For a bijective function $g : R \to R$, we say that a function $f : R \to R$ is its superinverse if it satisfies the following identity $(f \circ g)(x) = g^{-1}(x)$, where $g^{-1}$ is the inverse of $g$. Given $g(x) = x^3 + 9x^2 + 27x + 81$ and $f$ is its superinverse, find $|f(-289)|$.
1. Given the function \( g(x) = x^3 + 9x^2 + 27x + 81 \), we need to find its inverse \( g^{-1}(x) \). Notice that:
   \[
   g(x) = (x+3)^3 + 54
   \]
   To find the inverse, we solve for \( x \) in terms of \( y \):
   \[
   y = (x+3)^3 + 54
   \]
   Subtract 54 from both sides:
   \[
   y - 54 = (x+3)^3
   \]
   Take the cube root of both sides:
   \[
   (y - 54)^{1/3} = x + 3
   \]
   Subtract 3 from both sides:
   \[
   x = (y - 54)^{1/3} - 3
   \]
   Therefore, the inverse function is:
   \[
   g^{-1}(x) = (x - 54)^{1/3} - 3
   \]

2. We are given that \( f \) is the superinverse of \( g \), which means:
   \[
   (f \circ g)(x) = g^{-1}(x)
   \]
   This implies:
   \[
   f(g(x)) = g^{-1}(x)
   \]

3. We need to find \( |f(-289)| \). To do this, we first find \( x \) such that \( g(x) = -289 \):
   \[
   (x+3)^3 + 54 = -289
   \]
   Subtract 54 from both sides:
   \[
   (x+3)^3 = -343
   \]
   Take the cube root of both sides:
   \[
   x + 3 = -7
   \]
   Subtract 3 from both sides:
   \[
   x = -10
   \]

4. Now, we use the fact that \( f(g(x)) = g^{-1}(x) \) to find \( f(-289) \):
   \[
   f(g(-10)) = g^{-1}(-289)
   \]
   Since \( g(-10) = -289 \), we have:
   \[
   f(-289) = g^{-1}(-289)
   \]
   Using the inverse function we found earlier:
   \[
   g^{-1}(-289) = (-289 - 54)^{1/3} - 3
   \]
   Simplify inside the cube root:
   \[
   g^{-1}(-289) = (-343)^{1/3} - 3
   \]
   Since \( (-343)^{1/3} = -7 \):
   \[
   g^{-1}(-289) = -7 - 3 = -10
   \]

5. Therefore:
   \[
   f(-289) = -10
   \]
   The absolute value is:
   \[
   |f(-289)| = |-10| = 10
   \]

The final answer is \(\boxed{10}\)