TITLE: How to calculate $f(x)$ in this question?
QUESTION [2 upvotes]: I have a quation regarding composite functions.  
if $f\left( \dfrac{x^2+1} x \right)=\dfrac{x^4+1}{x^2}$, then what is $f(x)$?

REPLY [5 votes]: Hint: Solve $u=(x^2+1)/x$ for $x$ and then substitute $x=x(u)$ into $f((x^2+1)/x)=f(u)=\frac{x(u)^4+1}{x(u)^2}$.
It is important to note that $x\neq 0$ must hold. A useful hint is also that $x(u)^2=ux-1$ and similarly $x(u)^4=(ux-1)^2=u^2x^2-2ux+1=u^2(ux-1)-2ux+1$.

REPLY [1 votes]: HINT:
$$\left(\frac{x^2+1} x\right)^2=\frac{x^4+2x^2+1}{x^2}=\frac{x^4+1+2x^2}{x^2}$$
Alternatively you have $$\frac{x^2+1} x = x + \frac 1 x \\\frac{x^4+1}{x^2} = x^2 + \frac 1 {x^2}$$
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