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.