A) One-one and onto
B) one-one but NOT onto
C) onto but NOT one-one
D) Neither one-one nor onto
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. |
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