Question

Find the range of $f\left(x\right)=\frac{x}{1+x^2}$, Range is [-a,a], find a ?

Answer 1

$f\left(x\right)=\frac{x}{1+x^2}=\frac{1}{x+\frac{1}{x}}$
We know for all $x\in R,$ $|x+\frac{1}{x}|\ge 2$
Therefore $|f\left(x\right)|\le \frac{1}{2}\Rightarrow f\left(x\right)\in [-.5,.5]$