ANSWER
[tex]\lbrack\frac{\pi}{3},\text{ }\frac{5\pi}{3}\rbrack[/tex]
STEP-BY-STEP EXPLANATION
From the question provided, we were given the below function
[tex]g(x)=\text{ 2 cos 2x - 1 on the interval \lbrack{}0, 2}\pi\rbrack[/tex]
As you can see, we are asked to determine the x-intercepts on the interval given
[tex]\begin{gathered} \text{Note that} \\ g(x)\text{ }=\text{ 0} \\ \text{Then, we have the below equation} \\ 0\text{ = 2 cos2x - 1} \\ \text{Add 1 to the both sides} \\ 0\text{ + 1 = 2 cos 2x -1 + 1} \\ 1\text{ = 2 cos 2x} \\ \text{Divide both sides by 2} \\ \frac{1}{2}\text{ = }\frac{2\text{ cos 2x}}{2} \\ 0.5\text{ = cos 2x} \\ 2x=cos^{-1}(0.5) \\ 2x\text{ = 60} \\ x\text{ = }\frac{\pi}{3} \\ x\text{ = }\frac{5\pi}{3} \end{gathered}[/tex]
