Inverse functions: Linear, discrete
The one-to-one functions $g$ and $h$ are defined as follows.
\[
\begin{array}{l}
g(x)=3 x-2 \\
h=\{(-4,8),(5,6),(6,1),(8,3),(9,-8)\}
\end{array}
\]

Find the following.
$g^{-1}(x)=[0$
$\left(g^{-1} \circ g\right)(1)=\square$
$h^{-1}(6)=\square$

Step 1 ：Replace \(g(x)\) with \(y\) to get \(y = 3x - 2\).

Step 2 ：Swap \(x\) and \(y\) to get \(x = 3y - 2\).

Step 3 ：Solve for \(y\) by adding 2 to both sides to get \(x + 2 = 3y\).

Step 4 ：Divide both sides by 3 to get \(y = \frac{x + 2}{3}\).

Step 5 ：\(\boxed{g^{-1}(x) = \frac{x + 2}{3}}\) is the inverse function.

Step 6 ：Apply the function \(g\) to 1 to get \(g(1) = 3(1) - 2 = 1\).

Step 7 ：Apply the inverse function \(g^{-1}\) to 1 to get \(g^{-1}(1) = \frac{1 + 2}{3} = 1\).

Step 8 ：\(\boxed{\left(g^{-1} \circ g\right)(1) = 1}\).

Step 9 ：Find the pair in the set \(h\) where 6 is the second element, which is \((5,6)\).

Step 10 ：\(\boxed{h^{-1}(6) = 5}\).