Piecewise Functions

Definition and Structure

A piecewise function is a function defined by multiple sub-functions, each applying to a certain interval of the domain. The function's rule changes depending on the value of the independent variable.

• Piecewise notation: A piecewise function is typically written using braces to show different cases for different intervals.

• Example structure:

Evaluating Piecewise Functions

To evaluate a piecewise function at a specific value, determine which case applies to the input and use the corresponding formula.

• Step 1: Identify the interval or condition that matches the input value.

• Step 2: Substitute the input value into the appropriate formula.

• Step 3: Simplify to find the output.

Example: Evaluation at Specific Points

Given the piecewise function:

• Evaluate : Since , use .

• Evaluate : Since , use .

• Evaluate : Since , use .

Applications of Piecewise Functions

• Modeling real-world situations: Piecewise functions are used to describe scenarios where a rule changes at certain thresholds, such as tax brackets, shipping rates, or speed limits.

• Graphing: The graph of a piecewise function may have breaks, jumps, or different slopes depending on the intervals.

Summary Table: Piecewise Function Evaluation

Additional info: The variables , , and are parameters of the function and would be given specific values in a concrete example.