A function with a certain notation is a correspondence in which each completely x number is assigned to each number x from a certain set.
Usually functions are denoted in Latin letters. Consider any example f. The number y, which corresponds to the number x, is called the value of a given f at a particular point x. Represented as follows: f (x). The domain of the function f is D (f). The region that consists of all the values of the function f (x), where the argument x belongs to the region of definition, is called the region of values of f. It is written like this: E (f).
Most often, a function is specified using formulas. Moreover, if no additional restrictions are defined, the domain of the function that is specified by the formula will be considered the set of all values of the variable, and such a formula takes place.
The union of two sets is such a set, each element of which can belong and belongs to at least one of these sets.
To indicate numbers from the domain of the function x, choose some letter, which is called an independent variable or argument.
Areas that are often considered are those in which the range of values and the notation are not numerical sets.
When a function study is conducted, examples can be viewed using a graph. A function graph is a set of points on the coordinate plane, where the argument runs through the entire notation. In order for a subset of the coordinate plane to be a graph of some function, it is necessary that such a subset have at least one common point with any line that is parallel to the abscissa axis.
A function is called growing on the set if the highest value of the argument from such a set corresponds to the highest value of the function, and descending on the set if the highest value of the argument corresponds to the lowest value of the function.
In the process of investigating the rise and fall functions, it is necessary to designate the intervals of growth and decline of the maximum length.
A function is called paired if for any argument with its notation area it will be f (-x) = f (x), or unpaired if for any argument with its notation domain there will be f (-x) = - f (x). In addition, the graph of the paired function will be symmetric about the ordinate axis, and the graph of the unpaired function will be symmetric about the point (0; 0).
In order to avoid errors when a function is being investigated, it is necessary to learn how to find the characteristic features. To do this, do the following steps:
1. Find the scope.
2. To check for paired or unpaired, as well as periodicity.
3. It is necessary to find the intersection points of the function graph with the ordinate and abscissa.
4. At this stage, you need to find the gaps where the function has positive values, and where are negative. Such spaces are called spaces with constant signs. That is, you need to establish where the graph lies - above or below the abscissa axis.
5. Significantly facilitate the task of constructing a graph of information about at what intervals the function grows and at which it falls. Such gaps are called growth gaps and descent gaps.
6. Now you need to find those values of the function at the points where growth is replaced by a descent, or vice versa.
Such a study of the function makes it possible to construct a graph. In addition, it is necessary to find extremum points. What it is?
A point will be a minimum point if f (x)> f (x0) is valid for all argument values from a certain range of a point.
A point is a maximum point if the inequality f (x) <f (x0) holds for all values of the argument from a certain range of the point. Most often, the graph at the points of the extremum has the form of a hump, and the minimum point is the depression. The maximum and minimum points are the extremum points, and the value of the function at the points is the extremum. Examination of the function on the extremum is of great help in plotting.