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. $$ f(x)=\left\{\begin{array}{ll} x+1 & \text { for } x \geq 3 \\ x+2 & \text { for } x<3 \end{array}\right. $$

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.
$$
f(x)=\left\{\begin{array}{ll}
x+1 & \text { for } x \geq 3 \\
x+2 & \text { for } x<3
\end{array}\right.
$$

Key Concepts

Monotonicity (Increasing, Decreasing, Constant Intervals)

The concept of monotonicity deals with identifying intervals where a function consistently increases, decreases, or remains constant. This analysis is crucial for understanding the general trends of the function and for applications in optimization and calculus, as each segment of a piecewise function might exhibit different monotonic behavior.

Graphing Functions

Graphing functions involves plotting the ordered pairs on the coordinate plane to visualize the function’s behavior. For piecewise functions, graphing requires plotting each segment over its specific interval and ensuring accuracy regarding continuity or breaks where the definitions change.

Ordered Pairs and Tables of Values

Ordered pairs are a way to represent the relationship between inputs and their corresponding outputs in a function. Creating tables of ordered pairs helps in organizing and visualizing the function's values, which is especially useful when dealing with piecewise functions, as it clarifies how each part of the function behaves on its respective domain.

Domain and Range

The domain of a function is the set of all possible input values for which the function is defined, while the range is the set of all possible outputs the function can produce. In the context of piecewise functions, each segment may contribute differently to the overall domain and range, necessitating separate considerations before combining the results.

Piecewise Functions

Piecewise functions are functions that are defined by different expressions over various intervals of the independent variable. This means the behavior and formula used change based on the input value, requiring careful consideration of each segment’s definition when analyzing the function overall.

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