For the first time, students of educational schools usually come up with the concept of function in the 7th grade when they begin to study the course of algebra as a separate area of ​​mathematics. The study of functions begins, as a rule, without entering complex definitions and terms, which is quite logical. The most important thing at the familiarization stage is to provide students with the opportunity to familiarize themselves, on elementary examples, with a new mathematical object that they have not encountered before.
The study of functions begins with linear relationships, the graph of which is a straight line. Students study the mathematical notation of the dependence of one variable on another and understand which variable in the function is independent and which is dependent. In parallel with this, students begin plotting graphs on the coordinate plane, on which they previously marked only points.
The next function students learn is direct proportionality. Initially, in the course of algebra, the authors of many manuals distinguish this dependence separately from a linear function, noting some important properties of the function that are inherent in this dependence.
After considering the elementary functions of the students, they are introduced to generalized concepts that characterize numerical dependencies. First of all, this is writing y = f (x). Further, several lessons are necessarily devoted to the practical application of the obtained theoretical knowledge, within the framework of which the applied nature of the definition and any particular property of a function characterizing a particular process is considered.
In grade 8, students first encounter quadratic equations. After mastering the skills of solving equations of this type, the program provides for the study of a quadratic function and its main characteristics. Pupils learn not only to plot the dependence according to the presented equation, but also to analyze the presented image, identifying the main properties of the function and forming its mathematical description.
The 9th grade algebra course expands many of the functions known to students. Possessing a rather significant theoretical base devoted to mathematical analysis, students get acquainted with inverse proportionality and linear-fractional function, and also study the differences in the representation of equations and functions on the graphic plane . In the latter case, attention is focused on the fact that the equation graph can have several values ​​of the dependent variable for one argument - an independent variable. Functional dependence is characterized by a unique correspondence between independent and dependent variables.
At the senior level of the school, students study complex functional dependencies and learn to build graphs, based not on the table of values ​​“argument - function”, but on the properties of the function. This is due to the fact that the behavior of complex functions is rather difficult to predict “offhand”, and it is quite difficult to calculate a certain set of values. Therefore, to determine the nature of the behavior of a function, its main characteristics are described: areas of definition and value, asymptotes, monotonicity, maximum and minimum points, convexity, etc. Particular attention should be paid to such a property as parity. The even and odd functions have a special behavior: the first characteristic means that the graph of the function is symmetrical about the ordinate axis, the second - about the origin.
This concludes the study of the basics of mathematical analysis in a high school course. Further study of numerical dependencies will be necessarily presented in the course of higher mathematics, as well as in the disciplines devoted to statistical data processing. The latter often use an element such as distribution functions.