First slide

Functions (XII)

Question

If $f : R \to R$ given by $f (x) = a x + \sin x + a,$ then $f$ is one-one and onto for all

Moderate

A

$a \in R$
B

$a \in R ~ [- 1, 1]$
C

$a \in R ~ {0}$
D

$a \in R ~ {- 1}$

Solution

For $a \neq 0$ , the range of f is $R$ . $f$ is differentiable function so f is one-one and only if f is monotonic.
$f^{'} (x) = a + \cos x$
If $a > 1, f' (x) > 0$ i.e. $f$ is increasing
If $a < - 1, f' (x) < 0$ i.e. $f$ is decreasing
Thus $f$ is monotonic if $a \in R ~ [- 1, 1] .$

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