Decompose the function f, of, g, of, x, equals, cosine, x, to the power 5f(g(x))=cosx 
5
  into f, of, xf(x) and g, of, xg(x).

We are given the composite function \( f(g(x)) = \cos(x^5) \). Our goal is to decompose this function into \( f(x) \) and \( g(x) \).

To do this, we'll identify the inner function \( g(x) \) and the outer function \( f(x) \).

### Step 1: Identify \( g(x) \)
The function inside the cosine is \( x^5 \), so we can define the inner function as:
\[
g(x) = x^5
\]

### Step 2: Identify \( f(x) \)
The outer function applies the cosine to the result of \( g(x) \), which is \( \cos(g(x)) \). Since \( g(x) = x^5 \), the outer function must simply apply the cosine function. So:
\[
f(x) = \cos(x)
\]

### Conclusion
Thus, the decomposition of \( f(g(x)) = \cos(x^5) \) is:
\[
g(x) = x^5
\]
\[
f(x) = \cos(x)
\]

Would you like more details on function composition or anything else? Here are five related questions for deeper understanding:

1. How would you decompose a function like \( f(g(x)) = \sin(2x + 3) \)?
2. What are the conditions for two functions to form a valid composite function?
3. How do you find the derivative of a composite function using the chain rule?
4. Can every function be decomposed into two functions? Why or why not?
5. How would you graph the composite function \( f(g(x)) = \cos(x^5) \)?

**Tip**: When decomposing a function, look for an "inner" function being acted upon by an "outer" function. The outer function typically operates on the result of the inner function.

Decompose the function f(g(x)) = cos(x^5) into f(x) and g(x).

Learn how to decompose the composite function f(g(x)) = cos(x^5) into its individual functions f(x) and g(x). This solution involves concepts from trigonometry and function composition, suitable for students in Grades 10-12.

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