Let $f(x)=3x+4$ and $g(x)=2x-3$.  If $h(x)=f(g(x))$, then what is the inverse of $h(x)$?
\[h(x)=f(g(x))=3(2x-3)+4=6x-5.\]Let's replace $h(x)$ with $y$ for simplicity, so \[y=6x-5.\]In order to invert $h(x)$ we may solve this equation for $x$.  That gives \[y+5=6x\]or  \[x=\frac{y+5}{6}.\]Writing this in terms of $x$ gives the inverse function of $h$ as  \[h^{-1}(x)=\boxed{\frac{x+5}{6}}.\]
\frac{x+5}{6}
Level 5
Algebra
Problem: Let $f(x)=3x+4$ and $g(x)=2x-3$.  If $h(x)=f(g(x))$, then what is the inverse of $h(x)$?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the inverse of the function h(x), we first need to determine the composition of the functions f and g. We start by substituting g(x) into f(x). 

1. Calculate g(x):
   g(x) = 2x - 3.

2. Substitute g(x) into f(x):
   h(x) = f(g(x)) = f(2x - 3) = 3(2x - 3) + 4.

3. Simplify the expression:
   h(x) = 6x - 9 + 4 = 6x - 5.

Now we have h(x) = 6x - 5. 

Next, we need to find the inverse of h(x). The process for finding the inverse involves the following steps:

1. Replace h(x) with y:
   y = 6x - 5.

2. Solve for x in terms of y:
   y + 5 = 6x,
   x = (y + 5)/6.

3. Replace x with h^(-1)(y) to express the inverse function:
   h^(-1)(y) = (y + 5)/6.

4. To express the inverse function in terms of x, we write:
   h^(-1)(x) = (x + 5)/6.

Thus, the final answer is h^(-1)(x) = (x + 5)/6.<|END_OF_LATENT|> The final answer is $\frac{x+5}{6}$.