The set of values of parameter 'a' for which the function $f: R→ R$ defined by $f (x) = ax + \sin x$ is bijective, is

$R-[-1,1]$

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 $a ∈R -[-1,1]$.