The problem asks us to find the function $g(x)$ such that $h(x) = (x+9)^6 = f(g(x))$, given that $f(x) = x^6$.

1. Problem Description

The problem asks us to find the function $g(x)$ such that $h(x) = (x+9)^6 = f(g(x))$, given that $f(x) = x^6$.

2. Solution Steps

We are given that $h(x) = (x+9)^6$ and $f(x) = x^6$. We want to find $g(x)$ such that $f(g(x)) = h(x)$.

Since $f(x) = x^6$, we have $f(g(x)) = (g(x))^6$.

Therefore, we need to find $g(x)$ such that $(g(x))^6 = (x+9)^6$.

By taking the sixth root of both sides, we get $g(x) = x+9$.

3. Final Answer

$g(x) = x+9$