We discuss the domains of piecewise functions - which must be explicitly given.
The domains of piecewise functions are given explicitly, which can actually be a little misleading at times. In particular, they don’t really have a notion of “natural domain” because the domain must always be given.
Nonetheless there are still some key observations to make about the domain of piecewise functions. Consider the following piecewise function example:
Notice how there are certain values in the domain where the piecewise functions transitions from one function to another. In this case, at the values and the function types change. Importantly however, the inequalities on either side of these values don’t overlap. For example, on one side of the value we have that and on the other side we have , so the actual value is only used in one of the inequalities, not both.
It is also possible that the value itself isn’t used in either inequality, like in the case where . Notice here that we have and , but the value itself is not included in either place.
Thus, our domain for the function spans across the entire set of possible values - in particular all such that except for , which means the domain is .
These points, where the piecewise function transitions between functions, are often called pivot points or transition points. Regardless, these points are crucial to observe closely and determine which, if any, function is evaluated at the point. This is the most common way to artificially create discontinuities as well as atypical function behavior, and as such they are extensively useful in calculus - not to mention modeling various problems.
