Periodic Functions Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Periodic Functions.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A periodic function repeats its values at regular intervals: f(x + T) = f(x) for all x, where T > 0 is the period — the length of one complete cycle.

The same pattern over and over. Like a heartbeat or the seasons.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: The period is the smallest positive T for which f(x + T) = f(x). Knowing one period means knowing the entire function's behavior for all real inputs.

Common stuck point: Amplitude (height) and period (width) are independent properties.

Sense of Study hint: Try tracing one full cycle on the graph: start at any point and find where the pattern repeats exactly. That horizontal distance is the period.

Worked Examples

Example 1

easy
Verify that f(x) = \sin(x) is periodic with period 2\pi by checking the definition f(x + p) = f(x).

Solution

  1. 1
    Recall the identity: \sin(x + 2\pi) = \sin x \cos 2\pi + \cos x \sin 2\pi.
  2. 2
    Substitute known values \cos 2\pi = 1 and \sin 2\pi = 0: \sin(x + 2\pi) = \sin x \cdot 1 + \cos x \cdot 0 = \sin x.
  3. 3
    Since f(x + 2\pi) = f(x) for all x, and 2\pi is the smallest such positive number, the period is p = 2\pi.

Answer

f(x) = \sin(x) has period 2\pi
A function is periodic if it repeats its values at regular intervals. The sine function satisfies f(x+2\pi)=f(x) for all real x, making 2\pi its fundamental period.

Example 2

medium
Find the period of g(x) = \cos(3x) and sketch one complete cycle.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
The function h(x) = \tan(x) has period \pi. What is \tan\!\left(\frac{5\pi}{4}\right) given that \tan\!\left(\frac{\pi}{4}\right) = 1?

Example 2

hard
A function satisfies f(x+4) = f(x) for all x, and is defined on [0,4) by f(x) = x^2 - 4x + 3. Find f(13.5).
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Background Knowledge

These ideas may be useful before you work through the harder examples.

trigonometric functions