Give a rule for each piecewise-defined function. Also give the domain and range.

Verified step by step guidance

1. Identify the different pieces of the piecewise-defined function. Each piece will have its own rule (usually a function expression) and a specific domain interval where it applies.
2. Write down the function rule for each piece, clearly indicating the domain interval for which that rule is valid. For example, a piece might be defined as \(f(x) = 2x + 3\) for \(x \leq 1\).
3. Combine all the pieces into one piecewise function notation, using curly braces and specifying the domain for each piece. For example: \[ f(x) = \begin{cases} 2x + 3 & \text{if } x \leq 1 \\ -x + 4 & \text{if } x > 1 \end{cases} \]
4. Determine the overall domain of the piecewise function by combining the domains of all individual pieces. This is usually the union of all intervals where the pieces are defined.
5. Find the range of the piecewise function by analyzing the output values of each piece over its domain. Consider the behavior of each function piece (increasing, decreasing, constant) and the endpoints of the intervals to identify the minimum and maximum values.

Key Concepts

Piecewise-Defined Functions

A piecewise-defined function is a function composed of multiple sub-functions, each applying to a specific interval or condition in the domain. Understanding how to write and interpret these rules is essential for analyzing the function's behavior over different parts of its domain.

Domain of a Function

The domain of a function is the complete set of input values (x-values) for which the function is defined. For piecewise functions, the domain is often divided into intervals corresponding to each piece, and identifying these intervals is crucial for correctly describing the function.

Range of a Function

The range of a function is the set of all possible output values (y-values) the function can produce. Determining the range for piecewise functions involves analyzing the outputs of each piece over its domain interval and combining these results to find the overall range.