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). 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. (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 a called the slope of a 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.
Function y=f(x) is such a dependence of the variable y on the variable x when each valid value of the variable x corresponds to a single value of the variable y .
Function scope D(f) is the set of all possible values of the variable x .
Function range E(f) is the set of all valid values of the variable y .
Function Graph y=f(x) is the set of plane points whose coordinates satisfy the given functional dependence, that is, points of the form M (x; f(x)) . The graph of a function is a line on a plane.
If b=0 , then the function will take the form y=kx and will be called direct proportionality.
D(f) : x \in R;\enspace E(f) : y \in R
The graph of a linear function is a straight line.
The slope k of the straight line y=kx+b is calculated using the following formula:
k= tg \alpha , where \alpha is the angle of inclination of the straight line to the positive direction of the Ox axis.
1) The function increases monotonically for k > 0 .
For example: y=x+1
2) The function monotonically decreases as k< 0 .
For example: y=-x+1
3) If k=0 , then giving b arbitrary values, we get a family of straight lines parallel to the axis Ox .
For example: y=-1
Inverse proportionality
Inverse proportionality is called a function of the form y=\frac (k)(x), where k is a non-zero real number
D(f) : x \in \left \( R/x \neq 0 \right \); \: E(f) : y \in \left \(R/y \neq 0 \right \).
Function Graph y=\frac (k)(x) is a hyperbole.
1) If k > 0, then the graph of the function will be located in the first and third quarters of the coordinate plane.
For example: y=\frac(1)(x)
2) If k< 0 , то график функции будет располагаться во второй и четвертой координатной плоскости.
For example: y=-\frac(1)(x)
Power function
Power function is a function of the form y=x^n , where n is a non-zero real number
1) If n=2 , then y=x^2 . D(f) : x \in R; \: E(f) : y \in; main period of the function T=2 \pi
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.
The logarithmic function has found wide application 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 (the exact mathematical definition of 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 of the entire graph, but only of its part located in the final parts 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; наименьшее значение функция 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 give a finite number of values - say, x 1 , x 2 , x 3 ,..., x k and make a table that includes the selected values of the function.
The table looks like this:
| x | x 1 | x2 | x 3 | ... | x k |
| y | f(x1) | f(x2) | f(x3) | ... | f(xk) |
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:
| x | -2 | -1 | 0 | 1 | 2 |
| y | -1 | 0 | 1 | 2 | 3 |
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 of the absolute value of a number, one can write
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 when 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 top of the parabola has coordinates (1; -1), its graph intersects the abscissa axis at points 0 and 2. On the interval (0; 2) the function takes negative values, therefore this part of the graph reflect 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, a 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.
| x | -1,5 | - | -0,5 | 0 | 0,5 | 1,5 | 2 | |
| f(x) = x | -1,5 | - | -0,5 | 0 | 0,5 | 1,5 | 2 | |
| g(x) = sinx | 1 | 0 | -1 | 0 | 1 | 0 | -1 | 0 |
| y = x + sinx | 1-1,5 | - | -1-0,5 | 0 | 1+0,5 | 1+1,5 | 2 |
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
If a set of numbers is given X and the way f, by which for each value XЄ X matches only one number at. Then it is considered given function y = f(X), in which domain X(usually referred to D(f) = X). Lots of Y all values at, for which there is at least one value XЄ X, such that y = f(X), such a set is called set of values functions f(most commonly referred to E(f)= Y).
Or single variable dependence at from another X, for which each value of the variable X from a certain set D matches the single value of the variable at, is called function.
The functional dependence of the variable y on x is often emphasized by the notation y(x), which is read by y from x.
Domain functions at(X), i.e., the set of values of its argument X, denoted by the symbol D(y), which is read de from y.
Range of values functions at(X), i.e., the set of values that the function y takes on is denoted by the symbol E(at), which read e from Y.
The main ways to define a function are:
a) analytical(using the formula y = f(X)). This method also includes cases where the function is given by a system of equations. If a function is given by a formula, then its domain of definition is all those values of the argument for which the expression written on the right side of the formula has values.
b) tabular(using a table of corresponding values X and at). In this way, the temperature regime or exchange rates are often set, but this method is not as clear as the next one;
in) graphic(using a chart). This is one of the most visual ways to set a function, since changes are immediately "read" according to the graph. If the function at(X) is given by the graph, then its domain of definition D(y) is the projection of the graph onto the x-axis, and the range of values E(at) - projection of the graph on the y-axis (see figure).
G) verbal. This method is often used in problems, or rather in the description of their conditions. Usually this method is replaced by one of the above.
Functions y = f(X), xЄ X, and y = g(X), xЄ X, are called identically equal on a subset M FROM X if for each x 0 Є M fair equality f(X 0) = g(X 0).
Function Graph y = f(X) can be represented as a set of such points ( X; f(X)) on the coordinate plane, where X is an arbitrary variable, from D(f). If a f(X 0) = 0, where X 0 then the point with coordinates ( x 0; 0) is the point at which the graph of the function y = f(X) intersects with the O axis x. If 0Є D(f), then the point (0; f(0)) is the point at which the graph of the function at = f(x) intersects with the O axis at.
Number X 0 of D(f) functions y = f(X) is the zero of the function, when f(X 0) = 0.
Gap M FROM D(f) this is constancy interval functions y = f(X) if either for an arbitrary xЄ M right f(X) > 0, or for an arbitrary XЄ M right f(X) < 0.
There is appliances, which draw graphs of dependencies between quantities. These are barographs - devices for fixing the dependence of atmospheric pressure on time, thermographs - devices for fixing the dependence of temperature on time, cardiographs - devices for graphic recording of the activity of the heart. The thermograph has a drum, it rotates evenly. The paper wound on the drum is touched by a recorder, which, depending on the temperature, rises and falls and draws a certain line on the paper.
From the representation of a function by a formula, you can move on to its representation in a table and graph.
When studying mathematics, it is very important to understand what a function is, its domains and meanings. With the help of the study of functions to an extremum, many problems in algebra can be solved. Even problems in geometry sometimes come down to considering the equations of geometric figures on a plane.
Parallel transfer.
TRANSFER ALONG THE Y-AXIS
f(x) => f(x) - b
Let it be required to plot the function y \u003d f (x) - b. It is easy to see that the ordinates of this graph for all values of x on |b| units less than the corresponding ordinates of the graph of functions y = f(x) for b>0 and |b| more units - at b 0 or up at b To plot the function y + b = f(x), plot the function y = f(x) and move the x-axis to |b| units up for b>0 or by |b| units down at b
TRANSFER ALONG THE X-AXIS
f(x) => f(x + a)
Let it be required to plot the function y = f(x + a). Consider a function y = f(x), which at some point x = x1 takes the value y1 = f(x1). Obviously, the function y = f(x + a) will take the same value at the point x2, the coordinate of which is determined from the equality x2 + a = x1, i.e. x2 = x1 - a, and the equality under consideration is valid for the totality of all values from the domain of the function. Therefore, the graph of the function y = f(x + a) can be obtained by parallel displacement of the graph of the function y = f(x) along the x-axis to the left by |a| ones for a > 0 or to the right by |a| units for a To plot the function y = f(x + a), plot the function y = f(x) and move the y-axis to |a| units to the right for a>0 or |a| units to the left for a
Examples:
1.y=f(x+a)
2.y=f(x)+b
Reflection.
GRAPHING OF A FUNCTION OF THE VIEW Y = F(-X)
f(x) => f(-x)
Obviously, the functions y = f(-x) and y = f(x) take equal values at points whose abscissas are equal in absolute value but opposite in sign. In other words, the ordinates of the graph of the function y = f(-x) in the region of positive (negative) values of x will be equal to the ordinates of the graph of the function y = f(x) with negative (positive) x values corresponding in absolute value. Thus, we get the following rule.
To plot the function y = f(-x), you should plot the function y = f(x) and reflect it along the y-axis. The resulting graph is the graph of the function y = f(-x)
GRAPHING OF A FUNCTION OF THE VIEW Y = - F(X)
f(x) => - f(x)
The ordinates of the graph of the function y = - f(x) for all values of the argument are equal in absolute value, but opposite in sign to the ordinates of the graph of the function y = f(x) for the same values of the argument. Thus, we get the following rule.
To plot the function y = - f(x), you should plot the function y = f(x) and reflect it about the x-axis.
Examples:
1.y=-f(x)
2.y=f(-x)
3.y=-f(-x)
Deformation.
DEFORMATION OF THE GRAPH ALONG THE Y-AXIS
f(x) => kf(x)
Consider a function of the form y = k f(x), where k > 0. It is easy to see that for equal values of the argument, the ordinates of the graph of this function will be k times greater than the ordinates of the graph of the function y = f(x) for k > 1 or 1/k times less than the ordinates of the graph of the function y = f(x) for k ) or decrease its ordinates by 1/k times for k
k > 1- stretching from the Ox axis
0 - compression to the OX axis
GRAPH DEFORMATION ALONG THE X-AXIS
f(x) => f(kx)
Let it be required to plot the function y = f(kx), where k>0. Consider a function y = f(x), which takes the value y1 = f(x1) at an arbitrary point x = x1. Obviously, the function y = f(kx) takes the same value at the point x = x2, the coordinate of which is determined by the equality x1 = kx2, and this equality is valid for the totality of all values of x from the domain of the function. Consequently, the graph of the function y = f(kx) is compressed (for k 1) along the abscissa axis relative to the graph of the function y = f(x). Thus, we get the rule.
To plot the function y = f(kx), plot the function y = f(x) and reduce its abscissa by k times for k>1 (shrink the graph along the abscissa) or increase its abscissa by 1/k times for k
k > 1- compression to the Oy axis
0 - stretching from the OY axis
The work was carried out by Alexander Chichkanov, Dmitry Leonov under the supervision of Tkach T.V., Vyazovov S.M., Ostroverkhova I.V.
©2014
