Question

Let $f : R \to R$ be defined by $f(x) = \frac{x}{1+ x^2} , x \in R$. Then the range of $f$ is :

• A. (-1, 1) - {0}
• B. $\left[- \frac{1}{2} , \frac{1}{2}\right]$
• C. $- \left[- \frac{1}{2} , \frac{1}{2}\right]$
• D. $R - [-1, 1]$

Answer 1

Answer: (B)

$f(0) = 0$ & $f(x)$ is odd.
Further, if $x > 0$ then
$f\left(x\right) = \frac{1}{x+ \frac{1}{x}} \in \left(0, \frac{1}{2}\right] $
$f\left(x\right)\in \left[-\frac{1}{2} , \frac{1}{2}\right] $