Let A =
Question:

Let $\mathrm{A}=\{x \in \mathbf{R}: x$ is not a positive integer $\}$. Define a function f: $\mathbf{A} \rightarrow \mathbf{R}$ as $f(x)=\frac{2 x}{x-1}$, then $f$ is:

  1. (1) not injective

  2. (2) neither injective nor surjective

  3. (3) surjective but not injective

  4. (4) injective but not surjective


Correct Option: , 4

Solution:

As $A=\{x \in R: x$ is not a positive integer $\}$

A function $f: A \rightarrow R$ given by $f(x)=\frac{2 x}{x-1}$

$f\left(x_{1}\right)=f\left(x_{2}\right) \Leftrightarrow x_{1}=x_{2}$

So, $f$ is one-one.

As $f(x) \neq 2$ for any $x \in A \Rightarrow f$ is not onto.

Hence $f$ is injective but not surjective.

 

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