Question

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

Answer 1

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

$\Rightarrow f\left(y_x\right)=\frac{1}{x}+\frac{1}{\left(1/x\right)}\Rightarrow x+\frac{1}{x}=f\left(1/x\right)$ ...(2)

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

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

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

$\left(f\right(x){)}^3=f(x^3)+3f(\frac{1}{x})$ from (1) and (2)