Question

locate the discontinuities of the function. (let n represent an arbitrary integer.)

f(x) = \frac{1}{\sqrt{1 - \sin(2x)}}

x =

illustrate by graphing.

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:

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