Question

graph the inverse circular function.
$y = \tan^{-1}(x)$

Explanation:

Step1: Analyze the domain and range

The domain of \(y = \tan^{-1}(x)\) is \((-\infty,\infty)\) and the range is \((-\frac{\pi}{2},\frac{\pi}{2})\). The function \(y=\tan^{-1}(x)\) is an increasing function.

Step2: Check the behavior as \(x\to\pm\infty\)

As \(x\to\infty\), \(y = \tan^{-1}(x)\to\frac{\pi}{2}\) (horizontal asymptote \(y=\frac{\pi}{2}\)). As \(x\to-\infty\), \(y=\tan^{-1}(x)\to-\frac{\pi}{2}\) (horizontal asymptote \(y =-\frac{\pi}{2}\)). Also, when \(x = 0\), \(y=\tan^{-1}(0)=0\).

Answer:

A.