Let function f : R R be defined by f ( x ) = 2 x + sin x for x R , then f is [2002]

Question

Let function $f : R \to R$ be defined by $f (x) = 2 x + \sin x$ for $x \in R,$ then $f$ is [2002]

Smart Achievers · Accepted Answer

(1)

$Given: f (x) = 2 x + \sin x,$ $x \in R$

$\Rightarrow f' (x) = 2 + \cos x .$

Now, $- 1 \leq \cos x \leq 1$

$\Rightarrow 1 \leq 2 + \cos x \leq 3$

$∴$ $f' (x) > 0,$ $\forall x \in R$

$\Rightarrow f (x) is strictly increasing and therefore one-one.$

$Also,$ $\lim_{x \to \infty} f (x) = \infty$ and $\lim_{x \to - \infty} f (x) = - \infty$

$∴$ $Range of f (x) = R = domain of f (x) \Rightarrow f (x) is onto.$

$Hence,$ $f (x) is one-one and onto.$

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