous system- physiology of organs ... ... Philosophical Encyclopedia
function- A team or group of people, and the tools or other resources they use to carry out one or more processes or activities. For example, customer support. This term also has another meaning: ... ... Technical Translator's Handbook
Cm … Synonym dictionary
- (lat. functio). In physiology: the administration by some organ of its own inherent actions, such as breathing, digestion. 2) in mathematics: a quantity that depends on another variable. Dictionary foreign words included in the ... ... Dictionary of foreign words of the Russian language
Function- 1. Dependent variable; 2. Correspondence y \u003d f (x) between variables, due to which each considered value of a certain quantity x (argument or independent variable) corresponds to a certain value ... ... Economic and Mathematical Dictionary
Function- (from the Latin functio performance, implementation), 1) activity, duty, work; external manifestation of the properties of an object in a given system of relations (for example, the function of the sense organs, the function of money). 2) Function in sociology role, ... ... Illustrated encyclopedic Dictionary
- (from lat. functio execution implementation), ..1) activity, duty, work; external manifestation of the properties of any object in a given system of relations (for example, the function of the sense organs, the function of money) 2)] Function in sociology is the role that ... ... Big Encyclopedic Dictionary
FUNCTION, in mathematics, one of the basic concepts, an expression that defines a regular relationship between two sets of variables, which consists in the fact that each element of one set corresponds to a certain, unique ... ... Scientific and technical encyclopedic dictionary
- (function) Relationship between two or more variables. If y is a function of x and is written as y=f(x), then if the value of the argument x is known, the function allows you to show how to find the value of y. If y is a single-valued function of x, then ... ... Economic dictionary
- (from Latin I perform, I perform) a center, a concept in the methodology of functional and structural-functional analysis of c. The concept of "F." began to be actively used in the social sciences from Tue. floor. 19th century in connection with the penetration first ... ... Encyclopedia of cultural studies
