1. Even and odd. The function f(x) is called even if its values ​​are symmetric with respect to the OY axis, i.e. f(-x) = f(x). The function f(x) is called odd if its value is reversed when the variable x changes to -x, i.e. f(-x) = -f(x). Otherwise, the function is called a generic function.

2. Monotony. A function is called increasing (decreasing) on ​​the interval X, if the larger value of the argument from this interval corresponds to the larger (smaller) value of the function, i.e. at x1< (>)x2, f(x1)< (>) f(x2).

3. Periodicity. If the value of the function f(x) repeats after a certain period T, then the function is called periodic with a period T ≠ 0, i.e. f(x + T) = f(x). Otherwise non-periodic.

4. Limited. The function f (x) is called bounded on the interval X if there exists such a positive number M > 0 that for any x, belonging to the interval X, | f(x) |< M. В противном случае функция называется неограниченной.

1) Function scope and function range.

The scope of a function is the set of all valid valid values ​​of the argument x(variable x) for which the function y = f(x) defined. The range of a function is the set of all real values y that the function accepts.

In elementary mathematics, functions are studied only on the set of real numbers.

2) Function zeros.

Zero of the function is the value of the argument at which the value of the function is equal to zero.

3) Intervals of sign constancy of a function.

The intervals of constant sign of a function are such sets of argument values ​​on which the values ​​of the function are only positive or only negative.

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.

Decreasing function (in some interval) - a function in which a larger value of the argument from this interval corresponds to a smaller value of the function.

5) Even (odd) functions.

An even function is a function whose domain of definition is symmetric with respect to the origin and for any X from the domain of definition the equality f(-x) = f(x). Schedule even function symmetrical about the y-axis.

An odd function is a function whose domain of definition is symmetric with respect to the origin and for any X from the domain of definition the equality f(-x) = - f(x). Schedule odd function symmetrical about the origin.

6) Limited and unlimited functions.

A function is called bounded if there exists a positive number M such that |f(x)| ≤ M for all values ​​of x . If there is no such number, then the function is unbounded.

7) Periodicity of the function.

A function f(x) is periodic if there exists a non-zero number T such that for any x from the domain of the function, f(x+T) = f(x). This smallest number is called the period of the function. Everything trigonometric functions are periodic. (Trigonometric formulas).

19. Basic elementary functions, their properties and graphs. Application of functions in the economy.

Basic elementary functions. Their properties and graphs

1. Linear function.

Linear function is called a function of the form , where x is a variable, and and b are real numbers.

Number but called slope factor straight line, it is equal to the tangent of the angle of inclination of this straight line to the positive direction of the x-axis. The graph of a linear function is a straight line. It is defined by two points.

Linear Function Properties

1. Domain of definition - the set of all real numbers: D (y) \u003d R

2. The set of values ​​is the set of all real numbers: E(y)=R

3. The function takes a zero value for or.

4. The function increases (decreases) over the entire domain of definition.

5. The linear function is continuous on the entire domain of definition, differentiable and .

2. Quadratic function.

A function of the form, where x is a variable, the coefficients a, b, c are real numbers, is called quadratic.

In the opinion of some scientists, the main purpose of graphs is their significance for heuristic activities - illustrations for the presentation of the theory and, above all, indication of examples and counterexamples for proving or refuting connections between various properties of functions, i.e. the use of mathematical bilingualism developed in accordance with the requirements of the standard of "bilingual" thinking.

Wide application found logarithmic function in astronomy : For example, the brightness of the stars changes along it, if we compare the characteristics of the brightness marked by the eye and with the help of instruments, then we can draw up the following graph: Here, along the vertical axis, we plot the brightness of the stars in Hipparchus units (distribution of stars according to subjective characteristics (by eye) into 6 groups) , and on the horizontal - instrument readings. The graph shows that the objective and subjective characteristics are not proportional, and the device registers an increase in brightness not by the same amount, but by 2.5 times. This dependence is expressed by a logarithmic function.

Consider how they are built.

We choose a rectangular coordinate system on the plane and plot the values ​​of the argument on the abscissa axis X, and on the y-axis - the values ​​of the function y = f(x) .

Function Graph y = f(x) the set of all points is called, for which the abscissas belong to the domain of the function, and the ordinates are equal to the corresponding values ​​of the function.

In other words, the graph of the function y \u003d f (x) is the set of all points in the plane, the coordinates X, at which satisfy the relation y = f(x) .

On fig. 45 and 46 are graphs of functions y = 2x + 1 And y \u003d x 2 - 2x .

Strictly speaking, one should distinguish between the graph of a function (exact mathematical definition which was given above) and the drawn curve, which always gives only a more or less accurate sketch of the graph (and even then, as a rule, not the entire graph, but only its part located in the final part of the plane). In what follows, however, we will usually refer to "chart" rather than "chart sketch".

Using a graph, you can find the value of a function at a point. Namely, if the point x = a belongs to the scope of the function y = f(x), then to find the number f(a)(i.e. the function values ​​at the point x = a) should do so. Need through a dot with an abscissa x = a draw a straight line parallel to the y-axis; this line will intersect the graph of the function y = f(x) at one point; the ordinate of this point will be, by virtue of the definition of the graph, equal to f(a)(Fig. 47).

For example, for the function f (x) \u003d x 2 - 2x using the graph (Fig. 46) we find f(-1) = 3, f(0) = 0, f(1) = -l, f(2) = 0, etc.

A function graph visually illustrates the behavior and properties of a function. For example, from a consideration of Fig. 46 it is clear that the function y \u003d x 2 - 2x takes positive values ​​when X< 0 and at x > 2, negative - at 0< x < 2; smallest value function y \u003d x 2 - 2x accepts at x = 1 .

To plot a function f(x) you need to find all points of the plane, coordinates X , at which satisfy the equation y = f(x). In most cases, this is impossible, since there are infinitely many such points. Therefore, the graph of the function is depicted approximately - with greater or lesser accuracy. The simplest is the multi-point plotting method. It consists in the fact that the argument X attach finite number values ​​- say, x 1 , x 2 , x 3 ,..., x k and make up a table that includes the selected values ​​of the function.

The table looks like this:

Having compiled such a table, we can outline several points on the graph of the function y = f(x). Then, connecting these points with a smooth line, we get an approximate view of the graph of the function y = f(x).

However, it should be noted that the multi-point plotting method is very unreliable. In fact, the behavior of the graph between the marked points and its behavior outside the segment between the extreme points taken remains unknown.

Example 1. To plot a function y = f(x) someone compiled a table of argument and function values:

The corresponding five points are shown in Fig. 48.

Based on the location of these points, he concluded that the graph of the function is a straight line (shown in Fig. 48 by a dotted line). Can this conclusion be considered reliable? Unless there are additional considerations to support this conclusion, it can hardly be considered reliable. reliable.

To substantiate our assertion, consider the function

.

Calculations show that the values ​​of this function at points -2, -1, 0, 1, 2 are just described by the above table. However, the graph of this function is not at all a straight line (it is shown in Fig. 49). Another example is the function y = x + l + sinx; its meanings are also described in the table above.

These examples show that in its "pure" form, the multi-point plotting method is unreliable. Therefore, to plot a given function, usually proceed as follows. First, the properties of this function are studied, with the help of which it is possible to construct a sketch of the graph. Then, by calculating the values ​​of the function at several points (the choice of which depends on the set properties of the function), the corresponding points of the graph are found. And, finally, a curve is drawn through the constructed points using the properties of this function.

We will consider some (the most simple and frequently used) properties of functions used to find a sketch of a graph later, but now we will analyze some commonly used methods for plotting graphs.

Graph of the function y = | f(x) |.

It is often necessary to plot a function y = |f(x)|, where f(x) - given function. Recall how this is done. By definition absolute value numbers can be written

This means that the graph of the function y= | f(x) | can be obtained from the graph, functions y = f(x) as follows: all points of the graph of the function y = f(x), whose ordinates are non-negative, should be left unchanged; further, instead of the points of the graph of the function y = f(x), having negative coordinates, one should construct the corresponding points of the graph of the function y = -f(x)(i.e. part of the function graph
y = f(x), which lies below the axis X, should be reflected symmetrically about the axis X).

Example 2 Plot a function y = |x|.

We take the graph of the function y = x(Fig. 50, a) and part of this graph with X< 0 (lying under the axis X) is symmetrically reflected about the axis X. As a result, we get the graph of the function y = |x|(Fig. 50, b).

Example 3. Plot a function y = |x 2 - 2x|.

First we plot the function y = x 2 - 2x. The graph of this function is a parabola, the branches of which are directed upwards, the vertex of the parabola has coordinates (1; -1), its graph intersects the x-axis at points 0 and 2. On the interval (0; 2), the function takes negative values, therefore, it is this part of the graph that will be reflected symmetrically about the x-axis. Figure 51 shows a graph of the function y \u003d |x 2 -2x |, based on the graph of the function y \u003d x 2 - 2x

Graph of the function y = f(x) + g(x)

Consider the problem of plotting the function y = f(x) + g(x). if graphs of functions are given y = f(x) And y = g(x) .

Note that the domain of the function y = |f(x) + g(х)| is the set of all those values ​​of x for which both functions y = f(x) and y = g(x) are defined, i.e. this domain of definition is the intersection of the domains of definition, the functions f(x) and g(x).

Let the points (x 0, y 1) And (x 0, y 2) respectively belong to the function graphs y = f(x) And y = g(x), i.e. y 1 \u003d f (x 0), y 2 \u003d g (x 0). Then the point (x0;. y1 + y2) belongs to the graph of the function y = f(x) + g(x)(for f(x 0) + g(x 0) = y 1+y2),. and any point of the graph of the function y = f(x) + g(x) can be obtained in this way. Therefore, the graph of the function y = f(x) + g(x) can be obtained from function graphs y = f(x). And y = g(x) by replacing each point ( x n, y 1) function graphics y = f(x) dot (x n, y 1 + y 2), where y 2 = g(x n), i.e., by shifting each point ( x n, y 1) function graph y = f(x) along the axis at by the amount y 1 \u003d g (x n). In this case, only such points are considered. X n for which both functions are defined y = f(x) And y = g(x) .

This method of plotting a function graph y = f(x) + g(x) is called the addition of graphs of functions y = f(x) And y = g(x)

Example 4. In the figure, by the method of adding graphs, a graph of the function is constructed
y = x + sinx .

When plotting a function y = x + sinx we assumed that f(x) = x, but g(x) = sinx. To build a function graph, we select points with abscissas -1.5π, -, -0.5, 0, 0.5, , 1.5, 2. Values f(x) = x, g(x) = sinx, y = x + sinx we will calculate at the selected points and place the results in the table.

Based on the results obtained, we will construct points that we will connect with a smooth curve, which will be a sketch of the graph of the function y = x + sinx .

Function graphs can be built not only by hand on points, but also with the help of various programs (excel, maple), as well as programming in Pascal. Having studied the Pascal language, you will simultaneously improve your knowledge of computer science, but you will also quickly be able to build various graphs of functions. examples of functions in Pascal will help you understand the syntax of the language and build the first graphs yourself.

Basic properties of functions.

1) Function scope and function range .

The scope of a function is the set of all valid valid values ​​of the argument x(variable x) for which the function y = f(x) defined.
The range of a function is the set of all real values y that the function accepts.

In elementary mathematics, functions are studied only on the set of real numbers.

2) Function zeros .

Function zero is the value of the argument, at which the value of the function is equal to zero.

3) Intervals of sign constancy of a function .

Function constant-sign intervals are such sets of argument values ​​on which the function values ​​are only positive or only negative.

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.

Decreasing function (in some interval) - a function in which a larger value of the argument from this interval corresponds to a smaller value of the function.

5) Even (odd) functions .

An even function is a function whose domain of definition is symmetric with respect to the origin and for any X from the domain of definition the equality f(-x) = f(x). The graph of an even function is symmetrical about the y-axis.

An odd function is a function whose domain of definition is symmetric with respect to the origin and for any X from the domain of definition the equality f(-x) = - f(x). The graph of an odd function is symmetrical about the origin.

6) Limited and unlimited functions .

A function is called bounded if there exists a positive number M such that |f(x)| ≤ M for all values ​​of x . If there is no such number, then the function is unbounded.

7) Periodicity of the function .

A function f(x) is periodic if there exists a non-zero number T such that for any x from the domain of the function, f(x+T) = f(x). This smallest number is called the period of the function. All trigonometric functions are periodic

Function $f(x)=|x|$

$|x|$ - module. It is defined as follows: If the real number is non-negative, then the modulo value is the same as the number itself. If it is negative, then the value of the modulus coincides with the absolute value of the given number.

Mathematically, this can be written as follows:

Example 1

Function $f(x)=[x]$

The function $f\left(x\right)=[x]$ is a function of the integer part of a number. It is found by rounding the number (if it is not an integer itself) "down".

Example: $=2.$

Example 2

Let's explore and plot it.

1. $D\left(f\right)=R$.
2. Obviously, this function only takes integer values, i.e. $\ E\left(f\right)=Z$
3. $f\left(-x\right)=[-x]$. Therefore, this function will be of general form.
4. $(0,0)$ is the only point of intersection with the coordinate axes.
5. $f"\left(x\right)=0$
6. The function has break points (function jumps) for all $x\in Z$.

Figure 2.

Function $f\left(x\right)=\(x\)$

The function $f\left(x\right)=\(x\)$ is the function of the fractional part of a number. It is found by "discarding" the integer part of this number.

Example 3

Exploring and plotting a function graph

Function $f(x)=sign(x)$

The function $f\left(x\right)=sign(x)$ is a sign function. This function shows what sign a real number has. If the number is negative, then the function has the value $-1$. If the number is positive, then the function is equal to one. If the value of the number is zero, the value of the function will also take on the zero value.

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