Definition:Inverse Tangent/Real/Arctangent

Definition
From Shape of Tangent Function, we have that $\tan x$ is continuous and strictly increasing on the interval $\left({-\dfrac \pi 2 \,.\,.\, \dfrac \pi 2}\right)$.

From the same source, we also have that:
 * $\tan x \to + \infty$ as $x \to \dfrac \pi 2 ^-$
 * $\tan x \to - \infty$ as $x \to -\dfrac \pi 2 ^+$

Let $g: \left({-\dfrac \pi 2 \,.\,.\, \dfrac \pi 2}\right) \to \R$ be the restriction of $\tan x$ to $\left({-\dfrac \pi 2 \,.\,.\, \dfrac \pi 2}\right)$.

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

This function is called arctangent of $x$ and is written $\arctan x$.

Thus:
 * The domain of $\arctan x$ is $\R$
 * The image of $\arctan x$ is $\left({-\dfrac \pi 2 \,.\,.\, \dfrac \pi 2}\right)$.

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

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

The $\arctan$ 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 $\tan^{-1} x$, which (as pointed out) can be confusing and misleading.

Sometimes, $\operatorname{Tan}^{-1}$ (with a capital $\text{T}$) is taken to mean the same as $\arctan$.