First slide

Functions (XII)

Question

$Let the function f : R \to R be defined by f (x) = 2 x + \sin x for x \in R . Then f is$

Moderate

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

Solution

$Given that f (x) = 2 x + \sin x, x \in R$
$\begin{aligned} or f^{'} (x) = 2 + \cos x \\ But - 1 \leq \cos x \leq 1 \\ or 1 \leq 2 + \cos x \leq 3 \\ ∴ f^{'} (x) > 0 \forall x \in R \end{aligned}$
$Therefore, f (x) is strictly increasing and, hence, one-one. Also, as x \to \infty, f (x) \to \infty, and x \to - \infty, f (x) \to - \infty . Therefore,$
$Range of f (x) = R = co - domain of f (x)$
$Hence, f (x) is onto.$
$Thus, f (x) is one-one and onto.$

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