Question

If $f\left(x\right)=x-\frac{1}{x}$ then prove that ${\left(f\left(x\right)\right)}^3=f\left(x^3\right)+3f\left(\frac{1}{x}\right)$

Answer 1

$f\left(x\right)=x-\frac{1}{x}f\left(x^3\right)=x^3-\frac{1}{x^3}f\left(\frac{1}{x}\right)=\frac{1}{x}-x=x^3-\frac{1}{x^3}-3(x-\frac{1}{x})={(x-\frac{1}{x})}^3={\left[f\left(x\right)\right]}^3$