Piecewise-Defined Functions

A piecewise function is a function that uses different expressions for different intervals of its domain.

These expressions apply to specific intervals, producing a continuous or segmented graph.

A piecewise function may include a straight line in one interval, a curve, or a horizontal segment in another.

One example of a piecewise function is the absolute value function. It measures a number's distance from zero, turning negatives positive and keeping positives the same.

Another example of a piecewise function is the step function, which stays constant over intervals and jumps suddenly at certain points.

Its graph looks like stairs, with open and closed dots marking excluded and included values, respectively.

A piecewise function can have any number of sections, each following its own rule.

Graphs of piecewise functions often show sharp corners, open dots, or jumps where the rules change.

These functions are useful for applications like shipping rates, which increase in steps as the package weight crosses certain limits and follow different rules for different input ranges.