Piecewise function

Definition

A piecewise function is a function that is defined by multiple sub-functions, each applying to a specific interval of the domain. This means that different formulas or expressions can be used to calculate the output based on the input's value. Piecewise functions are useful in modeling situations where different rules or conditions apply to different parts of the input range.

5 Must Know Facts For Your Next Test

1. Piecewise functions are often represented using a bracket notation, showing each sub-function alongside its corresponding interval.
2. They can model real-life situations such as tax brackets, where different rates apply at different income levels.
3. A piecewise function can be continuous or discontinuous, depending on how the sub-functions connect at their boundaries.
4. When graphing piecewise functions, it's important to check whether the endpoints of each interval are included (using solid dots) or excluded (using open circles).
5. Evaluating a piecewise function requires determining which sub-function to use based on the input value's location within the defined intervals.

Review Questions

• How does a piecewise function differ from a traditional function in terms of its definition and application?
  • A piecewise function differs from a traditional function because it uses multiple sub-functions, each applicable to specific intervals of the input values. Traditional functions are generally described by a single formula across their entire domain. In contrast, piecewise functions allow for flexibility in modeling situations where different conditions or rules apply, making them particularly useful in scenarios like tax calculations or shipping costs, where rates change based on thresholds.
• Discuss how you would evaluate a piecewise function when given an input value that falls within one of its defined intervals.
  • To evaluate a piecewise function for a given input value, first identify which interval that value belongs to by comparing it with the defined ranges in the function. Once the correct interval is determined, use the corresponding sub-function to calculate the output. It’s also crucial to consider if the endpoint of that interval is included or excluded to ensure the proper formula is applied.
• Analyze how piecewise functions can represent real-world scenarios and provide an example to illustrate your point.
  • Piecewise functions effectively represent real-world scenarios by capturing situations where conditions change based on certain thresholds. For example, consider a utility company that charges different rates for electricity based on usage: the first 100 kWh might cost $0.10 per kWh, while usage above that could be charged at $0.15 per kWh. A piecewise function can model this by defining separate equations for the different usage ranges, allowing for clear calculations and understanding of costs incurred at varying levels of consumption.

Related terms

Domain:

The set of all possible input values (x-values) for a function.

Range:

The set of all possible output values (y-values) that a function can produce.

Continuity: A property of functions where they do not have breaks, jumps, or holes in their graph, allowing for smooth transitions between values.