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]`

Answer 1

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]`