I'm hoping that this isn't an obvious question, but I was curious about possible notations for what interval a piecewise function is defined on. For example, consider the following function (where $x \in \mathbb{R}$):
$$\begin{cases}
x & 0 \leq x \leq 5 \\
x^2 & 5 < x < 10 \\
x^3 & 10 < x \le 20 \\
0 & x < 0 \lor x > 20
\end{cases}$$
The output is on the intervals $[0, 5]$ (when $0 \leq x \leq 5$), $(25, 100)$ (when $5 < x < 10$), and $(1000, 8000]$ when $10 < x \le 20$. Is there a "better" way of notating just the interval of the output? For example, would writing something like $[[0, 5], (25, 100), (1000, 8000]]$ have any particular meaning? For the record, yes, I do realize that this proposed notation actually has somewhat less information than the original sentence because I'm not specifying when each interval will actually "occur," but suppose that I don't care about specifying that for what I'm communicating.
The original context for this was a Stack Overflow post asking how to extend a set of numbers in the range of [0, 1] to a set in the range [-1, 1] (where $[0, 0.5) \to [-1, 0]$ and $[0.5, 1] \to [0, 1]$. One proposed answer was
$$\begin{cases}
x – 1 & 0 \leq x < 0.5 \\
x & 0.5 \le x \le 1
\end{cases}$$
Clearly, this is
actually on the range of $(-0.5, 0]$ (if ($0 \le x < 0.5$) or $[0.5, 1]$ (if $0.5 \le x \le 1$).
What is the most concise notation for stating what the actual interval of this proposed function is?
Best Answer
In your example $[[0, 5], (25, 100), (1000, 8000]]$, one way of writing this is $$[0,5]\cup(25,100)\cup(1000,8000].$$
$\cup$ means union (of sets).