MathsMediumClass 11

Periodic Functions — Period, fundamental period

Sets, Relations & Functions

JEE Qs

Hard

min

Always verify periodicity and find the fundamental period by applying the definition f(x+T)=f(x) and ensuring T is the smallest positive value; do not blindly use LCM for all sums of functions.

🧮 Key Formulas

A function f(x) is periodic if there exists a positive real number T such that f(x + T) = f(x) for all x in the domain of f. The smallest such positive T is called the fundamental period.

If f(x) has a fundamental period T, then g(x) = f(ax + b) has a fundamental period T/|a|.

If f(x) has a fundamental period T, then 1/f(x) and sqrt(f(x)) (if defined) are also periodic with period T (though domain might change).

If f(x) has period T1 and g(x) has period T2, then f(x) +/- g(x) has a period equal to LCM(T1, T2) if T1/T2 is a rational number. If T1/T2 is irrational, the sum/difference may not be periodic.

Standard periods: sin(x), cos(x), sec(x), cosec(x) have period 2π. tan(x), cot(x) have period π.

For integer n: sin^n(x), cos^n(x) have period π if n is even, and 2π if n is odd. (Always verify this with fundamental period definition).

The fractional part function {x} has a fundamental period of 1.

✅ Key Points for JEE

1The fundamental period is the *smallest positive* value of T for which f(x+T) = f(x) holds for all x in the domain. A function can have multiple periods, but only one fundamental period.
2The LCM method for finding the period of a sum/difference of functions (f(x) +/- g(x)) is only applicable if the ratio of their individual periods (T1/T2) is a rational number. If the ratio is irrational, the sum/difference may not be periodic.
3To prove a function is periodic, always revert to the definition f(x+T) = f(x). For complicated functions, directly solving this equation for the smallest T is the most reliable method.
4A constant function, f(x) = c, is periodic but does not have a fundamental period, as any positive real number T satisfies f(x+T)=f(x).
5For a function f(x) that is periodic with period T, the graph repeats itself over intervals of length T. This property is useful for sketching and understanding function behavior.

⚠️ Common Mistakes

✕Confusing 'period' with 'fundamental period'. Any multiple of the fundamental period is also a period, but only the smallest positive one is fundamental.
✕Incorrectly applying the LCM rule for sum/difference of periods when the ratio of individual periods is irrational (e.g., assuming sin(x) + cos(sqrt(2)x) has a period).
✕Assuming that if f(x) has period T, then f(g(x)) also has period T. This is generally false; the period of composite functions needs careful evaluation.
✕Not checking the domain of the function while determining periodicity (e.g., sqrt(sin x) is only defined when sin x >= 0).

NCERT Chapters

Class 11 Maths Ch 2: Relations and Functions
Class 11 Maths Ch 3: Trigonometric Functions