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The value of

Question:

Let $A=\{x \varepsilon R: x$ is not a positive integer $\}$

Define a function $f: \mathrm{A} \rightarrow \mathrm{R}$ as $f(\mathrm{x})=\frac{2 \mathrm{x}}{\mathrm{x}-1}$ then $f$ is

  1. injective but not surjective

  2. not injective

  3. surjective but not injective

  4. neither iniective nor suriective

Correct Option: 1

Solution:

$f(\mathrm{x})=2\left(1+\frac{1}{\mathrm{x}-1}\right)$

$f^{\prime}(\mathrm{x})=-\frac{2}{(\mathrm{x}-1)^{2}}$

$\Rightarrow f$ is one-one but not onto

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