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The given function is \( F(x) = \cos^2(x + 9) \), and the problem asks to find functions \( f \), \( g \), and \( h \) such that \( F = f \circ g \circ h \), i.e., \( F(x) = f(g(h(x))) \).

To break down \( F(x) \):

1. We see that \( F(x) = \cos^2(x + 9) \). This suggests we can work backwards by identifying the innermost function and gradually applying transformations.
   
Let's define the following functions:

- \( h(x) = x + 9 \), which represents the shift inside the cosine function.
- \( g(x) = \cos(x) \), which applies the cosine operation to the result of \( h(x) \).
- \( f(x) = x^2 \), which squares the result of \( g(x) \).

So, the composition is:

\[
F(x) = f(g(h(x))) = f(\cos(h(x))) = f(\cos(x + 9)) = \cos^2(x + 9)
\]

Thus, the functions are:

- \( f(x) = x^2 \)
- \( g(x) = \cos(x) \)
- \( h(x) = x + 9 \)

Would you like more details on function compositions or have any questions? Here are some related questions to consider:

1. What are the steps to decompose a function in terms of composition?
2. How does the order of composition affect the final result?
3. Can other function combinations also lead to the same result for \( F(x) \)?
4. How can we verify that our chosen functions are correct?
5. What are some real-life applications of function composition?

**Tip:** When working with function compositions, always start by identifying the innermost transformation and work outward to match the overall expression.

Given F(x) = cos²(x + 9), find functions f, g, and h such that F = f ∘ g ∘ h.

Learn how to decompose the function F(x) = cos²(x + 9) into three functions f, g, and h using function composition techniques. This problem involves key trigonometric concepts and function transformations. Suitable for students in Grades 11-12 studying Pre-Calculus or Calculus.

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