Let f: R → R be a function such that f(x) = x ^ 2 + 2x + 1 x ^ 2 +...

Given,

$f\left(x\right) = \dfrac{{x}^{2} + 2x + 1}{{x}^{2} + 1}$

${f}^{'} ( x ) = \dfrac{\left({x}^{2} + 1\right) ( 2x + 2 ) - ( {x}^{2} + 2x + 1 ) 2x}{{\left({x}^{2} + 1\right)}^{2}}$

$⇒ {f}^{'} ( x ) = \dfrac{2{x}^{3} + 2{x}^{2} + 2x + 2 - 2{x}^{3} - 4{x}^{2} - 2x}{{\left({x}^{2} + 1\right)}^{2}}$

$⇒ {f}^{'} ( x ) = \dfrac{2 ( 1 - {x}^{2} )}{{\left({x}^{2} + 1\right)}^{2}} = - \dfrac{2 ( x + 1 ) ( x - 1 )}{{\left({x}^{2} + 1\right)}^{2}}$

Clearly $f ( x )$ is one-one in \left(- \infty , - 1\right)$and also in$( 1, \infty )$but$f ( x )$is not one-one in \left(- \infty , \infty\right)$