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Determination of period requires analysis based on properties and methods of functions, including transformation of graphs. We can use following guidelines to determine period :
1 : The definition of periodicity is helpful in determining period of generic function like f(x) under certain condition whose form i.e. rule is not given. The definition is also used to determine whether given function is periodic or not in the first place.
2 : Various generalizations about periods of trigonometric functions, their even and odd exponentiations etc. help to determine periods of individual terms in a trigonometric expression.
3 : Transformation of graphs or functions is an important method to determine periods for function expression, which involves arithmetic operations like addition, subtraction, multiplication, division or negation. These operations are applied either on independent variable or function itself.
4 : One of important concepts for determining period of function involving multiple periodic terms is that individual function should repeat simultaneously. This gives rises to LCM rule. It is defined for terms which appear as sum or difference in the function. However, LCM concept can be extended to division or multiplication of periodic terms as well.
5 : Exceptions to LCM rule are important. Even function and function comprising of cofunctions are two notable exceptions to LCM rule. Besides, LCM of irrational periods of different kinds is not possible. This fact is used to determine periodic nature of function involving irrational periods. If LCM can not be determined, then given function is not periodic in the first place.
6 : Composition of function gof(x) has same period as that of the period of f(x) under certain conditions : either domain of f(x) is proper subset of domain of g(x) or g(x) is a bijection (one-one function).
We are required to determine period of a generic function like f(x) based on certain conditional relation. In such situation, we are required to manipulate given condition in such a manner that we ultimately get a relation of type f(x+T)= f(x). Generally, this requires substitution of independent variable with expression which results in existing expressions. This enables us to use given relation repeatedly. This concept is better understood by working with an example.
Problem : A function f(x) satisfies equation given :
Determine its period.
Solution : Here, main strategy is to replace “x” such that we get expression on RHS which is same as the expression on LHS. Replacing “x” by “x+1” and replacing “x” by “x-1” separately in given equation, we get two equations :
Adding these two equations, we get term on RHS, which is same as LHS :
Replacing “x” by “x+2” in above equation, we have :
Note that we have not replaced “x” by “x-2”, because that yields a related which has argument form of “x-4”. As definition of periodic function involves addition of a positive constant being added to independent variable, we opt to replace “x” by “x+2”. Now, adding two preceding equations,
Replacing “x” by “x+2” so that one of two term becomes f(x), we have :
Replacing “x” by “x+6”, we have :
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