How to define area of values

How to define area of values

It is necessary to solve functions in everyday life not often, but when you face such need, to orient quickly can be difficult. Begin with definition of area of values.


1. Remember that function is such dependence of variable Y on variable X at which to each value of variable X there corresponds the unique value of variable Y. Variable X is an independent variable or an argument. Variable Y is a dependent variable. It is considered also that variable Y is function from variable X. Values of function are equal to values of a dependent variable.

2. For descriptive reasons write down expressions. If the dependence of variable Y on variable X is function, then for short it is written down so: y=f(x). (Read: at equal f from x.) By a f (x) symbol designate the value of function corresponding to the value of an argument equal x.

3. A range of definition of the f (x) function is called "the set of all valid values of an independent variable x at which function is defined (it makes sense)". Designate: D (f) (English Define – to define.) Example: The f (x) function = 1x+1 is defined for all valid values x meeting a condition x +1 ≠ 0, i.e. x ≠-1. Therefore D (f) = (-∞;-1) U (-1; ∞).

4. Area of values of the y=f (x) function is called "the set of all valid values which are occupied by independent variable y". Designation: E(f) (English Exist – to exist). Example: Y=x2 - 2x+10; as x2 - 2x +10 = x2 - 2x +1+9+(x-1)2 +9, the smallest value of a variable at =9 at x =1, therefore E(y) = [9; ∞)

5. All values of an independent variable display themselves a function range of definition. All values which are accepted by a dependent variable reflect area of values of function.

6. The area of values of function completely depends on its range of definition. In case the range of definition is not set, so it changes from minus infinity to plus infinity, thereby, search of value of function on the ends of a piece comes down to a slip of a limit of this function from minus and plus of infinity. Respectively, if function is set by a formula and its range of definition is not specified, then it is considered that the range of definition of function consists of all values of an argument at which the formula makes sense.

7. For finding of a set of values of functions it is necessary to know the main properties of elementary functions: range of definition, area of value, monotony, continuity, differentiability, parity, oddness, frequency, etc.

Author: «MirrorInfo» Dream Team