Question

If $f^2\left(x\right)\cdot f\left(\frac{1-x}{1+x}\right)=x^3,[x\ne -1,1$ and $f\left(x\right)\ne 0],$ then find $\left|\right[f(-2)\left]\right|$ (where $\left[\right]$ is the greatest integer function).

Answer 1

$f^2\left(x\right).f\left(\frac{1-x}{1+x}\right)=x^3$ ...(i)
Replacing x by $\frac{1-x}{1+x}$, we get $f^2\left(\frac{1-x}{1+x}\right)f\left(x\right)={\left(\frac{1-x}{1+x}\right)}^3$ ....(ii)
By using Eqs.(i) and (ii), we get $f^3\left(x\right)=x^6{\left(\frac{1-x}{1+x}\right)}^3$
$\Rightarrow f\left(x\right)=x^3\left(\frac{1-x}{1+x}\right)$
$f(-2)=\frac{8}{3}$
$\Rightarrow \left[f\right(-2\left)\right]=2\Rightarrow \left[\right[f(-2)\left]\right]=2$