What is a function? Functional dependence, or function, is such a relationship between two variables, in which each value of the independent variable. Main properties of the function

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 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. Thus, 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:

The second function definition.

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 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.

LECTURE 1. FUNCTIONAL DEPENDENCE.

1. The concept of a function

The concept of a function, along with the concept of a number and a variable, is one of the most important concepts of modern mathematics. In natural science and technology, we often encounter dependencies of some quantities on others with the so-called functional dependencies.

The functional dependence of one quantity (y) on another (x) means that each value of x corresponds to a single value of y . The value x is called the independent variable, and y is called the dependent variable, or a function of this variable. It is also said that x is an argument to the function y .

The term "function" was first introduced in 1692 by Gottfried Wilhelm Leibniz.

1. The area S of a square is a function of the length a of its side: S = a2. 2. The volume V of the ball can be expressed in terms of the radius R of the ball:

V = 4 3 πR3 .

3. The volume of a cone V with a given height h depends on the radius r of its base:

V = 1 3 πr2 h.

4. Let the path z traveled by a freely falling body depend on time t,

elapsed since the fall began. This dependence is expressed by the formula z = gt 2 2 (g free fall acceleration).

Definition 1. If each value that the variable x can take, according to some rule or law, is assigned one specific value of the variable y, then they say that y is a single-valued function of x, and designate y \u003d f (x) .

The set of all values ​​of the argument x for which the function y = f (x) is defined is called the domain of this function (O.O.F.).

The set of all values ​​taken by the variable y is called the function range (F.F.F.) of the function y = f (x) .

A function is called even if for any x from the domain of definition the equality f (−x) = f (x) holds.

A function is called odd if for any x from the domain of definition the equality f (−x) = −f (x) holds.

A function is called periodic with period T > 0 if for any x from the region

Solution. The domain of definition of the arcsine is the set of points from the interval [−1, 1]. Therefore, the problem reduces to solving the inequality

So, f (−x) = −f (x), i.e., the function is odd.

f (x + 2π) = tan(x + 2π) sin(3x + 2π) + ctg(2x + 2π) =

Tg x sin 3x + ctg 2x = f(x).

So, f (x + 2π) = f (x), i.e., the function is periodic with a period of 2π.

2. Ways to set a function

The analytical way is to define a function using formulas or equations.

For example: y = sin x, y = x2 , y2 + x2 = 1, etc.

If the equation by which the function is defined is not resolved with respect to y , then the function is called implicit. When such a solution is possible, the implicit function can be reduced to an explicit form, i.e., to the form y = f (x) .

For example, the equation 2x + 3y − 5 = 0 can be considered as an implicit function. Solving it for y , we get the same function, but in an explicit form:

y = 5 − 2x.

Note that in the analytical way of specifying a function, there are cases when the function is specified not by one, but by several formulas, for example:

The table way is a way of defining a function using a table. Examples of such a task are tables of trigonometric functions, logarithms, etc. The tabular method of specifying a function is widely used in various experiments and observations. Tables are easy to handle, but the disadvantage of this method is that the function is not specified for all values ​​of the argument.

Graphic way. The graph of the function y = f (x) is the set of points (x, y) of the XOY plane, the coordinates of which are related by the relation y = f (x) .

The advantage of the graphical method of defining a function is its clarity. The graphical way of setting the function is used in the operation of various recorders. In medicine, for example, the work of the heart is analyzed using a cardiograph.

Power, exponential, logarithmic, trigonometric, inverse trigonometric, constant (constant) functions are called basic elementary functions.

Graphs of basic elementary functions

3. Multivalued Functions

Sometimes we have to consider a situation where each value of the independent variable x is associated with several values ​​of y. In this case, they say that

for √ x > 0: each positive corresponds to two real numbers,

differing in signs, the squares of which are equal to x. As for the Arcsinx function, it assigns to each x value from the interval [−1, 1] an infinite set of y values ​​that can be written by the formula

y = (−1)k arcsin x + πk, (k = 0, 2, . . .).

If it is necessary to consider a function as multi-valued, then this must be specifically specified.

4. Inverse function

If the equation y = f (x) can be uniquely resolved with respect to x , then the function x = g(y) is said to be inverse with respect to y = f (x) . Denoted x = f −1 (y) . Moreover, y ≡ f (f−1 (y)).

Sometimes standard notation is followed: x is understood as an independent variable, and y is a function, that is, a dependent variable. In this case, the inverse function should be written as y = g(x) .

For example, we can say that the functions y = 2x and y = log2 x are mutually inverse. In order to obtain a graph of the inverse function y = g(x) from the graph of this function y = f (x), it is enough to display the first graph symmetrically with respect to the bisector of the 1st and 3rd coordinate angles.

Example 5. Given a function y = 1 − 2 − x . Find the inverse function.

2−x = 1 − y, x = − lg(1 − y) .lg 2

Function domain (O.O.F.) −∞< y < 1 .

5. Complicated function

Let the variable y depend on the variable u , which in turn depends on the variable x: y = f (u), u = ϕ(x) . Then when x changes, u will change, and therefore y will also change. Hence, y is a function of x: y = f (ϕ(x)). This function is called a complex function (or a function of a function), the variable u is called an intermediate one. This complex function is also called the superposition of the functions f and ϕ .

Example 6. Given a function f (x) = arccos(lg(x)) . Find a) f (10 1 ); b) f(1); c) f (10).

a) f (10 1 ) = arccos(lg(10 1 )) = arccos(−1) = π.

b), c) calculate independently.

Any function that is obtained from basic elementary functions by a finite number of superpositions and four arithmetic operations is called an elementary function. For example, a polynomial of degree n is an elementary function.

6. Parametric way of defining a function

A function is said to be specified parametrically if the dependence of y on x is specified using the parameter t: where t runs through some numerical values.

If you build these points on the XOY plane, you can see that with a continuous change in t, we get a circle of radius one with the center at the origin. Or you can do it differently, exclude the parameter t, then x2 + y2 = cos2 t + sin2 t = 1.

7. Plotting functions

Consider the simplest transformations of function graphs.

1. The graph of the function y = f (x + a) is obtained from the graph of the function y = f (x) by parallel shifting it along the Ox axis by |a| scale units in the opposite direction of a.

2. Graph of the function y = f (kx) (k >

''compressing'' it toward the Oy axis by a factor of k for k > 1 and ''stretching'' it away from the Oy axis by a factor of 1/k for

k< 1.

3. The graph of the function y = kf(x) (k > 0) is obtained from the graph of the function y = f(x)

''stretching'' it from the Ox axis by a factor of k for k > 1 and ''compressing'' it toward the Ox axis by a factor of 1/k for

k< 1.

4. The graph of the function y = f (x) + b is obtained from the graph of the function y = f (x) by shifting it parallel along the Oy axis by |b| scale units in the same direction as b.

5. The graph of the function y = −f (x) is symmetrical to the graph of the function y = f (x) about the Ox axis.

Consider plotting the function y = kf (mx + b) + a by transforming the graph of the function y = f (x). Let us first perform the identical transformation

Now, successively applying transformations 1 - 5, we build the desired graph of the function.

Example 8. Graph the function y = 3 sin(2x + 4) by transforming the graph of the function y = sin x.

Solution. Let's perform the identical transformation

y = 3 sin(2x + 4) = 3 sin 2(x + 2).

We will plot the function in the following order. 1. We build a graph of the function y \u003d sin x on the segment.

2. The graph of the function y = 2 sin x will be obtained by compressing the graph of the function y = sin x twice along the x-axis.

3. To plot the function y \u003d sin 2 (x + 2), it is necessary to move the graph of the function y \u003d sin 2x to the left along the abscissa axis by two units.

4. The graph of the function y \u003d 3 sin 2 (x + 2) is obtained from the graph of the function y \u003d sin 2 (x + 2) by stretching it along the y-axis three times.

I. y = sin x. II. y = sin 2x.

III. y = sin 2(x + 2). IV. y = 3 sin 2(x + 2).

Definition: A numerical function is a correspondence that maps a single number y to each number x from some given set.

Designation:

where x is an independent variable (argument), y is a dependent variable (function). The set of values ​​x is called the domain of the function (denoted D(f)). The set of values ​​y is called the range of the function (denoted by E(f)). The graph of a function is the set of points in the plane with coordinates (x, f(x))

Ways to set a function.

1. analytical method (using a mathematical formula);
2. tabular method (using a table);
3. descriptive method (using a verbal description);
4. graphical method (using a graph).

Basic properties of the function.

1. Even and odd

A function is called even if
– the domain of definition of the function is symmetric with respect to zero
f(-x) = f(x)

The graph of an even function is symmetrical about the axis 0y

A function is called odd if
– the domain of definition of the function is symmetric with respect to zero
– for any x from the domain of definition f(-x) = -f(x)

The graph of an odd function is symmetrical about the origin.

2. Periodicity

The function f(x) is called periodic with a period if for any x from the domain of definition f(x) = f(x+T) = f(x-T) .

The graph of a periodic function consists of infinitely repeating identical fragments.

3. Monotony (increase, decrease)

The function f(x) increases on the set P if for any x 1 and x 2 from this set, such that x 1

The function f(x) is decreasing on the set P if for any x 1 and x 2 from this set, such that x 1 f(x 2) .

4. Extremes

The point X max is called the maximum point of the function f (x) if for all x from some neighborhood X max , the inequality f (x) f (X max) is satisfied.

The value Y max =f(X max) is called the maximum of this function.

X max - maximum point
Max has a maximum

The point X min is called the minimum point of the function f (x) if for all x from some neighborhood X min, the inequality f (x) f (X min) is satisfied.

The value of Y min =f(X min) is called the minimum of this function.

X min - minimum point
Y min - minimum

X min , X max - extremum points
Y min , Y max - extrema.

5. Function zeros

The zero of the function y = f(x) is the value of the argument x at which the function vanishes: f(x) = 0.

X 1, X 2, X 3 are zeros of the function y = f(x).

Tasks and tests on the topic "Basic properties of a function"

• Function properties - Numerical functions Grade 9

  Lessons: 2 Assignments: 11 Tests: 1

• Properties of logarithms - Exponential and logarithmic functions Grade 11

  Lessons: 2 Assignments: 14 Tests: 1

• Square root function, its properties and graph - Square root function. Square root properties Grade 8

  Lessons: 1 Assignments: 9 Tests: 1

• Power functions, their properties and graphs - Degrees and roots. Power functions Grade 11

  Lessons: 4 Assignments: 14 Tests: 1

• Functions - Important topics for repeating the exam in mathematics

  Tasks: 24

Having studied this topic, you should be able to find the domain of definition of various functions, determine the monotonicity intervals of a function using graphs, and examine functions for even and odd. Consider the solution of such problems on the following examples.

Examples.

1. Find the domain of the function.

Solution: the scope of the function is found from the condition

Definition of a function, setting area and set of values. Definitions related to function notation. Definitions of complex, numeric, real, monotonic and multivalued functions. Definitions of maximum, minimum, upper and lower bounds for bounded functions.

Definition
Function y=f (x) the law (rule, mapping) is called, according to which, each element x of the set X is associated with one and only one element y of the set Y .

The set X is called function scope.
Set of elements y ∈ Y, which have preimages in the set X , is called set of function values(or range).

Domain functions are sometimes called set of definitions or set of tasks functions.

Element x ∈ X called function argument or independent variable.
y element ∈ Y called function value or dependent variable.

The mapping f itself is called function characteristic.

The characteristic f has the property that if two elements and from the definition set have equal values: , then .

The character denoting the characteristic can be the same as the character of the function value element. That is, you can write it like this: At the same time, it is worth remembering that y is an element from the set of function values, and is a rule according to which element x is associated with element y .

The process of calculating the function itself consists of three steps. In the first step, we select an element x from the set X . Further, with the help of the rule , the element x is associated with the element of the set Y . In the third step, this element is assigned to the y variable.

Private value of the function name the value of the function for the selected (private) value of its argument.

The graph of the function f is called a set of pairs.

Complex functions

Definition
Let the functions and be given. Moreover, the domain of the function f contains a set of values ​​of the function g. Then each element t from the domain of the function g corresponds to an element x , and this x corresponds to y . This correspondence is called complex function: .

A complex function is also called composition or superposition of functions and is sometimes referred to as:

In mathematical analysis, it is generally accepted that if the characteristic of a function is denoted by one letter or symbol, then it sets the same correspondence. However, in other disciplines, there is another way of notation, according to which mappings with the same characteristic, but different arguments, are considered different. That is, the mappings and are considered distinct. Let's take an example from physics. Suppose we are considering the dependence of the momentum on the coordinate . And let we have the dependence of the coordinate on time . Then the dependence of momentum on time is a complex function. But for brevity, it is denoted as follows:. With this approach, and are different functions. With the same values ​​of the arguments, they can give different values. In mathematics, this notation is not accepted. If a reduction is required, then a new characteristic should be entered. For example . Then it is clearly seen that and are different functions.

Valid Functions

The scope of the function and the set of its values ​​can be any sets.
For example, numerical sequences are functions whose domain of definition is the set of natural numbers, and the set of values ​​is real or complex numbers.
The cross product is also a function, since for two vectors and there is only one value of the vector . Here the domain of definition is the set of all possible pairs of vectors . The set of values ​​is the set of all vectors.
The boolean expression is a function. Its domain of definition is the set of real numbers (or any set in which the operation of comparison with the element “0” is defined). The set of values ​​consists of two elements - “true” and “false”.

Numerical functions play an important role in mathematical analysis.

Numeric function is a function whose values ​​are real or complex numbers.

Real or real function is a function whose values ​​are real numbers.

Maximum and minimum

Real numbers have a comparison operation. Therefore, the set of values ​​of the real function can be limited and have the largest and smallest values.

The actual function is called limited from above (from below), if there is such a number M that the following inequality holds for all:
.

The number function is called limited, if there exists a number M such that for all :
.

Maximum M (minimum m) function f , on some set X is called the value of the function for some value of its argument , for which for all ,
.

top face or exact upper bound real, bounded from above, the function is called the smallest of the numbers that limits the range of its values ​​from above. That is, this is a number s for which, for all and for any , there is such an argument, the value of the function of which exceeds s′ : .
The upper bound of the function can be denoted as follows:
.

The upper bound of a function unbounded from above

bottom face or precise lower bound real, bounded from below, the function is called the largest of the numbers that limits the range of its values ​​from below. That is, this is a number i for which for all and for any , there is such an argument , the value of the function from which is less than i′ : .
The lower bound of a function can be denoted as follows:
.

The lower bound of a function unbounded from below is the point at infinity.

Thus, any real function, on a non-empty set X , has an upper and lower bound. But not every function has a maximum and a minimum.

As an example, consider a function defined on an open interval.
It is limited, on this interval, from above by the value 1 and below - the value 0 :
for all .
This function has top and bottom faces:
.
But it has no maximum and minimum.

If we consider the same function on the interval , then it is bounded above and below on this set, has upper and lower bounds, and has a maximum and a minimum:
for all ;
;
.

Monotonic functions

Definitions of Increasing and Decreasing Functions
Let the function be defined on some set of real numbers X . The function is called strictly increasing (strictly decreasing)
.
The function is called non-decreasing (non-increasing), if for all such that the following inequality holds:
.

Definition of a monotonic function
The function is called monotonous if it is non-decreasing or non-increasing.

Multivalued Functions

An example of a multivalued function. Its branches are marked with different colors. Each branch is a feature.

As follows from the definition of the function, each element x from the domain of definition is associated with only one element from the set of values. But there are mappings in which the element x has several or an infinite number of images.

As an example, consider the function arcsine: . It is the inverse of the function sinus and is determined from the equation:
(1) .
For a given value of the independent variable x belonging to the interval , this equation satisfies infinitely many values ​​of y (see figure).

Let us impose a restriction on the solutions of Eq. (1). Let
(2) .
Under this condition, the given value corresponds to only one solution of equation (1). That is, the correspondence defined by equation (1) under condition (2) is a function.

Instead of condition (2), one can impose any other condition of the form:
(2.n) ,
where n is an integer. As a result, for each value of n , we will get our own function, different from the others. Many of these functions are multivalued function. And the function determined from (1) under condition (2.n) is branch of a multivalued function.

This is a collection of functions defined on some set.

Multivalued function branch is one of the functions included in the multivalued function.

single valued function is a function.

References:
O.I. Demons. Lectures on mathematical analysis. Part 1. Moscow, 2004.
L.D. Kudryavtsev. Course of mathematical analysis. Volume 1. Moscow, 2003.
CM. Nikolsky. Course of mathematical analysis. Volume 1. Moscow, 1983.