Let the function f: R→R be defined by f(x)=2x+sinx for x∈R, then f is

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$L e t t h e f u n c t i o n f : R \to R b e d e f i n e d b y f (x) = 2 x + \sin x f o r x \in R, t h e n f i s$

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Onto but not one – one

one – one but not onto

Neither one – one nor onto

one – one and onto

answer is A.

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Detailed Solution

$\begin{array}{l} f (x) = 2 x + \sin x \\ f^{1} (x) = 2 + \cos x > 0 [∵ - 1 \leq \cos x \leq 1] \\ ∴ f i s I n c r e \sin g f u n c t i o n \\ ∴ f i s o n e - o n e \\ ∵ - 1 \leq \sin x \leq 1 \\ \forall x \in R - \infty < 2 x + \sin x < \infty \\ ∴ R a n g e = (- \infty, \infty) = C o d o m a i n \\ ∴ f i s o n t o \end{array}$