The domain of the function \(f(x) = \displaystyle\frac{1}{\cos^{-1} x}\) is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \([-1, 1)\)
Two conditions are needed: \(\cos^{-1} x\) must be defined, and the denominator must not be 0.
\(\cos^{-1} x\) is defined for
\(-1 \le x \le 1\)
\(\cos^{-1} x = 0\) means \(x = \cos 0\):
\(x = 1\)
So only \(x = 1\) has to be removed. At the other end, \(\cos^{-1}(-1) = \pi \ne 0\), so \(x = -1\) stays.
Hence the domain is \([-1, 1)\).
Why the other options are incorrect
Option (1) \([-1, 1]\) — Incorrect. At \(x = 1\), \(\cos^{-1} 1 = 0\) and the fraction \(\frac{1}{0}\) is not defined.
Option (2) \((-1, 1]\) — Incorrect. It keeps \(x = 1\), where the denominator is 0, and leaves out \(x = -1\), where \(f(-1) = \frac{1}{\pi}\) is defined.
Option (4) \((-1, 1)\) — Incorrect. It leaves out \(x = -1\), where \(f(-1) = \frac{1}{\pi}\) is defined.