If f : R → R and is defined by f(x) = 1 / (2 - cos 3x ) for each x ∈ R, then the range of f is

Question

If f : R → R and is defined by f(x) = \(\frac{1}{2-cos 3x}\) for each x ∈ R, then the range of f is

(a) ( \(\frac{1}{3}\), 1) 

(b) [ \(\frac{1}{3}\), 1] 

(c) (1, 2) 

(d) [1, 2]

Answer 1

Answer : (b) [ \(\frac{1}{3}\),1] 

f(x) = \(\frac{1}{2-cos\,3x}\) 

– 1 ≤ cos 3x ≤ 1 

⇒ – 1 ≤ – cos 3x ≤ 1 (Multiplying all terms by – 1) 

⇒ 2 – 1 ≤ 2 – cos 3x ≤ 3 (Adding 2 to each term) 

⇒ 1 ≤ 2 – cos 3x ≤ 3 

⇒ \(1\geq \frac{1}{2-3\,cos\,3x}\geq \frac{1}{3}\)

Range of f is  [ \(\frac{1}{3}\),1]