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.

### Example 1

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 :