Graph each of the following functions. Check your results...

Graph each of the following functions. Check your results using a graphing calculator.
$$f(x)=\left\{\begin{array}{ll}
x+1, & \text { for } x \leq-3 \\
-1, & \text { for }-3<x<4 \\
\frac{1}{2} x, & \text { for } x \geq 4
\end{array}\right.$$

Key Concepts

Continuity and Discontinuity

When graphing piecewise functions, it is important to analyze continuity at the boundary points of the intervals. This involves checking whether the function's value approaches the same value from both sides of a boundary, which determines if there is a jump (discontinuity) or if the function is continuous at that point.

Graphing Functions

Graphing functions involves plotting the individual segments according to their respective expressions and domain restrictions. This process allows you to visualize the overall behavior of the function, including identifying intervals of continuity and any abrupt changes or jumps.

Using Graphing Calculators for Verification

Graphing calculators are useful tools for verifying hand-drawn graphs. They provide a visual confirmation of the function's behavior across different intervals and help in identifying any potential discrepancies in the manually plotted graph.

Piecewise Functions

A piecewise function is defined by different expressions over different intervals of the domain. It is important to understand how each segment of the function behaves within its prescribed domain to accurately analyze and graph the function.

Domain Restrictions

Each segment of a piecewise function has its own domain restrictions. Knowing these restrictions is crucial when determining which expression applies for a given input and ensuring that the graph correctly reflects the defined intervals.

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