The most part of the school program of mathematics is occupied by a research of functions, in particular, check on parity and oddness. This method is an important component of process of studying nature of behavior of function and creation of its schedule.
Instruction
1. Properties of parity and oddness of function is defined proceeding from influence of the sign of an argument on its value. This influence is displayed on a function graph in a certain symmetry. In other words, the property of parity if f (-x) = f(x), i.e. the sign of an argument does not affect value of function, and oddness if equality of f is fair (-x) = - f is carried out (x).
2. Odd function graphically looks a point of intersection of coordinate axes symmetric relatively, even – concerning ordinate axis. The parabola x², odd – f = x³ can be an example of even function.
3. Example No. 1issledovat on parity function x² / (4·x \-1).: Substitute in this function – x instead of x. You will see that the sign of function will not change as the argument in both cases is present at even degree which neutralizes the negative sign. Therefore, the studied function is even.
4. Example No. 2proverit function on parity and oddness: f = - x \+ 5 · x. The decision: As well as in the previous example, substitute – x instead of x: f (-x) = - x² – 5·x. It is obvious that f(x) ≠ f (-x) and f (-x) ≠ - f(x), therefore, function has no properties of neither parity, nor oddness. Such function is called indifferent or function of a general view.
5. It is possible to investigate function on parity and oddness in also evident way at creation of the schedule or finding of a range of definition of function. In the first example a range of definition is the set x ∈ (-∞; 1/2) ∪ (1/2; + ∞). The function graph is symmetric concerning Oy axis, so function even.
6. It is aware of mathematics at first studyof properties of elementary functions, and then the gained knowledge transfers to a research of more difficult functions. Power functions with the whole indicator, indicative a type of a^x are elementary at a> 0, logarithmic and trigonometrical functions.