https://file.aiquickdraw.com/mathbot/file/17269558518553jlunm7z.jpg

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!

Decompose the function f(g(h(x))) = cos(x^8) - 9 into f(x), g(x), and h(x).

Learn how to decompose the function f(g(h(x))) = cos(x^8) - 9 into f(x), g(x), and h(x). This problem involves function composition, exponentiation, and trigonometric functions. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation