Research function for beginners
A function with a certain domain of designation is a correspondence for which to each number x from a certain set there is associated a certain fully defined number y.
Usually, functions are denoted by Latin letters. Consider any example f. The number y that corresponds to the number x is called the value of the given f at a particular point x. Represent this: f (x). The domain of the function f is D (f). An area that consists of all values of the function f (x), where the argument x belongs to the domain of definition, is called the range of values of f. It is written as E (f).
Most often, the function is specified using formulas. Moreover, if additional constraints are not defined, the domain of the function designation, which is given by the formula, will be the set of all values of the variable, and such a formula holds.
A union of two sets is a set, each element of which can belong to and belongs to at least one of these sets.
To denote numbers from the domain of the designation of the function x, select a letter, which is called an independent variable or argument.
Often considered are such areas in which the range of values and the scope of notations are not numerical sets.
When a function study is conducted, examplescan be viewed using a graph. The graph of a function is the set of points on the coordinate plane, where the argument "runs through" the whole domain of the 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 straight line that is parallel to the axis of abscissas.
A function is said to grow on a set ifthe higher value of the argument from such a set corresponds to the higher value of the function, and the descending one on the set, if the lower value of the function corresponds to the higher value of the argument.
In the process of investigating the function, growth and descent must be marked by the intervals of growth and decline of maximum length.
A function is called a pair if for anyThe argument with its notation area will be f (-x) = f (x), or unpaired if for any argument with the notation area it will be f (-x) = -f (x). In addition, the graph of the pair function will be symmetrical about the ordinate axis, and the graph of the unpaired function is symmetric with respect to the point (0; 0).
In order to avoid mistakes, when the function is being studied, it is necessary to learn to find characteristic features. To do this, you need to do the following:
1. Find the notation area.
2. Carry out a check for the pairing or the same incompatibility, as well as the periodicity.
3. It is necessary to find the points of intersection of the graph of the function with the ordinate and abscissa.
4. At this stage, you need to find gaps where the function has positive values, and where - the negative. Such intervals are called intervals with constant signs. That is, you need to establish where the graph lies - above or below the abscissa axis.
5. Substantially facilitate the task of plotting the information about the intervals in which the function grows, and on what falls. Such intervals are called growth intervals and intervals of descent.
6. Now we need to find those values of the function at points where growth is replaced by descent, or vice versa.
Such a study of the function makes it possible to construct a graph. In addition, it is necessary to find the extremum points. What it is?
The point will be a minimum point if for all values of the argument from some range of the point the inequality f (x)> f (x0) is valid.
A point is a maximum point if for allof the values of the argument from a certain range of the point, the inequality f (x) <f (x0) is valid. Most often the graph at the extremum points has the form of a hump, and the minimum point is a depression. The points of maximum and minimum are extremum points, and the value of the function at points is an extremum. The study of the function at the extremum renders a great help in plotting the graph.
