To find a function range of definition, it is necessary to calculate borders of one or several intervals containing points in which it makes sense. This first action at the solution of tasks on the mathematical analysis of behavior of functions.
Instruction
1. The task of any function is an indication of the rule according to which elements of two sets are connected with each other. The first is called a function range of definition. These are such permissible values of her argument which correspond to certain elements of the second set, area of values of function.
2. It is considered that function is set if both of these sets are known. Sometimes a range of definition is the infinite interval (-∞; + ∞), but at the majority a case there are some restrictions which are imposed by the making function expression elements. For example, at it there can be such mathematical concepts as a root, degree, a logarithmic or trigonometrical subfunction and so forth.
3. The algorithm of finding of a range of definition of function consists of three stages: definition of type or types of restrictions, drawing up and solution of the corresponding inequalities, record of an interval or intervals of permissible values of an argument.
4. There are six types of subfunctions which presence at the main expression can impose restriction for the field of its definition. This radicand, a power function, a logarithm, expression split up also some trigonometrical functions below the line.
5. Write down inequalities according to the revealed restrictions: - function under the sign of a root, i.e. in fractional degree with even number in a denominator: f (x) ≥ 0; - function in degree of an indicator of other function of the same argument: f (x)> 0; - logarithm of log_a f (x): f (x)> 0; - relation of two functions f (x) / g (x): g (x) ≠ 0; - tg f (x) and ctg f (x): f (x) ≠ π\• k + π/2; - arcsin f (x) and arccos f (x):-1 ≤ f (x) ≤ 1.
6. Solve inequalities and write down the interval closed or opened depending on whether its borders are the pricked-out points or ranges of definition belong. Say designations about it: the square bracket means entry into an interval, and round - an exception. For example, if the area is set by an interval (1; 3], for its elements double inequality 1 is carried out