Piece-wise defined functions
A typical introduction of piece-wise defined functions in a calculus textbook looks something like this: there are functions that take on different formulas depending on the intervals (pieces) of their domains, for example: [math]\displaystyle{ f(n) = \begin{cases} x^2, & \mbox{if }x\le 1 \\ 3-2x, & \mbox{if }x \gt 1 \end{cases} }[/math]
(examples may vary).
Below we try to give a bit more motivation to piece-wise functions (which is not just a nuance for students, but rather indispensable tool in mathematics and applications).
Pool table
A pool table has dimensions 10 ft by 5 ft. A ball is hit from a corner towards the longer side at the angle 30 degrees (with respect to the shorter side). Describe the trajectory of the ball before it hits the side second time.
Ball trajectory
Water freezing