https://file.aiquickdraw.com/mathbot/file/17294589598069p9jeoug.jpg

Let's work through the problem shown in the image.

### Part (a): Explain the meaning of \( f(f(x)) \)
You correctly marked **C**: "the function of the function of x." This is correct. Mathematically, \( f(f(x)) \) represents a composition of the function \( f \) with itself. That means you apply the function \( f \) to \( x \), and then take that result and apply \( f \) again.

### Part (b): Find \( f(f(x)) \) for \( f(x) = 3x^2 \)

Given:
\[
f(x) = 3x^2
\]

We need to find \( f(f(x)) \), which means substituting \( f(x) \) back into itself:
\[
f(f(x)) = f(3x^2)
\]
Now, substitute \( 3x^2 \) into the function \( f \):
\[
f(3x^2) = 3(3x^2)^2
\]
Simplify \( (3x^2)^2 \):
\[
(3x^2)^2 = 9x^4
\]
Thus:
\[
f(3x^2) = 3 \times 9x^4 = 27x^4
\]
So, the final answer is:
\[
f(f(x)) = 27x^4
\]

Would you like further clarification or details on any part of this process?

### Related Questions:
1. How do you find the composition of two different functions, not just the same function like in this case?
2. What happens if a function \( f \) is linear? How would \( f(f(x)) \) behave?
3. Can you always compose a function with itself? What are the restrictions, if any?
4. How would the result change if \( f(x) \) were a cubic function instead of a quadratic one?
5. What real-world situations might require using function composition?

### Tip:
When working with function composition, it's helpful to visualize it step by step: first apply the inner function, then the outer function.

Explain the meaning of f(f(x)). Find f(f(x)) for f(x) = 3x^2.

Learn how to solve the function composition problem f(f(x)) for f(x) = 3x^2. This problem covers function composition and algebraic manipulation for high school students in Grades 9-12.

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