A piecewise definition is a way of defining a function by specifying different formulas or expressions for different intervals or domains of the independent variable. This allows for the representation of functions that have different behaviors or characteristics in different regions.
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Piecewise definitions are commonly used to model real-world phenomena that exhibit different behaviors in different regions or conditions.
The different formulas or expressions in a piecewise definition are applied based on the value of the independent variable, which determines the appropriate interval or domain.
Piecewise functions can be used to represent absolute value functions, which have different behaviors for positive and negative inputs.
Identifying the correct interval or domain is crucial when evaluating a piecewise function, as the function may have different values or behaviors depending on the input.
Graphing piecewise functions often involves connecting the different segments or pieces of the function, resulting in a graph with distinct regions or shapes.
Review Questions
Explain how a piecewise definition can be used to represent an absolute value function.
An absolute value function can be defined piecewise by specifying different formulas or expressions for the positive and negative regions of the input variable. For example, the absolute value function $|x|$ can be defined as a piecewise function where $f(x) = x$ for $x \geq 0$, and $f(x) = -x$ for $x < 0$. This piecewise definition allows the function to represent the distance of the input from zero, regardless of the sign of the input.
Describe the role of the domain and intervals in a piecewise definition.
In a piecewise definition, the domain and intervals of the independent variable determine which formula or expression should be used to evaluate the function. The different formulas or expressions are applied based on the value of the input, which corresponds to a specific interval or domain of the function. Identifying the correct interval or domain is crucial when working with piecewise functions, as the function may have different values or behaviors depending on the input.
Analyze how the graphical representation of a piecewise function reflects its underlying definition.
The graphical representation of a piecewise function often consists of distinct segments or regions, each corresponding to a different formula or expression in the piecewise definition. The transitions between these segments, known as 'corners' or 'kinks,' indicate where the function changes its behavior based on the input value. By examining the graph of a piecewise function, one can gain insights into the underlying structure and domains of the different formulas or expressions that define the function.