Find​ f(x) and​ g(x) such that

​h(x)=​(f◦​g)(x).

​h(x)=​(3−9​x)2

Suppose that

​g(x)=3−9x.

To find \( f(x) \) and \( g(x) \) such that \( h(x) = (f \circ g)(x) \), we use the definition of composition:

\[
h(x) = f(g(x)).
\]

We are given:
\[
h(x) = (3 - 9x)^2 \quad \text{and} \quad g(x) = 3 - 9x.
\]

### Step 1: Substitute \( g(x) \) into \( h(x) \)
\[
h(x) = f(g(x)) = f(3 - 9x).
\]

Thus:
\[
f(3 - 9x) = (3 - 9x)^2.
\]

### Step 2: Identify \( f(u) \) where \( u = g(x) \)
Let \( u = 3 - 9x \). Then:
\[
f(u) = u^2.
\]

### Final Answer:
- \( g(x) = 3 - 9x \),
- \( f(x) = x^2 \).

These satisfy the given composition \( h(x) = (f \circ g)(x) \). 

---

Would you like a deeper explanation or a graph of these functions?  
Here are 5 related questions to explore:  
1. What is the domain of \( f(x) \) and \( g(x) \)?  
2. How does the composition of functions work conceptually?  
3. What are examples of other \( h(x) \) functions and their decompositions?  
4. How does this apply to real-world problems, such as nested relationships?  
5. Can we reverse the process to find \( g(x) \) given \( f(x) \) and \( h(x) \)?  

**Tip:** Always check the domain and range of each function in composition problems to ensure they are compatible.

Find f(x) and g(x) such that h(x) = (f◦g)(x). h(x) = (3−9x)². Suppose that g(x) = 3−9x.

Learn how to find functions f(x) and g(x) such that their composition results in h(x) = (3−9x)². The solution includes the use of function composition and algebraic techniques. Suitable for students in Grades 9-11.

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