Let $f:R \to R$ be a function such that $f(x) = {{{x^2} + 2x + 1} \over {{x^2} + 1}}$. Then

<p>$f(x) = {{{x^2} + 2x + 1} \over {({x^2} + 1)}}$</p> <p>$$ \Rightarrow f'(x) = {{({x^2} + 1)(2x + 2) - ({x^2} + 2x + 1)(2x)} \over {{{({x^2} + 1)}^2}}}$$</p> <p>$\Rightarrow f'(x) = {{2 - 2{x^2}} \over {{{({x^2} + 1)}^2}}}$</p> <p>$$ = {{2\left( {1 + x} \right)\left( {1 - x} \right)} \over {{{\left( {{x^2} + 1} \right)}^2}}}$$</p> <p>$\therefore$ $f'(x) = 0$ at x = 1 and x = -1. So, x = 1 and x = -1 are point of maxima/minima.</p> <p><b>Option A :</b> $f(x)$ is many-one in $( - \infty , - 1)$. <br/><br/>As x = -1 is a point of maxima/minima. So it is a boundary point in the range $( - \infty , - 1)$. So, $f(x)$ is one-one in $( - \infty , - 1)$. <br/><br/>$\therefore$ Option A is incorrect.</p> <p><b>Option B :</b> $f(x)$ is one-one in $( - \infty ,\infty )$. <br/><br/>As, x = 1 and x = -1 are point of maxima/minima which is present inside the range $( - \infty ,\infty )$. So, $f(x)$ is many-one function in $( - \infty ,\infty )$. <br/><br/>$\therefore$ Option B is incorrect.</p> <p><b>Option C :</b> $f(x)$ is one-one in $[1,\infty )$ but not in $( - \infty ,\infty )$. <br/><br/>As x = 1 is a point of maxima/minima. So it is a boundary point in the range $[1,\infty )$. So, $f(x)$ is one-one in $[1,\infty )$. <br/><br/>As, x = 1 and x = -1 are point of maxima/minima which is present inside the range $( - \infty ,\infty )$. So, $f(x)$ is many-one function in $( - \infty ,\infty )$ . <br/><br/>$\therefore$ Option C is correct.</p> <p><b>Option D :</b> $f(x)$ is many-one in $(1,\infty )$. <br/><br/>As x = 1 is a point of maxima/minima. So it is a boundary point in the range $[1,\infty )$. So, $f(x)$ is one-one in $[1,\infty )$. <br/><br/>$\therefore$ Option D is incorrect.</p> <p><b>Note :</b></p> <p><b>Methods to Check One-One Function :</b> <br/><br/>(i) If a function is one-one, any line parallel to $x$-axis cuts the graph of the function maximum at one point. <br/><br/>(ii) Any continuous function which is entirely increasing or decreasing (no maxima/minima is present) in whole domain will always be one-one function. </p>