Question

What is the domain of ${\tan }^{-1}\left(x\right)$?

Answer 1

To find the domain of ${\tan }^{-1}\left(x\right)$:

The tangent function $\tan x$ is a many to one periodic function, so to define an inverse function requires that we restrict its domain (or restrict the range of the inverse function).

To define ${\tan }^{-1}\left(x\right)$ as a function we can restrict the domain of $\tan x$ to $\left(-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right)$.

The function $\tan x$ is one to one, continuous and unbounded over this interval, so has a well defined ${\tan }^{-1}\left(x\right):\mathbf{\text{R}}\to \left(-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right)$ that is also continuous and one to one.

Hence, the domain of the function ${\tan }^{-1}\left(x\right)$ is all real number.