Definition:Inverse Cotangent/Real/Arccotangent

Definition
From Nature 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 $$\arccot x$$.

Thus:
 * The domain of $$\arccot x$$ is $$\R$$;
 * The image of $$\arccot x$$ 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 $$\arccot$$ 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 C) is taken to mean the same as $$\arccot$$, although this can also be confusing due to the visual similarity of upper and lower case c.