Question

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

Answer 1

$LHS=\left(f\right(x){)}^3=(x+\left(\frac{1}{x}\right){)}^3$

$=x^3+\left(\frac{1}{x^3}\right)+3\cdot x\cdot \left(\frac{1}{x}\right)(x+(\frac{1}{x}\left)\right)$

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

$=f\left(x^3\right)+3f\left(\frac{1}{x}\right)=RHS$

Hence Proved!