The range of definition of expression is a set of values at which this expression makes sense. It is the best of all to look for a range of definition by process of elimination - rejecting all values at which expression loses the mathematical meaning.
Instruction
1. The first stage of finding of a range of definition of expression can be made an exception of division into zero. If at expression there is a denominator which can address in zero, it is necessary to find all values at which he addresses in zero and to exclude them. Example: 1/x. The denominator addresses in zero at x = 0. 0 will not enter an expression range of definition. (x-2)/((x^2)-3x+2). The denominator addresses in zero at x = 1 and x = 2. These values will not be included into an expression range of definition.
2. In expression also various irrationalities can enter. If roots of even degrees enter expressions, then radicands have to be not negative. Examples: 2+v(x-4). From here, x? 4 - range of definition of this expression. x^(1/4) is a fourth root from x. Therefore, x? 0 - range of definition of this expression.
3. In expressions at which there are logarithms it is necessary to remember that the basis of a logarithm of an is defined at a> 0 except for a=1. Expression under the sign of a logarithm has to be more than zero.
4. If at expression there are functions of an arcsine or an arccosine, then the area of values of the expression which is under the sign of this function has to be limited-1 at the left and 1 on the right. From here it is also necessary to find a range of definition of this expression.
5. Both division, and, for example, a square root can appear in expression. When finding a range of definition of all expression it is necessary to consider all moments which can lead to restriction of this area. Having excluded all improper values, it is necessary to write down a range of definition. The range of definition can accept also any valid values in the absence of specific points.