What is the range of the function tan inverse?

The range of the function ( \tan^{-1}(x) ), or the inverse tangent function, is the set of values that the function can output as ( x ) takes on all real numbers. The ( \tan^{-1}(x) ) function maps any real number input to an angle in radians.

This function uniquely assigns a value between ( -\frac{\pi}{2} ) and ( \frac{\pi}{2} ) (exclusive) to every real number input, which corresponds to the angles for which the tangent values exist. Here, ( -\frac{\pi}{2} ) signifies a vertical asymptote where the function approaches but never actually reaches this value, and ( \frac{\pi}{2} ) behaves similarly on the upper end.

The interval ( (-\frac{\pi}{2}, \frac{\pi}{2}) ) captures all possible output values of ( \tan^{-1}(x) ), thus defining the complete range of the function. This makes it clear why this choice accurately represents the range of the inverse tangent function.