What is the domain of the function tan inverse?

The function ( \tan^{-1}(x) ), also known as the inverse tangent function or arctangent, takes all real numbers as input. This means that the domain of the function extends from negative infinity to positive infinity.

In mathematical terms, the domain is represented as ( (-\infty, \infty) ), indicating that there are no restrictions on the values that ( x ) can take. As one might recall, the tangent function itself is defined for all real numbers except at odd multiples of ( \frac{\pi}{2} ), where it is undefined due to vertical asymptotes. However, the arctangent function is designed to reverse this behavior, allowing it to accept any real number as input and produce a corresponding angle in the range.

This characteristic of the arctangent function is important to understand—it is not limited to any particular interval, unlike other trigonometric functions or their inverses, which may have more restricted domains. Therefore, the answer reflecting the complete set of possible inputs for the function ( \tan^{-1}(x) ) is ( (-\infty, \infty) ).