The range of ‘a’ for which f(x) = ax + cos x is one-one, is :

(–∞, –1] ∪ [ 1, ∞)

If a function is continuously increasing or decreasing the f'(x) ≥ 0 or f'(x) ≤ 0 and equal to zero only at one point.

⇒ f'(x) = a – sin x ≥ 0 or a – sin x ≤ 0

⇒ a ≥ sin x or a ≤ sin x

⇒ a ≥ 1 or a ≤ –1

(–∞, –1] ∪ [ 1, ∞)

Hence (2) is the correct answer.

Common mistake: If f(x) in continuously increasing or decreasing the f(x) will be a one-one function f'(x) = a – sin x > 0 or a – sin x < 0 ⇒ either a > sin x or a < sin x

⇒ either a > 1 or a < –1 ⇒ (–∞, –1) ∪ (1, ∞)