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Question
Range of f(x) = `1/(1 - 2 cosx)` is ______.
Options
`[1/3, 1]`
`[-1, 1/3]`
`(-oo, -1] ∪ [1/3, oo)`
`[- 1/3, 1]`
Solution
Range of f(x) = `1/(1 - 2 cosx)` is `[-1, 1/3]`.
Explanation:
Given that: `1/(1 - 2 cosx)`
We know that – 1 ≤ cos x ≤ 1
⇒ 1 ≥ cos x ≥ – 1
⇒ – 1 ≤ – cos x ≤ 1
⇒ – 2 ≤ – 2 cos x ≤ 2
⇒ – 2 + 1 ≤ 1 – 2 cos x ≤ 2 + 1
⇒ – 1 ≤ 1 – 2 cos x ≤ 3
⇒ – 1 ≤ `1/(1 - 2 cosx) ≤ 1/3`
⇒ `– 1 ≤ "f"(x) ≤ 1/3`
So the range of f(x) = `[-1, 1/3]`
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