$f:[0,1] \to [0,1]$ be continuous bijection , $g \in C[0,1]$ and such that $\int_0^1g(x)(f(x))^{6n}dx=0, \forall n\ge 0$ , then $g=0$?

Question

Let $f:[0,1] \to [0,1]$ be continuous bijection , $g:[0,1] \to \mathbb R$ be continuous such that

$\int_0^1g(x)(f(x))^{6n}dx=0, \forall n\ge 0$ , then is it true that $g(x)=0,\forall x \in [0,1]$ ?

I had done a problem where the condition were $\int_0^1g(x)x^ndx=0$ , but there the proof worked because by Weierstrass approximation theorem , the span of $\{1,x,x^2....\}$ is dense in $C[0,1]$ , for this problem I have no such idea . Please help . Thanks in advance

Answer 1

Replace $f^6$ with $h$, and then consider the subalgebra of $C[0,1]$ generated by $h$. This separates points, thus is dense by Stone-Weierstrass. Now proceed as usual.