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 $f$.
$$
f(x)=\left\{\begin{aligned}
x^{2}+3, & \text { if } x<0 \\
3-x, & \text { if } x \geq 0
\end{aligned}\right.
$$

Key Concepts

Domain Analysis and Boundary Considerations

Understanding the domain restrictions and the boundaries between different pieces of a function is essential for accurate evaluation and graphing. Analyzing how the function behaves at these boundaries helps in checking for continuity and in correctly depicting whether a boundary point is included or excluded.

Piecewise-Defined Functions

A piecewise-defined function is one that is specified by different formulas for different intervals of its input values. This concept requires students to pay attention to which formula applies based on the value of the input, ensuring that evaluations and manipulations use the correct part of the definition.

Function Evaluation

Evaluating a function involves substituting a given input value into the function's formula. With piecewise-defined functions, it is crucial to first identify which segment of the domain the input falls into, ensuring that the correct expression is used for computation.

Graphical Representation of Functions

Graphing a function entails plotting the set of (x, y) points and connecting them to visualize how the function behaves over its domain. For piecewise functions, each segment is graphed separately according to its definition, making sure to correctly indicate endpoints (using open or closed circles) determined by whether the value is included in the segment.

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