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.
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Piecewise functions are useful for modeling situations where the behavior of a function changes at certain points or intervals.
The domain of a piecewise function is the union of the domains of its individual sub-functions.
The range of a piecewise function is the union of the ranges of its individual sub-functions.
Piecewise functions are often represented using a table or a formula with conditional statements.
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.