Identify the smallest positive value (T) such that (f(x+T)=f(x)) for all (x) in the domain
Check whether the function repeats its values at regular intervals
For basic trigonometric functions, use known periods:
(sin x) and (cos x): (2pi)
(tan x) and (cot x): (pi)
(sec x) and (csc x): (2pi)
For transformed trigonometric functions (f(x)=asin(bx+c)+d) or (acos(bx+c)+d), use (T=frac{2pi}{|b|})
For transformed tangent or cotangent functions (f(x)=atan(bx+c)+d) or (acot(bx+c)+d), use (T=frac{pi}{|b|})
For sums of periodic functions, find the periods of each part
Determine the least common multiple of the individual periods if one exists
If no common repeating interval exists, the function is not periodic
Verify the candidate period by substituting (x+T) into the function and simplifying to match (f(x))
Use graphing or pattern recognition to confirm the repeating interval if needed