make a complete graph of the following function on the given interval. use a graphing utility to check your…

Explanation:

Step1: Analyze function behavior

The function (y = f(x)=2x - 2\tan x). The derivative (y'=2-2\sec^{2}x=2(1 - \sec^{2}x)=- 2\tan^{2}x\leqslant0). The function is non - increasing on the interval (\left(-\frac{3\pi}{2},\frac{3\pi}{2}\right)). Also, (\tan x) has vertical asymptotes at (x =-\frac{\pi}{2},\frac{\pi}{2}). When (x = 0), (y=2\times0 - 2\tan(0)=0).

Step2: Evaluate at endpoints

As (x\to-\frac{3\pi}{2}^{+}), (\tan x\to-\infty), so (y = 2x-2\tan x\to+\infty). As (x\to\frac{3\pi}{2}^{-}), (\tan x\to+\infty), so (y = 2x - 2\tan x\to-\infty).

Step3: Match with graphs

Based on the non - increasing nature, passing through the origin ((0,0)) and behavior near asymptotes, we can identify the correct graph.

Answer:

(Without seeing the actual content of the graphs A, B, C, D, we can't give a specific option. But the above steps can be used to analyze and choose the correct one among them. If you can describe the graphs in more detail, we can further determine the answer.)