If given a set of numbers X and the method is indicated f, according to which for each value XЄ X only one number is assigned at. Then it is considered given function y = f(X), in which domain X(usually denoted D(f) = X). A bunch 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 meanings functions f(most often denoted E(f)= Y).
Or dependence of one variable at from another X, at which each value of the variable X from a certain set D corresponds to a single variable value at, called function.
The functional dependence of the variable y on x is often emphasized by the notation y(x), which is read as a letter 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 by de from igrek.
Range of values functions at(X), i.e., the set of values that the function y takes is denoted by the symbol E(at), which is read from the game.
The main ways to specify a function are:
A) analytical(using formula y = f(X)). This method also includes cases when the function is specified by a system of equations. If a function is given by a formula, then its domain of definition consists of 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). Temperature conditions or exchange rates are often set in this way, but this method is not as visual as the next one;
V) graphic(using a graph). This is one of the most visual ways of specifying a function, since changes are immediately “read” from 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 onto the ordinate axis (see figure).
G) verbal. This method is often used in problems, or more precisely in describing 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 WITH X, if for each x 0 Є M equality is true f(X 0) = g(X 0).
Graph of a function y = f(X) can be represented as a set of such points ( X; f(X)) on the coordinate plane, where X- arbitrary variable, from D(f). If 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 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, then when f(X 0) = 0.
Interval M WITH D(f) This interval of sign constancy 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.
Eat devices, which draw graphs of dependencies between quantities. These are barographs - devices for recording the dependence of atmospheric pressure on time, thermographs - devices for recording the dependence of temperature on time, cardiographs - devices for graphically recording the activity of the heart. The thermograph has a drum that rotates evenly. The paper wound on the drum touches the recorder, which, depending on the temperature, rises and falls and draws a certain line on the paper.
From representing a function with a formula, you can move on to representing it with a table and graph.
When studying mathematics, it is very important to understand what a function is, its domains of definition and meaning. Using the study of extremum functions, you can solve many problems in algebra. Even problems in geometry sometimes come down to considering the equations of geometric figures on a plane.
1) Function domain and function range.
The domain of a function is the set of all valid valid argument values x(variable x), for which the function y = f(x) determined. The range of a function is the set of all real values y, which 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 constant sign of a function.
Intervals of constant sign of a function are 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.
A decreasing function (in a certain interval) is a function in which a larger value of the argument from this interval corresponds to a smaller value of the function.
5) Even (odd) function.
An even function is a function whose domain of definition is symmetrical 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 ordinate.
An odd function is a function whose domain of definition is symmetrical with respect to the origin and for any X from the domain of definition the equality is true 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 is a positive number M such that |f(x)| ≤ M for all values of x. If such a number does not exist, then the function is unlimited.
7) Periodicity of the function.
A function f(x) is periodic if there is a non-zero number T such that for any x from the domain of definition of the function the following holds: 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 economics.
Basic elementary functions. Their properties and graphs
1. Linear function.
Linear function is called a function of the form , where x is a variable, a and b are real numbers.
Number A called the slope of the line, it is equal to the tangent of the angle of inclination of this 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.
Properties of a Linear Function
1. Domain of definition - the set of all real numbers: D(y)=R
2. The set of values is the set of all real numbers: E(y)=R
3. The function takes a zero value when or.
4. The function increases (decreases) over the entire domain of definition.
5. A linear function is continuous over the entire domain of definition, differentiable and .
2. Quadratic function.
A function of the form, where x is a variable, coefficients a, b, c are real numbers, is called quadratic
1. Even and odd. The function f(x) is called even if its values are symmetrical about the OY axis, i.e. f(-x) = f(x). A function f(x) is called odd if its value changes to the opposite when the variable x changes by -x, i.e. f(-x) = -f(x). Otherwise, the function is called a general function.
2.Monotony. A function is said to be increasing (decreasing) on the interval X if a larger value of the argument from this interval corresponds to a larger (smaller) value of the function, i.e. at x1< (>) x2, f(x1)< (>) f(x2).
3. Frequency. If the value of the function f(x) repeats after a certain period T, then the function is called periodic with period T ≠ 0, i.e. f(x + T) = f(x). Otherwise non-periodic.
4. Limited. A function f (x) is called bounded on the interval X if there is a positive number M > 0 such that for any x belonging to the interval X, | f(x) |< M. В противном случае функция называется неограниченной.
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Lety- some function of a variablex; Moreover, it does not matter how this function is specified: a formula, a table, or some other way. What is important is the very fact of the existence of this functional dependence, which is written as follows:y = f(x). Letterf(the initial letter of the Latin word “functio” - function) does not denote any quantity, just like the letterslog, sin, tan in function recordsy=logx, y= sinx, y= tanx. They only talk about certain functional dependenciesyfromx. Recordy = f (x) isanyfunctional dependence. If two functional dependencies:yfromxAndzfromtdiffer from one another, they are written using different letters:y = f (x) Andz = F (t). If some dependencies are the same, then they are written with the same letterf: y = f (x) Andz = f (t). If the expression for a functional dependencyy = f (x) is known, then it can be written using both function notations. For example,y= sin x or f(x) = sin x. Both forms are completely equivalent. Sometimes another form of notation is used: y (x). This means the same thing as y = f (x).
Graphic representation of functions.
To represent a functiony = f(x) in the form of a graph, you need:
1) Write a number of values of the function and its argument into the table:
2) Transfer the coordinates of the function points from the table to the coordinate system,
marking the abscissa values on the selected scale
axesXand ordinate values on the axisY(Fig. 2). As a result, in our system
coordinates a series of points will be plottedA, B, C, . . . , F.
3) Connecting the dotsA, B, C, . . . , Fsmooth curve, we obtain a graph of the given
functional dependence.
Such a graphical representation of a function gives a clear idea of the nature of its behavior, but the accuracy achieved is insufficient. It is possible that intermediate points not plotted on the graph lie far from the drawn smooth curve. Good results also largely depend on a good choice of scales. Therefore it is necessary to determine graph of a function as locus of points , coordinates which M (x, y) are connected by a given functional dependence .
The domain of definition and the range of values of a function. In elementary mathematics, functions are studied only on the set of real numbers R. This means that the function argument can only take those real values for which the function is defined, i.e. it also accepts only real values. A bunch of X all valid valid argument values x, for which the function y= f(x) defined, called domain of the function. A bunch of Y all real values y, which the function accepts, is called function range. Now we can give a more precise definition of the function: rule (law) of correspondence between sets X and Y, by which for each element from the set X one and only one element from the set Y can be found is called a function.
