What is the range of the tangent function?

The tangent function, represented as tan(x), has a range that includes all real numbers. This is due to the periodic nature of the tangent function, which means that for any value of (y), there exists an (x) such that (tan(x) = y).

As (x) approaches (\frac{\pi}{2} + k\pi) (where (k) is any integer), the value of the tangent function approaches infinity, and as (x) approaches these points from either side, the function outputs negative or positive values that can be as low as negative infinity or as high as positive infinity.

Therefore, the tangent function can produce any real number, demonstrating that its range is indeed all real numbers. This is fundamentally different from other trigonometric functions like sine and cosine, which are restricted to values between -1 and 1, thereby distinguishing the behavior of the tangent function in terms of its range.