Question

Let $A=\{x\in R:x\text{is not a positive integer}\}$.
If a function $f:A\to R$ defined 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 $f\left(x\right)=\frac{2x}{x-1}$
$f\left(x\right)=2+\frac{2}{x-1}$
Let $x_1,x_2\in A$ such that $f\left(x_1\right)=f\left(x_2\right)$
$\Rightarrow \frac{1}{x_1-1}=\frac{1}{x_2-1}\Rightarrow x_1=x_2$
Hence, $f$ is an injective function.

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