The problem asks to decompose the function $f(g(h(x))) = \cos(x^8) - 9$ into three functions: $f(x)$, $g(x)$, and $h(x)$.

To do this, we'll follow these steps:

1. Innermost function $h(x)$: Look for the simplest part of the input expression for $\cos(x^8) - 9$. We notice the expression $x^8$, which suggests $h(x) = x^8$.

2. Next layer $g(x)$: After applying $h(x)$, we get $\cos(h(x)) = \cos(x^8)$, so $g(x) = \cos(x)$.

3. Outermost function $f(x)$: Finally, the entire function is $\cos(x^8) - 9$, meaning that after applying $g(x) = \cos(x)$, the last operation is subtracting 9. Therefore, $f(x) = x - 9$.

Decomposition:

• $h(x) = x^8$
• $g(x) = \cos(x)$
• $f(x) = x - 9$

Would you like further details on function composition or other related concepts?

Here are some related questions to deepen your understanding:

1. What is the definition of function composition?
2. How can you verify that the decomposition is correct?
3. What happens if you decompose a different function like $f(g(h(x))) = e^{x^2} - 7$?
4. How can chain rules apply to a composed function in calculus?
5. How does function composition relate to inverse functions?

Tip: When decomposing functions, always start by identifying the innermost operation and work your way out!