Periodic Function

Periodic Function

A function which has a graph that repeats itself identically over and over as it is followed from left to right. Formally, a function f is periodic if there exists a number p such that f(x + p) = f(x) for all x.

Key Formula

f(x + p) = f(x) \quad \text{for all } x

Where:

$f$ = The periodic function
$x$ = Any input value in the domain of f
$p$ = The period — the smallest positive number for which the equation holds

Worked Example

Problem: Show that f(x) = sin(x) is periodic and find its period.

Step 1: Recall the defining condition for a periodic function: we need a positive number p such that f(x + p) = f(x) for all x.

f(x + p) = f(x)

Step 2: Test p = 2π. Substitute into the sine function.

\sin(x + 2\pi) = \sin(x)

Step 3: This identity holds for every real number x because the sine function completes one full cycle every 2π radians. You can verify with specific values: sin(0) = 0 and sin(0 + 2π) = sin(2π) = 0.

\sin(0 + 2\pi) = \sin(0) = 0 \quad \checkmark

Step 4: Check that no smaller positive number works. For example, try p = π: sin(0 + π) = sin(π) = 0 = sin(0), but sin(π/2 + π) = sin(3π/2) = −1 ≠ 1 = sin(π/2). So π is not a period.

\sin\!\left(\frac{\pi}{2} + \pi\right) = -1 \neq 1 = \sin\!\left(\frac{\pi}{2}\right)

Step 5: Since 2π is the smallest positive value of p satisfying the condition, the period of sin(x) is 2π.

p = 2\pi

Answer: f(x) = sin(x) is periodic with period 2π.

Another Example

This example differs by applying the period formula to a transformed trigonometric function (tan instead of sin, with a horizontal compression factor), showing how the coefficient inside the argument affects the period.

Problem: Determine whether g(x) = tan(3x) is periodic, and if so, find its period.

Step 1: The basic tangent function tan(x) has period π, meaning tan(x + π) = tan(x) for all x in its domain.

\tan(x + \pi) = \tan(x)

Step 2: For a horizontal compression g(x) = tan(Bx), the period changes to π/|B|. Here B = 3.

p = \frac{\pi}{|B|} = \frac{\pi}{3}

Step 3: Verify the condition: substitute x + π/3 into g.

g\!\left(x + \frac{\pi}{3}\right) = \tan\!\left(3\!\left(x + \frac{\pi}{3}\right)\right) = \tan(3x + \pi) = \tan(3x) = g(x)

Step 4: The equation g(x + π/3) = g(x) holds for all x in the domain, and no smaller positive number satisfies it.

Answer: g(x) = tan(3x) is periodic with period π/3.

Frequently Asked Questions

How do you find the period of a periodic function?

For sine and cosine functions of the form f(x) = sin(Bx) or f(x) = cos(Bx), the period is 2π/|B|. For tangent and cotangent, the period is π/|B|. For a general function, you solve f(x + p) = f(x) for the smallest positive p. If the function is given as a graph, measure the horizontal distance between two consecutive identical points (such as two adjacent peaks).

What is the difference between a periodic function and a non-periodic function?

A periodic function has a repeating pattern: its output values cycle through the same sequence over and over at a fixed interval. A non-periodic function, such as f(x) = x² or f(x) = eˣ, never repeats its entire pattern. No positive number p exists that satisfies f(x + p) = f(x) for every x.

Are all trigonometric functions periodic?

The six standard trigonometric functions — sin, cos, tan, cot, sec, and csc — are all periodic. Sine and cosine have period 2π, while tangent and cotangent have period π. Secant and cosecant share the period of cosine and sine respectively (both 2π). However, inverse trigonometric functions like arcsin(x) are not periodic.

Periodic Function vs. Non-Periodic Function

	Periodic Function	Non-Periodic Function
Definition	f(x + p) = f(x) for some positive p and all x	No such positive p exists for all x
Graph behavior	Repeats identically at regular intervals	Does not repeat a complete cycle
Examples	sin(x), cos(x), tan(x)	x², eˣ, ln(x), √x
Domain	Typically all reals (or all reals minus isolated points)	Can be any domain

Why It Matters

Periodic functions are central to trigonometry, precalculus, and physics. You encounter them when modeling anything that repeats: sound waves, alternating current, the motion of a pendulum, seasonal temperatures, and tidal patterns. Understanding periodicity is also essential in calculus when you study Fourier series, which express complicated signals as sums of simple periodic functions.

Common Mistakes

Mistake: Confusing the period with the amplitude or frequency.

Correction: The period is the horizontal length of one complete cycle (measured along the x-axis). The amplitude is the vertical distance from the midline to a peak, and the frequency is the reciprocal of the period (number of cycles per unit). Keep these three quantities distinct.

Mistake: Forgetting to use the smallest positive p when stating the period.

Correction: Many values of p satisfy f(x + p) = f(x). For sin(x), both p = 2π and p = 4π work. The period is defined as the smallest such positive p. Always verify that no smaller value satisfies the condition before declaring the period.

Related Terms

Function — General concept that periodic functions are a type of
Period of a Periodic Function — The smallest repeating interval of a periodic function
Periodic Motion — Physical motion described by periodic functions
Graph of an Equation or Inequality — Visual representation showing the repeating pattern
Amplitude — Measures the height of oscillation in periodic functions
Frequency — Reciprocal of period; counts cycles per unit
Sine — The most fundamental periodic function
Tangent — Periodic function with period π instead of 2π