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?

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.