Identify whether the function repeats its values for some nonzero shift
Find the smallest positive number (T) such that (f(x+T)=f(x)) for all (x) in the domain
Check common periodic functions:
For (sin(x)) and (cos(x)), the period is (2pi)
For (tan(x)) and (cot(x)), the period is (pi)
For (sec(x)) and (csc(x)), the period is (2pi)
For (f(ax+b)), divide the base period by (|a|)
For (f(x)=sin(ax+b)) or (cos(ax+b)), period (=frac{2pi}{|a|})
For (f(x)=tan(ax+b)) or (cot(ax+b)), period (=frac{pi}{|a|})
For sums of periodic functions, find a common period of all parts
If the periods are (T_1) and (T_2), look for the least common multiple when the ratio is rational
For algebraic combinations, verify the periodicity directly by testing (f(x+T)=f(x))
For piecewise functions, check periodicity on each interval and across boundaries
If no positive (T) satisfies (f(x+T)=f(x)), the function is not periodic