For each piecewise-defined function, (a) calculate f(-1),...

For each piecewise-defined function,
(a) calculate $f(-1), f(0), f(1)$, and $f(2)$, and (b) sketch a graph of $f$.
$$
f(x)=\left\{\begin{aligned}
4 x-1, & \text { if } x<0 \\
2, & \text { if } x=0 \\
-3 x+5, & \text { if } x>0
\end{aligned}\right.
$$

Key Concepts

Piecewise-Defined Functions

Piecewise-defined functions are functions that use different expressions with distinct rules on different parts of their domains. This concept is central to understanding how a single function can behave in multiple ways depending on the input value, and it is defined by breaking the domain into specific intervals with associated function definitions.

Function Evaluation

Function evaluation involves computing the function's output for a given input. With piecewise functions, this means first determining which interval the input belongs to and then applying the corresponding rule. Mastery of this concept requires careful attention to the conditional boundaries that separate different pieces of the function.

Domain Partitioning

Domain partitioning is the process of segmenting the domain of a function into distinct intervals based on provided conditions. This enables one to correctly relate each piece of the function's definition to the appropriate input values, emphasizing the importance of understanding both the intervals and the related rule for accurate computation and analysis.

Graphing Piecewise Functions

Graphing piecewise functions involves plotting each segment of the function separately according to its defined domain interval. When doing so, it's important to denote whether endpoints are included or excluded (typically using filled or open circles), which reflects the continuity or discontinuity at the points where the function rules change.

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