Let $A=\{x \in R / x$ is not a positive integer). Let a function $f$ be defined as $f: A \rightarrow R$ so…
Let $A=\{x \in R / x$ is not a positive integer). Let a function $f$ be defined as $f: A \rightarrow R$ so that $f(x)=\frac{2 x}{x-1}$, then $f$ is
Not injective.
surjective but not injective.
neither injective nor surjective.
injective but not surjective.
Solution
$\begin{aligned} & f(x)=\frac{2 x}{x-1} \\ & \Rightarrow f^{\prime}(x)=\frac{(x-1) 2-2 x(1-0)}{(x-1)^2}=\frac{-2}{(x-1)^2}<0\end{aligned}$
$f(x)$ is strictly decreasing so $f(x)$ is injective
But $f(x)=4$
$\Rightarrow \frac{2 x}{x-1}=4$
$\Rightarrow x=2$ (which is a positive integer)
i.e., $4 \in R$ (co-domain of $f$ ) has no pre-image in $A$ (domain of $f$ )
So, ' $f$ ' is not surjective