How do I tell if a Function is periodic or not? If it is, how do I find its Base Period?

Are all trig functions periodic? (besides inverse functions, like arcsin/arcos)|||sin(x), cos(x), tan(x), and their inverses are all periodic. But just because a function consists only trig functions doesn't guarantee periodicity.



For example, sin(x^2) is not periodic, nor is sin(1/x). If f(x) = sin(x) + cos(sqrt(2)*x), then f is not periodic. In general, f(x) = sin(a*x) + cos(b*x) is periodic if a and b are both rational.



For a general function f(x), to find the base period, first find any period P such that f(x) = f(x+P) for all x. Then, continue finding smaller periods of the form P/n for some integer n if they exist, or prove somehow that P is the smallest possible period. The method will vary for different types of functions.|||A function is periodic if (and only if) the value of the function (the output) repeats exactly after a period.





If the period is P, then


f(x+P) = f(x)


for ALL values of x.


And this should remain true for any integer multiple of the period.





f(x+7P) = f(x)


f(x+326P) = f(x)


and this must be true for ALL x.





This is really how you check is a function is periodic.





---





All the basic trigonometric functions are periodic (period = 2*pi or some harmonic) but I do not know if this is true for all mixed functions or for the versine functions.