Make a table listing ordered pairs for each function. Then...

Make a table listing ordered pairs for each function. Then sketch the graph and state the domain and range. Identify any intervals on which $f$ is increasing, decreasing, or constant. $$f(x)=\left\{\begin{array}{cll} x+3 & \text { for } & x \leq-2 \\ \sqrt{4-x^{2}} & \text { for } & -2<x<2 \\ -x+3 & \text { for } & x \geq 2 \end{array}\right.$$

Make a table listing ordered pairs for each function. Then sketch the graph and state the domain and range. Identify any intervals on which $f$ is increasing, decreasing, or constant.
$$f(x)=\left\{\begin{array}{cll}
x+3 & \text { for } & x \leq-2 \\
\sqrt{4-x^{2}} & \text { for } & -2<x<2 \\
-x+3 & \text { for } & x \geq 2
\end{array}\right.$$

Key Concepts

Piecewise Functions

A piecewise function is defined by different expressions over various intervals of its domain. Each piece of the function applies under specific conditions, and understanding these conditions is essential when evaluating the function or analyzing its behavior, including continuity, limits, and differentiability at the boundaries between pieces.

Ordered Pairs

Ordered pairs represent the input-output relationship of a function, typically written in the form (x, f(x)). They are the building blocks of graphing a function, allowing one to create a visual representation on the coordinate plane by plotting each pair as a point.

Graphing Functions

Graphing a function involves plotting the ordered pairs on a coordinate system and joining them according to the functional behavior. This process helps visualize how the function behaves over its domain and make it easier to identify trends, intercepts, and discontinuities.

Domain and Range

The domain of a function is the set of all input values (x-values) for which the function is defined, while the range is the set of all output values (f(x) values) produced by the function. Identifying these sets is crucial for understanding the extent and limitations of a function’s behavior.

Intervals of Increase and Decrease

Determining intervals where a function is increasing, decreasing, or constant involves analyzing how the function’s output changes as the input changes. Increasing intervals are regions where the function rises as x increases, decreasing intervals are where the function falls as x increases, and constant intervals are where the function's output remains the same despite changes in x. This analysis is key for understanding the overall behavior and shape of the function graph.

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