11. 0.33/1 points details graph y = tan⁻¹(x).

Explanation:

Step1: Recall domain and range

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

Step2: Analyze key - points

When $x = 0$, $y=\tan^{-1}(0) = 0$. As $x\to\infty$, $y\to\frac{\pi}{2}$ (but never reaches it), and as $x\to-\infty$, $y\to-\frac{\pi}{2}$ (but never reaches it). The function $y = \tan^{-1}(x)$ is an increasing function since the derivative $y'=\frac{1}{1 + x^{2}}>0$ for all real $x$.

Step3: Sketch the graph

Plot the point $(0,0)$ and draw a smooth increasing curve that approaches $y = \frac{\pi}{2}$ as $x\to\infty$ and $y=-\frac{\pi}{2}$ as $x\to-\infty$.

Answer:

Sketch a smooth increasing curve with domain $(-\infty,\infty)$, range $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ passing through the point $(0,0)$ and approaching $y=\pm\frac{\pi}{2}$ as $x\to\pm\infty$ respectively.