Show that \(f(x)+f\left(\frac1x\right)=0\)

Solution

Given: $$ f(x)=x^3-\frac1{x^3} $$

Replace \(x\) by \(\frac1x\):

$$ f\left(\frac1x\right) = \left(\frac1x\right)^3-\frac1{\left(\frac1x\right)^3} $$

$$ = \frac1{x^3}-x^3 $$

Now, $$ f(x)+f\left(\frac1x\right) = \left(x^3-\frac1{x^3}\right) + \left(\frac1{x^3}-x^3\right) $$

$$ =0 $$

Hence, $$ \boxed{f(x)+f\left(\frac1x\right)=0} $$