locate the discontinuities of the function. (let n represent an arbitrary integer.)\n\nf(x) =…

locate the discontinuities of the function. (let n represent an arbitrary integer.)\n\nf(x) = \\frac{1}{\\sqrt{1 - \\sin(2x)}}\n\nx = \n\nillustrate by graphing.

locate the discontinuities of the function. (let n represent an arbitrary integer.)\n\nf(x) = \\frac{1}{\\sqrt{1 - \\sin(2x)}}\n\nx = \n\nillustrate by graphing.

Answer

Explanation:

Step1: Analyze the domain of the function

For the function (y = f(x)=\frac{1}{\sqrt{1-\sin(2x)}}), the expression under the square - root must be positive, i.e., (1-\sin(2x)>0). But the function is discontinuous when (1 - \sin(2x)=0).

Step2: Solve the equation (1-\sin(2x) = 0)

We know that (\sin(2x)=1). Using the formula (\sin\theta = 1\Rightarrow\theta=\frac{\pi}{2}+2n\pi), where (n\in\mathbb{Z}).

Substitute (\theta = 2x), we get (2x=\frac{\pi}{2}+2n\pi).

Step3: Solve for (x)

Dividing both sides of the equation (2x=\frac{\pi}{2}+2n\pi) by (2), we have (x=\frac{\pi}{4}+n\pi), (n\in\mathbb{Z}).

Answer:

(x=\frac{\pi}{4}+n\pi)