Question

do you understand?

1. essential question what are the key features of piecewise-defined functions?

Explanation:

Piecewise-defined functions have distinct rules for different intervals of the input domain. Key features include:

1. Domain Partitioning: The overall domain is split into non-overlapping intervals (or points), each with its own function rule.
2. Multiple Function Rules: A unique expression defines the output for each specific sub-domain interval.
3. Potential Discontinuities: Gaps, jumps, or holes can occur at the boundary points between intervals, though some piecewise functions are continuous across all boundaries.
4. Specific Boundary Conditions: Clear definitions for whether boundary points belong to one interval or another (often using $\leq$, $\geq$, $<$, or $>$).
5. Combined Behavior: The function's overall graph and behavior are a combination of the individual rules for each interval.

Answer:

The key features of piecewise-defined functions are:

1. Their domain is divided into non-overlapping intervals (or single points).
2. A different function rule applies to each interval of the domain.
3. They may have discontinuities (jumps, gaps, holes) at the interval boundary points (or can be continuous across boundaries).
4. Explicit boundary conditions define which interval includes each domain boundary point.
5. Their overall behavior is a combination of the behavior of each individual segment function.