To find a range of definition and values of function f, it is necessary to define two sets. One of them is set of all values of an argument x, and another consists of objects f corresponding to them (x).
Instruction
1. At the first stage of any algorithm of a research of mathematical function it is necessary to find a range of definition. If not to make it, then all calculations will be a useless waste of time as on its basis the area of values is formed. Function is a certain law under which elements of the first set are put in compliance to another.
2. To find a function range of definition, it is necessary to consider its expression in terms of possible restrictions. It can be presence of fraction, a logarithm, arithmetic root, power function, etc. If it is several such elements, then for each of them make and solve the inequality to reveal critical points. If there is no restriction, then the range of definition represents all numerical space (-∞; ∞).
3. There are six types of restrictions: A power function of a type of f^ (k/n), where a degree denominator – even number. The expression standing under a root cannot be less than zero, therefore, inequality looks so: f ≥ 0. Function of a logarithm. On property the expression standing under its sign can be only strictly positive: f> 0. f/g fraction where g is function too. It is obvious that g ≠ 0.tg and ctg: x ≠ π/2 + π\• k as in these points these trigonometrical functions do not exist (cos or sin standing in a denominator address in zero) .arcsin and arccos:-1 ≤ f ≤ 1. Restriction is imposed by area of values of these functions. A power function with degree in the form of other function of the same argument: f^g. Restriction is presented in the form of inequality f> 0.
4. To find area of value of function, substitute in its expression all points from a range of definition by search of one behind another. There is a concept of a set of values of function on an interval. These two terms should be distinguished, except for a case when the set interval coincides with a range of definition. Otherwise this set is a subset of area of values.