Question
Let function $$f:R \to R$$ be defined by $$f\left( x \right) = 2x + \sin x$$ for $$x \in R,$$ then $$f$$ is
A.
one-to-one and onto
B.
one-to-one but NOT onto
C.
onto but NOT one-to-one
D.
neither one-to-one nor onto
Answer :
one-to-one and onto
Solution :
$$\eqalign{
& {\text{Given}}\,{\text{that}}\,f\left( x \right) = 2x + \sin x,\,\,x \in R \Rightarrow f'\left( x \right) = 2 + \cos x \cr
& {\text{But}}\, - 1 \leqslant \cos x \leqslant 1 \Rightarrow \,1 \leqslant 2 + \cos x \leqslant 3 \Rightarrow 1 \leqslant 2 + \cos x \leqslant 3 \cr
& \therefore f'\left( x \right) > 0,\forall x \in R \cr} $$
$$ \Rightarrow f\left( x \right)$$ is strictly increasing and hence one-one
Also as $$x \to \infty ,f\left( x \right) \to \infty \,{\text{and}}\,x \to - \infty ,f\left( x \right) \to - \infty $$
$$\therefore $$ Range of $$f\left( x \right) = R = $$ domain of $$f\left( x \right) \Rightarrow f\left( x \right)$$ is onto.
Thus, $$f\left( x \right)$$ is one-one and onto.