Functions Question 531
Question: Let function $ f:R\to R $ be defined by $ f(x)=2x+sinx $ for $ x\in R, $ then f is
Options:
A) One-one and onto
B) one-one but NOT onto
C) onto but NOT one-one
D) Neither one-one nor onto
Show Answer
Answer:
Correct Answer: A
Solution:
[a] Given that $ f(x)=2x+sinx,x\in R $
$ \Rightarrow f’(x)=2+cosx $ But $ -1\le \cos x\le 1 $
$ \Rightarrow 1\le 2+\cos x\le 3 $
$ \Rightarrow 1\le 2+\cos x\le 3 $
$ \therefore f’(x)>0,\forall x\in R $
$ \Rightarrow 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= $ domain of $ f(x)\Rightarrow f(x) $ is onto. Thus, f(x) is one-one and onto.
BETA
