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.