Question

$f:(-\infty ,\infty )\to (0,1]$ defined by $f\left(x\right)=\frac{1}{x^2+1}$ is

• A. one-one but not onto
• B. onto but not one-one
• C. bijective
• D. neither one-one nor onto

Answer 1

The correct option is B onto but not one-one

$f\left(x\right)=\frac{1}{x^2+1}$
$limx\to \pm f\left(x\right)=0$
$limx\to 0,f\left(x\right)=1$
So range of $f\left(x\right)$ is $[0,1]$ & that is codomain of $f\left(x\right).$ So $f\left(x\right)$ is onto.
but $f\left(x\right)$ is symmetric about $y-$axis it is not a monotonic function.
So it is not one - one.