Functions Ques 3
- Let function $f: R \rightarrow R$ be defined by $f(x)=2 x+\sin x$ for $x \in R$. Then, $f$ is
(2002, 1M)
(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
Show Answer
Answer:
Correct Answer: 3.(a)
Solution: (a) Given,
$ f(x)=2 x+\sin x $
$\Rightarrow \quad f^{\prime}(x)=2+\cos x \quad \Rightarrow \quad f^{\prime}(x)>0, \forall x \in R$
which shows $f(x)$ is one-one, as $f(x)$ is strictly increasing.
Since, $f(x)$ is increasing for every $x \in R$,
$\therefore \quad f(x)$ takes all intermediate values between $(-\infty, \infty)$.
Range of $f(x) \in R$.
Hence, $f(x)$ is one-to-one and onto.
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