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We are given the functions \( f(x) = \cos(x) \) and \( g(x) = 4x^3 + 10x^2 - 2 \). We need to calculate the composite functions \( f(g(x)) \) and \( g(f(x)) \).

### Step 1: Calculate \( f(g(x)) \)
The composite function \( f(g(x)) \) means we plug \( g(x) \) into \( f(x) \). Since \( f(x) = \cos(x) \), we replace \( x \) in \( f(x) \) with \( g(x) \).

\[
f(g(x)) = \cos(g(x)) = \cos(4x^3 + 10x^2 - 2)
\]

Thus, \( f(g(x)) = \cos(4x^3 + 10x^2 - 2) \).

### Step 2: Calculate \( g(f(x)) \)
Now, we compute \( g(f(x)) \), which means we plug \( f(x) = \cos(x) \) into \( g(x) \). So, in the function \( g(x) = 4x^3 + 10x^2 - 2 \), replace \( x \) with \( \cos(x) \).

\[
g(f(x)) = 4(\cos(x))^3 + 10(\cos(x))^2 - 2
\]

Thus, \( g(f(x)) = 4\cos^3(x) + 10\cos^2(x) - 2 \).

### Final Answers:
- \( f(g(x)) = \cos(4x^3 + 10x^2 - 2) \)
- \( g(f(x)) = 4\cos^3(x) + 10\cos^2(x) - 2 \)

Would you like further details or have any questions? Here are some related questions for deeper understanding:

1. How do we find the domain of composite functions like \( f(g(x)) \) or \( g(f(x)) \)?
2. What are common applications of composite functions in real-world problems?
3. How do you differentiate composite functions using the chain rule?
4. How do you graph composite functions step by step?
5. Can composite functions be inverted under certain conditions?

**Tip:** When working with composite functions, ensure that the output of the inner function is within the domain of the outer function to avoid undefined expressions.

Calculate the composite functions f(g(x)) and g(f(x)) for f(x) = cos(x) and g(x) = 4x^3 + 10x^2 - 2.

This problem explores how to calculate the composite functions f(g(x)) and g(f(x)) for the functions f(x) = cos(x) and g(x) = 4x^3 + 10x^2 - 2. Learn step-by-step how to solve these composite functions and understand the key mathematical concepts involved, including trigonometric and polynomial functions.

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