Question

\(f(x) = \frac{1}{7 - \cos x},\ x \in \mathbb{R}\). Then the range of \(f\) is

• A. \((-8, -7)\)
• B. \([-7, -4]\)
• C. \(\left(1, \frac{5}{4}\right)\)
• D. \(\left(\frac{5}{7}, 1\right)\)
• E. \(\left[\frac{1}{8}, \frac{1}{6}\right]\)

Hint:

For \(\frac{1}{a - \cos x}\) with \(a$>$1\), range is \(\left[\frac{1}{a+1}, \frac{1}{a-1}\right]\).

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
Range of \(\cos x\) is \([-1, 1]\). Then \(7 - \cos x \in [6, 8]\).
Step 2: Detailed Explanation:
Since \(f(x) = \frac{1}{7 - \cos x}\), minimum of denominator gives maximum of \(f\) and vice versa.
Denominator max \(8 \implies f_{min} = \frac{1}{8}\). Denominator min \(6 \implies f_{max} = \frac{1}{6}\). So range is \(\left[\frac{1}{8}, \frac{1}{6}\right]\).
Step 3: Final Answer:
Option (E) is correct.