Period of a function
Period of a Function
In mathematics, the period of a function is a fundamental concept in the study of periodic functions. A periodic function is a function that repeats its values at regular intervals or periods. The period is the length of the smallest interval that you can translate along the function's domain such that the function's shape repeats.
Definition
The period of a function $f(x)$ is the smallest positive number $P$ such that:
$$ f(x + P) = f(x) $$
for all $x$ in the domain of $f$.
If such a $P$ exists, then $f$ is said to be periodic with period $P$. If no such smallest positive number exists, then the function is not periodic.
Properties of Periodic Functions
- Repetition: The function repeats its values every $P$ units along the x-axis.
- Symmetry: Many periodic functions exhibit symmetry, such as sine and cosine functions.
- Additivity: If $P$ is a period of $f(x)$, then so is any integer multiple of $P$, i.e., $nP$ for $n \in \mathbb{Z}$.
Examples of Periodic Functions
- Sine and Cosine Functions: The sine and cosine functions have a period of $2\pi$.
- Tangent Function: The tangent function has a period of $\pi$.
- Square Wave: A square wave function can have various periods depending on its construction.
Table: Differences and Important Points
| Property | Description | Example |
|---|---|---|
| Period | The smallest positive value for which the function repeats | For $f(x) = \sin(x)$, the period is $2\pi$. |
| Amplitude | The height from the average value to the maximum value of the function | For $f(x) = \sin(x)$, the amplitude is $1$. |
| Frequency | The number of times the function repeats per unit interval | For $f(x) = \sin(x)$, the frequency is $\frac{1}{2\pi}$. |
| Phase Shift | The horizontal shift of the function from its usual position | For $f(x) = \sin(x - \frac{\pi}{2})$, the phase shift is $\frac{\pi}{2}$ to the right. |
Formulas Involving Period
For trigonometric functions, the period can be affected by scaling the variable $x$. For example:
- For $f(x) = \sin(kx)$, the period is $\frac{2\pi}{|k|}$.
- For $f(x) = \cos(kx)$, the period is $\frac{2\pi}{|k|}$.
- For $f(x) = \tan(kx)$, the period is $\frac{\pi}{|k|}$.
Examples to Explain Important Points
Example 1: Basic Sine Function
Consider the function $f(x) = \sin(x)$. The sine function has a period of $2\pi$ because:
$$ \sin(x + 2\pi) = \sin(x) $$
for all $x$.
Example 2: Scaled Cosine Function
Consider the function $f(x) = \cos(3x)$. The period of this function is $\frac{2\pi}{3}$ because:
$$ \cos(3(x + \frac{2\pi}{3})) = \cos(3x + 2\pi) = \cos(3x) $$
for all $x$.
Example 3: Tangent Function with Phase Shift
Consider the function $f(x) = \tan(x - \frac{\pi}{4})$. The period of the tangent function is $\pi$, and the phase shift is $\frac{\pi}{4}$ to the right. The function repeats its values every $\pi$ units along the x-axis, but it is shifted to the right by $\frac{\pi}{4}$.
Example 4: Non-Trigonometric Periodic Function
Consider the function $f(x) = \text{sign}(\sin(x))$, where $\text{sign}$ is the signum function. This function produces a square wave that alternates between $1$ and $-1$. The period of this function is $2\pi$, the same as the sine function, because it is derived from the sine function.
Understanding the period of a function is crucial for analyzing and graphing periodic behavior, which is common in many areas of science and engineering, such as signal processing, physics, and even finance.