Piecewise Functions

A piecewise function is a function that is defined by several sub-functions, each applying to a different interval of the main function's domain. These sub-functions are defined by specifying the domain over which each sub-function is valid. Piecewise functions are often used to model situations where different rules or formulas apply in different scenarios or intervals.

Example of a Piecewise Function:

Let's consider the following piecewise function:

In this example, the function \( f(x) \) is defined differently for two distinct intervals: when \( x \) is less than or equal to 3 and when \( x \) is greater than 3.

Key Concepts:

1. Domain and Intervals: Piecewise functions are defined over specific intervals, and it's important to understand and specify the domain for each sub-function within the piecewise function.
2. Continuity: When working with piecewise functions, it's important to consider the continuity of the function at the points where the sub-functions meet. This involves checking if the function values from the left and right of the intersection point are equal.
3. Graphing: Graphing a piecewise function involves plotting the graphs of the individual sub-functions over their respective domains, and ensuring a smooth transition between the sub-functions at the points of intersection.

Study Guide:

Here are some steps to help you better understand and work with piecewise functions:

1. Identify the different intervals over which the function is defined.
2. Write down the specific formula for each sub-function within the defined intervals.
3. Understand the domain and range of each sub-function and how they contribute to the overall piecewise function.
4. Check for continuity at the points of intersection between the sub-functions.
5. Practice graphing piecewise functions and understanding how the individual sub-functions come together to form the overall function.

By following these steps and practicing with different examples, you'll gain a solid understanding of piecewise functions and how to work with them effectively.