Question

Let function $f: R \rightarrow R$ be defined by $f(x)=2 x+\sin x$ for $x \in R$, then $f$ is

• A. one-one and onto
• B. one-one but NOT onto
• C. onto but NOT one-one
• D. neither one-one nor onto

Answer 1

Answer: (A)

Given that
$f(x)=2 x+\sin x, x \in R$
$ \Rightarrow f'(x)=2+\cos x$
But $-1 \leq \cos x \leq 1 $
$\Rightarrow 1 \leq 2+\cos x \leq 3$
$\Rightarrow 1 \leq 2+\cos x \leq 3 $
$\therefore f'(x) > 0, \forall x \in R$
$\Rightarrow f(x)$ is strictly increasing and hence one-one
Also as $x \rightarrow \infty, f(x) \rightarrow \infty$ and
$x \rightarrow-\infty, f(x) \rightarrow-\infty$
$\therefore $ Range of $f(x)=R=$ domain of $f(x) $
$\Rightarrow f(x)$ is onto.
Thus, $f(x)$ is one-one and onto.