Find the Range of the Function
The range of
\[ f(x)=\frac{1}{1-2\cos x} \]
is
(a) \(\left[\frac13,1\right]\)
(b) \(\left[-1,\frac13\right]\)
(c) \((-\infty,-1]\cup\left[\frac13,\infty\right)\)
(d) \(\left[-\frac13,1\right]\)
Since
\[ -1\le\cos x\le1 \]
therefore
\[ -2\le2\cos x\le2 \]
\[ -1\le1-2\cos x\le3 \]
Also,
\[ 1-2\cos x\ne0 \]
Let
\[ t=1-2\cos x \]
Then
\[ t\in[-1,0)\cup(0,3] \]
Hence,
\[ \frac1t\in(-\infty,-1]\cup\left[\frac13,\infty\right) \]
Therefore, range is
\[ \boxed{(-\infty,-1]\cup\left[\frac13,\infty\right)} \]
\[ \boxed{\text{Correct Answer: (c)}} \]