An investigation of a function is called a specialTask in the school course of mathematics, during which the main parameters of the function are identified and its schedule is constructed. Previously, the purpose of this study was to build a graph, today this problem is solved with the help of specialized computer programs.
But still it is not superfluous to get acquainted with the general scheme of the study of the function.
The domain of the function definition is located, i.e. Range of values of x for which the function takes a value.
Areas of continuity and points of discontinuity are determined. In this case, usually the areas of continuity coincide with the domain of definition of the function, it is necessary to investigate the left and right side-points of isolated points.
Verification of the presence of vertical asymptotes. If the function has discontinuities, then it is necessary to investigate the ends of the corresponding intervals.
The parity and oddness of a function are verified by definition. A function y = f (x) is said to be even if for any x in the domain of definition the equality f (-x) = f (x) is true.
The function is checked for periodicity. To do this, x changes to x + T and the smallest positive number T is sought. If such a number exists, then the function is periodic, and the number T is the period of the function.
The function is checked for monotony,Extremum points. In this case, the derivative of the function is equated to zero, the points found in this case are exposed on the number line and add to them points at which the derivative is not defined. The signs of the derivative on the resulting intervals determine the regions of monotonicity, and the transition points between different regions are extremums of the function.
The convexity of the function is investigated, the inflection points are found. The study is similar to monotonicity, but the second derivative is considered.
There are points of intersection with the axes OX and OY, with y = f (0) - intersection with the axis OY, f (x) = 0 - intersection with the axis OX.
Limits are defined at the ends of the domain of definition.
A function graph is constructed.
According to the graph, the range of values of the function and the boundedness of the function are determined.