Question

What is the domain of `arctan(x)`?

Answer 1

The domain of `arctan(x)` is the whole of `RR`, that is: `(-oo, oo)`

Explanation:

The 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 `arctan(x)` as a function we can restrict the domain of `tan(x)` to `(-pi/2, pi/2)`. The function `tan(x)` is one to one, continuous and unbounded over this interval, so has a well defined inverse `arctan(x): RR -> (-pi/2, pi/2)` that is also continuous and one to one.