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The problem is asking for the function \( g(x) \) such that the composition of functions \( f(g(x)) = h(x) \). We are given:

- \( h(x) = (x + 3)^6 \)
- \( f(x) = x^6 \)

The goal is to express \( h(x) \) as \( f(g(x)) \).

### Step 1: Recognize the Structure
We know that:
\[
h(x) = (x + 3)^6
\]
We need to match this to the form \( f(g(x)) \). Since \( f(x) = x^6 \), we have:
\[
f(g(x)) = g(x)^6
\]
Thus, we need \( g(x) \) such that:
\[
g(x)^6 = (x + 3)^6
\]

### Step 2: Find \( g(x) \)
By comparing the expressions, we see that:
\[
g(x) = x + 3
\]

### Final Answer:
The function \( g(x) \) is:
\[
g(x) = x + 3
\]

Would you like any further details or explanation on this process?

Here are 5 related questions to expand on this:

1. How do you verify that \( f(g(x)) \) correctly gives \( h(x) \)?
2. Can you explain how function composition works in different scenarios?
3. How would this change if \( f(x) \) was a different function, like \( f(x) = x^3 \)?
4. What are some other examples of functions that can be expressed as compositions?
5. How do you differentiate composite functions using the chain rule?

**Tip:** Always break down compositions by looking at the "inner" and "outer" functions separately.

The function h(x) = (x + 3)^6 can be expressed in the form f(g(x)), where f(x) = x^6, and g(x) is defined below.

Learn how to express the function h(x) = (x + 3)^6 in the form f(g(x)), where f(x) = x^6. This problem focuses on the composition of functions, using key algebraic concepts and function transformation. Suitable for students in Grades 10-12.

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