The set of values of parameter 'a' for which the function f: R → R defined by $f(x) = ax + \sin x$ is bijective, is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → $R-[-1,1]$
If f(x) is an injection, then
$f'(x) >0$ or, $f'(x) <0$ for all $x ∈R$
$⇒a + \cos x >0$ or, $a + \cos x < 0$ for all $x ∈R$
$⇒a > 1$ or, $a <-1$
$⇒a ∈R-[-1,1]$.
We observe that $f(x) → ∞$ as $x → ∞$ and $f(x) → -∞$ as $x → -∞$
Therefore, range of f = R. So, f is surjective for all values of a. Hence, f is a bijection if $a ∈ R -[−1, 1]$.