To find the period of a function, you need to identify the smallest positive value of \( p \) such that \( f(x + p) = f(x) \) for all \( x \) in the domain of the function. In other words, the period is the distance along the x-axis after which the function starts to repeat itself.
Here's how you can find the period of a function:
1. **Identify the Function**: Determine the function for which you want to find the period. Let's denote it as \( f(x) \).
2. **Identify the Form of the Function**: Determine the form of the function. Some common periodic functions include trigonometric functions (like sine and cosine), exponential functions, and periodic piecewise functions.
3. **Use Properties of the Function to Find the Period**:
- **Trigonometric Functions**: For functions like sine and cosine, the period is \( 2\pi \) for basic functions. However, if there's any transformation (like stretching or compressing) or phase shift, you need to adjust accordingly. The period \( p \) for a sine or cosine function can be calculated as \( p = \frac{{2\pi}}{{|b|}} \), where \( b \) is the coefficient of \( x \) in the function (if there's any).
- **Exponential Functions**: For exponential functions, the period is \( \frac{{2\pi}}{{|b|}} \), where \( b \) is the coefficient of \( x \) in the exponent.
- **Piecewise Functions**: For piecewise functions, determine the behavior of the function over different intervals and find the smallest positive value of \( p \) for which the function repeats itself.
4. **Verify**: Once you have found the period, verify that \( f(x + p) = f(x) \) for all \( x \) in the domain of the function.
5. **Adjustments for Different Domains**: Keep in mind that the period may differ based on the domain of the function. For example, trigonometric functions have a period of \( 2\pi \) in radians, but if the domain is specified in degrees, you need to adjust accordingly using the conversion factor \( \frac{{180^\circ}}{{\pi}} \).
By following these steps, you can find the period of a given function. Remember to consider any transformations or adjustments to the basic period based on the specific form of the function.