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, typically defined over a specific interval.
2. For each piece, determine the rule or expression that defines the function within that interval. This could be a linear function, quadratic function, or another type of function.
3. Specify the domain for each piece. The domain is the set of all input values (x-values) for which the piece is defined. This is usually given as an interval.
4. Determine the range for each piece. The range is the set of all possible output values (y-values) that the function can take on within the specified domain.
5. Combine the rules, domains, and ranges for each piece to fully describe the piecewise-defined function. Ensure that the domains of the pieces do not overlap unless specified by the problem.

Key Concepts

Piecewise-Defined Functions

A piecewise-defined function is a function that is defined by different expressions or rules over different intervals of its domain. Each piece of the function applies to a specific part of the input values, allowing for flexibility in modeling complex behaviors. For example, a function might be defined as f(x) = x^2 for x < 0 and f(x) = 2x + 1 for x ≥ 0.

Domain

The domain of a function is the complete set of possible input values (x-values) for which the function is defined. In the context of piecewise functions, the domain is often divided into intervals that correspond to the different pieces of the function. Understanding the domain is crucial for determining where each rule applies and ensuring that inputs are valid.

Range

The range of a function is the set of all possible output values (y-values) that the function can produce based on its domain. For piecewise-defined functions, the range can vary depending on the specific rules applied in each interval. Analyzing the range helps in understanding the overall behavior of the function and identifying any limitations on the output values.