The one-to-one functions $g$ and $h$ are defined as follows. \[ \begin{array}{l} g(x)=3 x-14 \\ h=\{(0,8),(2,-8),(7,2),(8,4)\} \end{array} \] Find the following.

Step 1 ：First, we need to find the function $f(x)$ such that $h(x)=f(g(x))$. Since $g(x)=3x-14$, we substitute the values of $h$ into $g(x)$ to find $f(x)$.

Step 2 ：We have $h=\{(0,8),(2,-8),(7,2),(8,4)\}$ and $g(x)=3x-14$. So, we get the following pairs: $(g(0),8)$, $(g(2),-8)$, $(g(7),2)$, $(g(8),4)$ which simplifies to $(-14,8)$, $(-8,-8)$, $(7,2)$, $(10,4)$.

Step 3 ：From these pairs, we can see that $f(x)=2x+22$.

Step 4 ：So, $h(x)=f(g(x))=2(3x-14)+22=6x-28+22=6x-6$.

Step 5 ：Let's replace $h(x)$ with $y$ for simplicity, so $y=6x-6$.

Step 6 ：In order to invert $h(x)$ we may solve this equation for $x$. That gives $y+6=6x$ or $x=\frac{y+6}{6}$.

Step 7 ：Writing this in terms of $x$ gives the inverse function of $h$ as $h^{-1}(x)=\boxed{\frac{x+6}{6}}$.