Piecewise Models and Restrictions

Some relationships cannot be captured by one formula. A parking garage might charge a flat rate for the first hour, then a different rate per hour after that. A piecewise function handles this by defining separate rules, each one valid only over its own interval. You write it as a list: this formula when x is in this range, that formula when x is in that range. The hardest part is the seam, the x-value where one piece ends and the next begins. If the two pieces give the same output at that shared point, the graph connects smoothly with no jump. If they do not match, the graph breaks there, which can be exactly what a real situation needs, like a price that jumps up at the start of a new hour.

Checking continuity at the seam means plugging the break point into both pieces and comparing outputs. You also need to track which piece owns the boundary point itself, shown by whether the inequality is strict or not, since a function can only output one value at each input.

This topic connects to the rest of the unit because it shows a function's formula is not the whole story. A polynomial like a volume formula is defined for every real number algebraically, but when it models a real object, only some inputs make physical sense. Cutting squares from a sheet of metal to fold a box only works while the cut is small enough to leave material behind and large enough to be a real cut. That squeeze turns an all-real-numbers domain into a tight interval, which matters when you graph or optimize the model.