Let A = x ∈ R: x is not a positive integer . Define a function f: A...

Given, function $f\left(x\right) = \dfrac{2x}{x - 1}$ and domain of $f\left(x\right)$ is \left(- \infty, 0\right]$<br /><br />Now${f}^{'}\left(x\right) = \dfrac{\left(x - 1\right)2 - 2x\left(1 - 0\right)}{{\left(x - 1\right)}^{2}}$</p><p>${f}^{′}\left(x\right) = \dfrac{- 2}{{\left(x - 1\right)}^{2}} < 0$<br /><br />$⇒ f$is always decreasing in the given domain.<br /><br /><img src="https://cdn-question-pool.Quantrex.app/pyq/jee_main/vlfmdnlgxdrppcsmpzgjykjjnwhbtvnjrxmprwyhzwoqluotzp.png" /><br /><br />Clearly, given function is one-one or injective.<br />Since, codomain$\left(- \infty, \infty\right) \ne$Range \left(0, 2\right)$ , hence $f$ is into.