A function f: X->Y is said to be a periodic function provided there exists a positive real number T such that f(x + T) = f(x), for all x belongs X. The least of all such positive numbers T is called the principal period or fundamental period or simply period of f.
- To check the periodicity of a function put f(T+x)=f(x) and solve this equation to find the positive values of t independent of x. If positive values of T independent of x are obtained, then f(x) is a periodic function and the least positive value of T is the period of the function f(x). If no positive value of T independent of x is obtained then f(x) is non-periodic function.
- A constant function is periodic but does not have a well-defined period.
- If f(x) is periodic with period p, then f(ax + b) where a, b ∈ R (a ≠ 0) is also period with period p/|a|.
- If f(x) is periodic with period p, then a f(x) + b where a, b ∈ R (a ≠ 0) is also periodic with period p.
- If f(x) is periodic with period p, then f (ax + b) where a, b ∈ R (a ≠ 0) is also period with period p/|a|
SOME IMPORTANT POINTS:
- LCM of p and q always exist if p/q is a rational quantity. If p/q is irrational then algebraic) combination of f and g is non-periodic.
- sinnx, cosnx, cosecnx and Secnx have period 2π if n is odd and π if n is even.
- tannx and cotnx have period π whether n is odd or even.
- If g is periodic then fog will always be a periodic function. Period of fog may or may not be the period of g.
- If f is periodic and g is strictly monotonic (other than linear) then fog is non-periodic.
There are two types of questions asked in the examination. You may be asked to test for periodicity of the function or to find the period of the function. In the former case you just need to show that f(x + T) = f(x) for same T (>0) independent of x whereas in the latter, you are required to find a least positive number T independent of x for which f(x +T)= f(x) is satisfied
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