Function types and their properties. Basic concepts and properties of functions

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

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.

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

Basic properties functions.

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) .

Schedule 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 - Numeric functions Grade 9
Lessons: 2 Assignments: 11 Tests: 1
Properties of logarithms - Demonstrative and logarithmic function Grade 11
Lessons: 2 Assignments: 14 Tests: 1
Square root function, its properties and graph - Function square root. 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 repetition of the exam 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.

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

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The scope and range of the function. In elementary mathematics, functions are studied only on the set real numbers R.This means that the function argument can only take on those real values for which the function is defined, i.e. it also only accepts real values. A bunch of X all valid valid values of the argument x, for which the function y= f(x) is defined, called function scope. A bunch of Y all real values y that the function accepts is called function range. Now you can give more precise definition features: rule(law) of correspondence between sets X and Y, by which for each element from the setX can find one and only one element from the set Y, is called a function.

It follows from this definition that a function is considered given if:

The scope of the function is set X ;

The scope of the function is set Y ;

The rule (law) of correspondence is known, and such that for each

Only one function value can be found for an argument value.

This requirement of uniqueness of the function is mandatory.

monotonic function. If for any two values of the argument x 1 and x 2 of the condition x 2 > x 1 follows f(x 2) > f(x 1), then the function f(x) is called increasing; if for any x 1 and x 2 of the condition x 2 > x 1 follows f(x 2) < f(x 1), then the function f(x) is called waning. A function that only increases or only decreases is called monotonous.

Limited and unlimited functions. The function is called limited if there is such positive number M what | f(x) | M for all values x . If no such number exists, then the function is unlimited.

EXAMPLES.

The function depicted in Fig. 3 is bounded, but not monotonic. The function in Figure 4 is just the opposite, monotonic, but unlimited. (Explain this please!)

Continuous and discontinuous functions. Function y = f (x) is called continuous at the pointx = a, if:

1) the function is defined for x = a, i.e. f (a) exist;

2) exists finite limit lim f (x) ;

x→a

(See "Limits of Functions")

3) f (a) = lim f (x) .

x→a

If at least one of these conditions is not met, then the function is called discontinuous at the point x = a.

If the function is continuous in all points of its domain of definition, then it is called continuous function.

Even and odd functions. If for any x f(- x) = f (x), then the function is called even; if it does: f(- x) = - f (x), then the function is called odd. Schedule even functionsymmetrical about the Y axis(Fig.5), a graph odd function Simmetric about the origin(Fig. 6).

Periodic function. Function f (x) - periodical if there is such non-zero number T what for any x from the scope of the function definition takes place: f (x + T) = f (x). Such least the number is called function period. All trigonometric functions are periodic.

EXAMPLE 1. Prove that sin x has a period of 2.

SOLUTION We know that sin ( x+ 2n) = sin x, where n= 0, ± 1, ± 2, …

Therefore, adding 2 n to the sine argument

Changes its value. Is there another number with this

Same property?

Let's pretend that P- such a number, i.e. equality:

Sin ( x+ P) = sin x,

Valid for any value x. But then it has

Location and x= / 2 , i.e.

sin(/2 + P) = sin / 2 = 1.

But according to the reduction formula sin ( / 2 + P) = cos P. Then

It follows from the last two equalities that cos P= 1, but we

We know that this is true only when P = 2n. Since the smallest

A non-zero number out of 2 n is 2, then this number

And there is a sin period x. It is proved similarly that 2 from n is , so this is the period sin 2 x.

Function nulls. The value of the argument for which the function is equal to 0 is called zero (root) functions. A function can have multiple zeros. For example, the function y = x (x + 1) (x-3) has three zeros: x= 0, x= -1, x= 3. Geometrically function null - is the abscissa of the point of intersection of the graph of the function with the axis X .

Figure 7 shows the graph of the function with zeros: x= a, x = b and x= c.

Asymptote. If the graph of a function approaches a certain straight line indefinitely as it moves away from the origin, then this straight line is called asymptote.

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 greater value 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 called a point local maximum, 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 called a point local minimum, 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
A function f(x) is called continuous at a point x 0 if the limit of the function as x → x 0 equals the value functions at this point, i.e. .