Question

If \( y = \tan^{-1} x \), then __________.

Hint:

Remember that the range of the arctangent function \( \tan^{-1} x \) is limited to \( -\frac{\pi}{2} \leq y \leq \frac{\pi}{2} \) to maintain a unique result.

Answer 1

Step 1: The function \( y = \tan^{-1} x \) is the inverse of the tangent function. By the definition of the inverse tangent (or arctangent), it returns the angle \( y \) such that: \[ \tan y = x. \] Step 2: The range of the arctangent function \( \tan^{-1} x \) is restricted to \( -\frac{\pi}{2} \leq y \leq \frac{\pi}{2} \). This is because the tangent function is periodic, and the principal value of the inverse tangent is chosen to lie within this interval to ensure it is a one-to-one function. Thus, we conclude: \[ -\frac{\pi}{2} \leq y \leq \frac{\pi}{2}. \]