Research of function call a special task in a school course of mathematics during which key parameters of function come to light and its schedule is under construction. Earlier creation of the schedule was an objective of this research, today this problem is solved with the help of specialized computer programs. But nevertheless not superfluous will be to study the general scheme of a research of function.
Instruction
1. There is a function range of definition, i.e. range of values x at which function accepts any value.
2. Areas of continuity and a point of a gap are defined. At the same time usually areas of continuity coincide with a function range of definition, it is necessary to investigate the left and right side-altars of the isolated points.
3. Existence of vertical asymptotes is checked. If function has gaps, then it is necessary to investigate the ends of the corresponding intervals.
4. The parity and oddness of function is checked by definition. Function y = f(x) is called even if for any x from a range of definition the equality of f is right (-x) = f (x).
5. Function is checked for frequency. For this purpose x T changes on x + and the smallest positive number T is looked for. If such number exists, then function is periodical, and number T is the function period.
6. Function is checked for monotony, there are extremum points. At the same time the derivative of function is equated to zero, the points found at the same time, expose on a numerical straight line and add to them points in which the derivative is not defined. Signs of a derivative on the turned-out intervals define areas of monotony, and transition points between different areas are function extrema.
7. The camber of function is investigated, there are inflection points. The research is made similar to a research on monotony, but at the same time the second derivative is considered.
8. There are points of intersection with axes of OX and OY, at this y = f(0) is crossing with an axis of OY, f(x) = 0 – crossing with OX axis.
9. Limits on the ends of a range of definition are defined.
10. The function graph is under construction.
11. The area of values of function and limitation of function is determined by the schedule.