Question

Describe your understanding of the meaning of a periodic function.

Short answer

A periodic function repeats its values at regular intervals, with a positive number called the period.

Step-by-step solution

Define a Function

A function is a relation between a set of inputs and a set of permissible outputs, where each input is related to exactly one output.

Introduce Periodic Functions

A periodic function is a function that repeats its values in regular intervals or periods. In other words, if a function is periodic, there exists a positive number, called the period, such that the function's value at any given point is the same as its value at that point plus the period.

Mathematical Formulation of Periodic Functions

Mathematically, a function \( f(x) \) is called periodic if there exists a positive number \( P \) such that \( f(x+P) = f(x) \) for all values of \( x \) in the domain of \( f(x) \). The smallest such positive number \( P \) is called the fundamental period of the function.

Examples of Periodic Functions

Common examples of periodic functions include trigonometric functions such as sine and cosine. For instance, the sine function, \( \text{sin}(x) \), is periodic with a period of \( 2\pi \) because \( \text{sin}(x + 2\pi) = \text{sin}(x) \) for all \( x \).

Visualize Periodic Functions

Visualizing periodic functions can help in understanding their behavior. When graphed, periodic functions exhibit repeating patterns. For instance, the graph of the sine function shows a wave that repeats every \( 2\pi \) units.

Key concepts

Function Definition

A function is a fundamental mathematical concept. It is essentially a relation that uniquely associates each element from one set (called the domain) to exactly one element in another set (called the co-domain). Think of it as a machine where each input has only one output.
For example, for a function \( f(x) \):

• \( f(2) = 4 \)
• \( f(3)= 9 \)

Each input value of \( x \) gives only one output. This one-to-one relationship is what defines a function.

Trigonometric Functions

Trigonometric functions are a specific category of functions that relate to angles and their ratios. They are essential in studying periodic phenomena. The most commonly used trigonometric functions include:

• \( \text{sin}(x) \) - Sine
• \( \text{cos}(x) \) - Cosine
• \( \text{tan}(x) \) - Tangent

These functions are periodic and are very useful in modeling wave patterns and oscillations. Understanding these can help you gain deeper insights into periodic functions.

Fundamental Period

The fundamental period of a function is the smallest positive number \( P \) for which the function repeats. For a periodic function \( f(x) \), this means that \( f(x + P) = f(x) \) for every value of \( x \).
For example, in the function \( \text{sin}(x) \), the period is \( 2\pi \) because \( \text{sin}(x + 2\pi ) = \text{sin}(x) \). This repetition pattern helps us predict the function's values over its domain.

Sine Function

The sine function, denoted as \( \text{sin}(x) \), is one of the basic trigonometric functions that describe a smooth, periodic oscillation. It is widely used in various fields, including physics, engineering, and music.
Mathematically, the sine function can be defined as:

• \( \text{sin}(x) = \frac{\text{opposite}}{\text{hypotenuse}} \) in a right-angled triangle

On a graph, the sine function produces a wave that repeats every \( 2\pi \) units horizontally. This repetitive pattern is a clear demonstration of its periodic nature.

Cosine Function

The cosine function, denoted as \( \text{cos}(x) \), is another fundamental trigonometric function closely related to the sine function. It also exhibits periodic behavior and shows a smooth wave on a graph.
The cosine function is defined as:

• \( \text{cos}(x) = \frac{\text{adjacent}}{\text{hypotenuse}} \) in a right-angled triangle

Similar to the sine function, the cosine function repeats every \( 2\pi \), making it a periodic function. The graph of \( \text{cos}(x) \) waves smoothly and helps in understanding various oscillatory motions.