Question

Determine whether the function f(x) = 2x+1/x-2 has inverse, if it exists find it.

Answer 1

$\large\underline{\sf{Solution-}}$

$\rm :\longmapsto\:Let \: f(x) \: = \dfrac{2x + 1}{x - 2} \: and \: x \ne \: 2$

To Check One - One

$\rm \: Let \: us \: consider \: two \: numbers \: x \: and \: y \: \ne \: 2 \: such \: that$

$\rm :\longmapsto\:\rm \: f(x) = f(y)$

$\rm :\longmapsto\:\dfrac{2x + 1}{x - 2} = \dfrac{2y + 1}{y - 2}$

$\rm :\longmapsto\: \cancel{2xy} + y - 4x - \cancel2 = \cancel{2xy} + x - 4y - \cancel2$

$\rm :\longmapsto\:5x = 5y$

$\rm :\longmapsto\:x = y$

$\bf\implies \:f(x) \: is \: one - one.$

To Check Onto :-

$\rm \: Let \: if \: possible \: there \: exist \: an \: element \: y \: \ne \: 2 \: such \: that$

$\rm :\longmapsto\:f(x) = y$

$\rm :\longmapsto\:\dfrac{2x + 1}{x - 2} = y$

$\rm :\longmapsto\:2x + 1 = xy - 2y$

$\rm :\longmapsto\:xy - 2x = 1 + 2y$

$\rm :\longmapsto\:x(y - 2) = 2y + 1$

$\rm :\longmapsto\:x = \dfrac{2y + 1}{y - 2} \in \: R - \{2 \}$

$\bf\implies \:f(x) \: is \: onto.$

$\rm :\longmapsto\:Since \: f(x) \: is \: one - one \: and \: onto$

$\rm :\implies\:f(x) \: is \: bijective$

$\rm :\implies\:f(x) \: is \: invertible$

$\rm :\implies\: {f}^{ - 1} (x) = \dfrac{2x + 1}{x - 2} \: for \: x \ne \: 2$