Question

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

• A. not injective
• B. injective but not surjective
• C. surjective but not injective
• D. neither injective nor subjective

Answer 1

The correct option is B injective but not surjective

Given $f\left(x\right)=\frac{2x}{x-1}$
$f^{\prime }\left(x\right)=\frac{-1}{(x-1{)}^2}<0\forall x\in A$
$\Rightarrow f$ is decreasing in its domain.
Hence, $f$ is injective.

Let $y=f\left(x\right)$
$\Rightarrow y=\frac{2x}{x-1}$
$\Rightarrow x=\frac{y}{y-2}$
If $y=3,$ then $x=\frac{3}{3-2}=1>0$
But it is given, $x$ is not a positive integer.
So, function is not surjective.