Question

If $f\left(x\right)(x+\frac{1}{x})=x^2+\frac{1}{x^2}$, find $f\left(x^3\right)$

Answer 1

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

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

We need to find $f\left(x^3\right)$ :

To find that we need to replace $x$ by $x^3$

Replacing and rewriting the equation ;

$\Rightarrow f\left(x\right)=\frac{x^6+\frac{1}{x^6}}{x^3+\frac{1}{x^3}}$

$\Rightarrow f\left(x\right)=\frac{x^{12}+1}{x^3(x^6+1)}$