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The question asks us to identify the outer function \( f(x) \) and the inner function \( g(x) \) for the expression \( y = \sin^7(x) \), such that \( y = f(g(x)) \).

### Step-by-Step Solution

1. **Identify the Structure of \( y \):**
   - The expression \( y = \sin^7(x) \) means \( y = (\sin(x))^7 \).
   - This can be seen as a composition of two functions: an exponentiation and a sine function.

2. **Define \( f(x) \) and \( g(x) \):**
   - If we let \( g(x) = \sin(x) \), then \( f(g(x)) = f(\sin(x)) \) would need to be \( (\sin(x))^7 \).
   - Therefore, we can define \( f(x) = x^7 \), which, when composed with \( g(x) = \sin(x) \), gives \( f(g(x)) = (\sin(x))^7 \), matching \( y = \sin^7(x) \).

3. **Conclusion:**
   - The correct choice is:
     \[
     f(x) = x^7, \quad g(x) = \sin(x)
     \]

So, the answer is the **third option**:
- \( f(x) = x^7 \), \( g(x) = \sin(x) \).

Would you like further details on this solution or have any other questions?

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Here are 5 related questions to deepen understanding:

1. How would the solution change if \( y = \sin^3(x) \) instead of \( y = \sin^7(x) \)?
2. If \( y = \sin(x^7) \), what would be \( f(x) \) and \( g(x) \) in that case?
3. Can you derive \( y' = \frac{d}{dx} \left(\sin^7(x)\right) \) using the chain rule?
4. What is the composition function if \( y = (\cos(x))^7 \)?
5. How does the chain rule apply when \( y = \sin^k(x) \) for any integer \( k \)?

**Tip:** When dealing with function compositions, look for operations applied in layers—inner functions are applied first, then outer functions.

Let \( y = \sin^7(x) \). Which of the following is the outer function, \( f(x) \), and the inner function, \( g(x) \), so that \( y = f(g(x)) \)? Options: 1. \( f(x) = \sin(x), g(x) = x^7 \) 2. \( f(x) = \sin^7(x), g(x) = x \) 3. \( f(x) = x^7, g(x) = \sin(x) \) 4. None of the above.

Understand how to determine the outer function \( f(x) \) and the inner function \( g(x) \) for the expression \( y = \sin^7(x) \) by analyzing function composition. This solution covers function composition, trigonometric functions, and exponentiation, suitable for Grades 10-12 students.

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