11. make a complete graph of f(x) = tan^(-1)(x^2 / √3).

Explanation:

Step1: Analyze domain

The domain of (y = f(x)=\tan^{- 1}\left(\frac{x^{2}}{\sqrt{3}}\right)) is all real - numbers since (x^{2}) is defined for all (x\in R) and (\frac{x^{2}}{\sqrt{3}}) is well - defined for all (x\in R). So, the domain is ((-\infty,\infty)).

Step2: Analyze range

We know that the range of the inverse - tangent function (y = \tan^{-1}(u)) is (\left(-\frac{\pi}{2},\frac{\pi}{2}\right)). Since (u=\frac{x^{2}}{\sqrt{3}}\geq0) for all (x\in R), the range of (y = \tan^{-1}\left(\frac{x^{2}}{\sqrt{3}}\right)) is (\left[0,\frac{\pi}{2}\right)).

Step3: Analyze symmetry

Replace (x) with (-x): (f(-x)=\tan^{-1}\left(\frac{(-x)^{2}}{\sqrt{3}}\right)=\tan^{-1}\left(\frac{x^{2}}{\sqrt{3}}\right)=f(x)). So, the function is even and symmetric about the (y) - axis.

Step4: Analyze critical points

Find the derivative using the chain rule. If (y = \tan^{-1}(u)) and (u=\frac{x^{2}}{\sqrt{3}}), then (\frac{dy}{du}=\frac{1}{1 + u^{2}}) and (\frac{du}{dx}=\frac{2x}{\sqrt{3}}). So, (y^\prime=f^\prime(x)=\frac{1}{1+\left(\frac{x^{2}}{\sqrt{3}}\right)^{2}}\cdot\frac{2x}{\sqrt{3}}=\frac{2x}{\sqrt{3}(1 + \frac{x^{4}}{3})}=\frac{2x}{\sqrt{3}+\frac{x^{4}}{\sqrt{3}}}). Set (y^\prime = 0), then (2x = 0), so (x = 0). When (x = 0), (y=\tan^{-1}(0)=0).

Step5: Analyze end - behavior

As (x\to\pm\infty), (\lim_{x\to\pm\infty}\tan^{-1}\left(\frac{x^{2}}{\sqrt{3}}\right)=\frac{\pi}{2}) because as (x\to\pm\infty), (\frac{x^{2}}{\sqrt{3}}\to\infty) and (\lim_{u\to\infty}\tan^{-1}(u)=\frac{\pi}{2}).

To graph the function:

• Plot the point ((0,0)) (the minimum point).
• Since the function is symmetric about the (y) - axis, we can find some additional points for (x>0) (e.g., when (x = 1), (y=\tan^{-1}\left(\frac{1}{\sqrt{3}}\right)=\frac{\pi}{6}); when (x=\sqrt{3}), (y=\tan^{-1}(1)=\frac{\pi}{4})).
• Sketch a smooth curve that approaches (y = \frac{\pi}{2}) as (x\to\pm\infty) and is symmetric about the (y) - axis.

We cannot provide a visual graph in this text - based format, but the above steps describe how to construct the graph of (y = \tan^{-1}\left(\frac{x^{2}}{\sqrt{3}}\right)).

Answer:

Follow the steps above to graph (y = \tan^{-1}\left(\frac{x^{2}}{\sqrt{3}}\right)) with domain ((-\infty,\infty)), range (\left[0,\frac{\pi}{2}\right)), symmetric about the (y) - axis, minimum at ((0,0)) and approaching (y=\frac{\pi}{2}) as (x\to\pm\infty).