Definition:Inverse Cotangent/Real/Arccotangent

Definition
From Shape of Cotangent Function, we have that $\cot x$ is continuous and strictly decreasing on the interval $\left({0 \,.\,.\, \pi}\right)$.

From the same source, we also have that:
 * $\cot x \to + \infty$ as $x \to 0^+$
 * $\cot x \to - \infty$ as $x \to \pi^-$

Let $g: \left({0 \,.\,.\, \pi}\right) \to \R$ be the restriction of $\cot x$ to $\left({0 \,.\,.\, \pi}\right)$.

Thus from Inverse of Strictly Monotone Function, $g \left({x}\right)$ admits an inverse function, which will be continuous and strictly decreasing on $\R$.

This function is called arccotangent of $x$ and is written $\operatorname{arccot} x$.

Thus:
 * The domain of $\operatorname{arccot} x$ is $\R$
 * The image of $\operatorname{arccot} x$ is $\left({0 \,.\,.\, \pi}\right)$.

Caution
There exists the a popular but misleading notation $\cot^{-1} x$, which is supposed to denote the inverse cotangent function.

However, note that as $\cot x$ is not an injection, it does not have an inverse.

The $\operatorname{arccot}$ function as defined here has a well-specified image which (to a certain extent) is arbitrarily chosen for convenience.

Therefore it is preferred to the notation $\cot^{-1} x$, which (as pointed out) can be confusing and misleading.

Sometimes, $\operatorname{Cot}^{-1}$ (with a capital $\text{C}$) is taken to mean the same as $\operatorname{arccot}$, although this can also be confusing due to the visual similarity between that and the lowercase $\text{c}$.