A piecewise-defined function is an amalgam of two or more rules, each operative on its own domain .  In general, these domains are contiguous, but that is not an essential element of the definition.

Some piecewise functions have special names that are used the way the name of the sine or exponential functions might be used.  Two, in particular, are $\left|x\right|$ and $\left[x\right]$, the first being the absolute-value function, and the second, the greatest-integer function.  These are met in the next two sections below.  The general manipulation of piecewise-defined functions is thus the object of this chapter.

	Chapter Glossary
	$$ The following terms in Chapter 10 are linked to the Maple Math Dictionary. $$ absolute value  additive inverse  denominator  domain  exponential  extension  greatest integer function  inequality  interval  negative  nonnegative  numerator  positive  real line  real number  sine  singularity  subset  $$

As was seen in Chapter 1, the absolute value function, written as

$f\left(x\right)\=\left|x\right|$ 

is always a nonnegative quantity.  For example, the absolute value of $-3$ is the positive number 3.

Viewing the change from $-3$ to 3 as "dropping the minus sign" obscures the abstract definition of this function.

It's much better to see the transition of $-3$ to 3 as attained by multiplying the negative number $-3$ by $-1$ to form +3, the additive inverse of $-3$.  Hence, the action of the absolute value function should be seen as