What is the name of the following function (if there is one)?
$f(x) = \begin{cases} x & \text{ if } -1 \leq x \leq 1\\ \frac 1x & \text{ if } x < -1 \text{ or } x >1\\ \end{cases} $
If this function has a name, how is it usually denoted?
What is the name of the following function (if there is one)?
$f(x) = \begin{cases} x & \text{ if } -1 \leq x \leq 1\\ \frac 1x & \text{ if } x < -1 \text{ or } x >1\\ \end{cases} $
If this function has a name, how is it usually denoted?
If there doesn't exist a standard name, I'd call it the "left one-right one-identity, alter-inverse" function on the reals. The "left one-right one identity" part means that from the "left one" $(-1)$ to "the right one" the function matches that of the identity function on the reals. The "alter-inverse" means "otherwise we have the inverse" (in Latin alter means "other"), since $1/x$ comes as the inverse function from the reals without 0 to the reals without 0. The function maps to $[-1, 1]$ which under real-number multiplication forms a commutative monoid [$([-1, 1], \ast)$ is the commutative monoid] with annihilator of $0$, just like the reals do. Have you found an inverse for your function?