Piecewise function

Definition

A piecewise function is a mathematical function that is defined by multiple sub-functions, each applying to a specific interval or condition within its domain. These functions are often used to model situations where a single formula cannot adequately describe the behavior of the function across its entire domain, leading to the need for different expressions based on different conditions.

5 Must Know Facts For Your Next Test

1. Piecewise functions can have different formulas for different intervals, which means that they can change their behavior based on the input value.
2. To evaluate a piecewise function at a certain point, you first determine which interval the point falls into and then apply the corresponding formula.
3. These functions are often represented using curly braces to separate the different pieces, clearly showing which formula applies to each interval.
4. When graphing piecewise functions, it is important to note whether the endpoints of the intervals are included (closed) or excluded (open) from the graph.
5. Many real-world situations, like tax brackets or shipping costs, can be modeled effectively using piecewise functions due to their varying rates or conditions.

Review Questions

• How do you determine which formula to use when evaluating a piecewise function?
  • To determine which formula to use for a piecewise function, you first need to identify the input value you want to evaluate. Then, check which interval this value falls into by comparing it with the specified conditions for each sub-function. Once you locate the correct interval, you can apply the corresponding formula to find the output value.
• Discuss how piecewise functions can be applied in real-world scenarios and why they are effective for modeling certain situations.
  • Piecewise functions are particularly effective in modeling real-world scenarios where conditions change based on specific criteria. For instance, tax rates often apply differently depending on income brackets, which can be represented as a piecewise function where each bracket corresponds to a different tax rate. This flexibility allows piecewise functions to accurately reflect situations where a single linear or polynomial function would be inadequate, providing a more accurate representation of various behaviors in fields like economics or logistics.
• Evaluate the impact of discontinuities in piecewise functions and how they affect graphing and analysis.
  • Discontinuities in piecewise functions can significantly impact both graphing and analysis because they indicate points where the behavior of the function abruptly changes. For example, if a sub-function has an open endpoint at a certain value while another starts at that value with a closed endpoint, it creates a jump discontinuity. When graphing these functions, it's crucial to represent these points accurately using open and closed dots to indicate whether values at those points are included in the function's domain. Understanding these discontinuities helps in analyzing the overall behavior and properties of piecewise functions.

Related terms

Domain:

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

Interval: A range of values between two endpoints, often used to specify the domain of a particular sub-function within a piecewise function.

Continuity: A property of a function where it is uninterrupted or unbroken across its domain, meaning it can be graphed without lifting the pencil off the paper.