the figure shows the graph of f(x)=a tan(bx), where a and b are constants, in the xy - plane. the graph of f…

Explanation:

Step1: Find the period of the tangent - function

The vertical asymptotes of (y = a\tan(bx)) are given by (bx=-\frac{\pi}{2}+n\pi) and (bx=\frac{\pi}{2}+n\pi), (n\in\mathbb{Z}). Given the vertical asymptotes (x =-\frac{3\pi}{2}) and (x=\frac{3\pi}{2}), the period (T) of the function (y = a\tan(bx)) is (T=\frac{3\pi}{2}-(-\frac{3\pi}{2}) = 3\pi). For the tangent function (y = a\tan(bx)), the period (T=\frac{\pi}{|b|}). So, (\frac{\pi}{|b|}=3\pi), which gives (|b|=\frac{1}{3}).

Step2: Recall the property of the tangent - function

The general solution of the equation (\tan(t)=k) (where (k) is a constant) is (t = t_0+\pi n), (n\in\mathbb{Z}), where (t_0) is a particular solution. For the function (y = a\tan(bx)), if (x_0) is a particular solution of (a\tan(bx)=k), then the general solution is (x=x_0 +\frac{\pi}{|b|}n). Since (|b|=\frac{1}{3}), the general solution of (f(x)=a\tan(bx)=0.813) is (x = x_0+3\pi n), where (x_0 = 3.545) is a particular solution.

Answer:

D. (x = 3.545+3\pi k), where (k) is any integer