I don't understand why I'm not able to manipulate this quite basic function into another form

the function is:

$\displaystyle \\x=\sqrt[3]{9}\\x^{3}=9\\f(x)=x^{3}-9$

But then this function is exactly the same:

$\displaystyle \\x=\sqrt[3]{9}\\x-\sqrt[3]{9}=0\\\text{If i try to produce a 9 and a}\;x^3\;\text{by cubing both sides}\\(x^3-\sqrt[3]{9})^3=x^3-3x^2\times\sqrt[3]{9}+3x\times(\sqrt[3]{9})^2-9\\\neq x^{3}-9$

Am I misunderstanding something basic here?