Plot several points, and sketch the graph of the function...

Plot several points, and sketch the graph of the function defined by the given expression.
$$
\left\{\begin{aligned}
x^{2}-4 & \text { if } x<2 \\
x+2 & \text { if }-2 \leq x<2 \\
x^{2} & \text { if } 2 \leq x
\end{aligned}\right.
$$

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Key Concepts

Piecewise Functions

Piecewise functions are defined by different expressions over various intervals of their domain. Understanding them involves recognizing that each segment of the function has its own rule that applies only within a specific range of x-values, which is essential for correctly interpreting and graphing the function.

Domain Partitioning

Domain partitioning refers to dividing the entire domain of a function into segments on which different rules apply. This is particularly important for piecewise functions, as it helps determine which expression to use for a given x-value, ensuring accurate evaluation and graphing of the function.

Graphing Functions

Graphing functions involves plotting a series of points determined by the function's rule and connecting them appropriately. For piecewise functions, this process includes graphing each segment over its respective interval and paying attention to the behavior at the endpoints.

Endpoint Behavior and Continuity

Analyzing endpoint behavior is crucial when working with piecewise functions to assess whether the segments meet smoothly or involve jumps. This concept involves checking the limits from both sides of the breakpoints to determine if the function is continuous or if there are any discontinuities.

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