Q86 / 180 · KCET 2021 · Mathematics

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

  • Correct answer
    A
    injective but not surjective.
  • B
    surjective but not injective.
  • C
    bijective.
  • D
    neither injective nor surjective.

Solution

Given function, $f(x)=\frac{2 x}{x-1}$ On differentiating w.r.t. $x$, we get $\begin{aligned} f^{\prime}(x) &=\frac{(x-1)(2)-2 x(1-0)}{(x-1)^{2}} \\ &=\frac{2 x-2-2 x}{(x-1)^{2}}=\frac{-2}{(x-1)^{2}} < 0 \end{aligned}$ Function is strictly decreasing. Function is injective $\frac{2 x}{x-1}=y \Rightarrow 2 x=y x-y$ $\Rightarrow \quad y=x(y-2)$ Let $\quad x=\frac{y}{y-2} \notin \pi$ for $y=2$ $f$ is not surjective.

Hence, the correct answer is option: 'A'.

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