Question
Question: If f : R → R where f(x) = ax + cosx is an invertible function then...
If f : R → R where f(x) = ax + cosx is an invertible function then
A
ae(-2, 1]∪[1, 2)
B
a ∈[-2, 2]
C
ae[-∞, -1]∪[1, ∞)
D
a ∈ [-1, 1]
Answer
ae−∞,−1∪[1, ∞)
Explanation
Solution
The function f(x) is clearly on to if a ≠ 0. Now f '(x)
= a − sinx. f '(x) > 0 ∀ x ∈ R
⇒ a ≥ sin x ∀ x ∈ R
⇒ a ≥ 1. For f'(x) ≤ 0. ∀ x ∈ R
we get a ≤ sin x ∀ x ∈ R
⇒ a ≤ -1 a ∈ (-∞, -1]∪[1, ∞)