If \( \mathrm{f}: \mathbb{R} \rightarrow \mathbb{R} \& \mathrm{f}(\...

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If \( \mathrm{f}: \mathbb{R} \rightarrow \mathbb{R} \& \mathrm{f}(\mathrm{x})=\frac{\sin ([\mathrm{x}] \pi)}{\mathrm{x}^{2}+2 \mathrm{x}+3}+2 \mathrm{x}-1+\sqrt{\mathrm{x}(\mathrm{x}-1)+\frac{1}{4}} \) (where \( [\mathrm{x}] \) denotes integral part of \( \left.\mathrm{x}\right) \), then \( \mathrm{f}(\mathrm{x}) \) is -
\( \mathrm{P} \)
(A) one-one but not onto
(B) one-one \& onto
W
(C) onto but not one-one
(D) neither one-one nor onto
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