Piecewise Functions

Definition

A piecewise function is a mathematical function that is defined by multiple sub-functions, each valid for a specific interval or domain. These sub-functions are joined together to create a complete function that can model complex real-world phenomena.

5 Must Know Facts For Your Next Test

1. Piecewise functions are useful for modeling situations where the behavior of a function changes at certain points or intervals.
2. The domain of a piecewise function is the union of the domains of its individual sub-functions.
3. The range of a piecewise function is the union of the ranges of its individual sub-functions.
4. Piecewise functions are often represented using a table or a formula with conditional statements.
5. Graphing piecewise functions involves plotting the individual sub-functions and connecting them at the appropriate points.

Review Questions

• Explain how the domain and range of a piecewise function are determined.
  • The domain of a piecewise function is the union of the domains of its individual sub-functions. This means that the domain of the overall function includes all the input values for which at least one of the sub-functions is defined. Similarly, the range of a piecewise function is the union of the ranges of its individual sub-functions, encompassing all the possible output values that the function can produce.
• Describe the key features of a piecewise function and how they are represented.
  • A piecewise function is composed of multiple sub-functions, each valid for a specific interval or domain. These sub-functions are typically represented using a table or a formula with conditional statements, where the appropriate sub-function is applied based on the input value. The graph of a piecewise function is constructed by plotting the individual sub-functions and connecting them at the appropriate points, creating a continuous representation of the overall function.
• Analyze how the behavior of a piecewise function can change at different intervals or domains, and the implications for modeling real-world situations.
  • The ability of piecewise functions to model complex, real-world phenomena stems from their flexibility in capturing changes in behavior at different intervals or domains. By defining multiple sub-functions, piecewise functions can accurately represent situations where the underlying relationships or governing principles shift based on the input values. This makes piecewise functions particularly useful for modeling a wide range of applications, such as engineering systems, economic models, and biological processes, where the behavior of the system may change depending on the specific conditions or input values.

Related terms

Domain:

The set of input values for which a function is defined.

Range:

The set of all possible output values a function can produce.

Interval: A set of consecutive real numbers that can be represented using inequalities.