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
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 $a ∈R -[-1,1]$.