https://file.aiquickdraw.com/mathbot/file/1726917249514c8omhldz.jpg

The problem asks to evaluate \( f(g(x)) \) for \( x = 12 \), where the functions are defined as:

\[
f(x) = x^3 + 4x
\]
\[
g(x) = \sqrt{3x}
\]
\[
h(x) = 8x + 2
\]

### Step 1: Evaluate \( g(12) \)
We first evaluate \( g(12) \):

\[
g(12) = \sqrt{3 \times 12} = \sqrt{36} = 6
\]

### Step 2: Plug \( g(12) = 6 \) into \( f(x) \)
Now, we substitute \( x = 6 \) into \( f(x) \):

\[
f(6) = 6^3 + 4 \times 6 = 216 + 24 = 240
\]

Thus, \( f(g(12)) = 240 \).

### Final Answer:
\[
f(g(12)) = 240
\]

Let me know if you'd like further clarification or additional steps!

---

Here are 5 follow-up questions for further understanding:
1. What is the derivative of \( f(x) = x^3 + 4x \)?
2. How would you compute \( f(h(x)) \) for a given \( x \)?
3. What happens if \( x = 0 \) in the function \( f(g(x)) \)?
4. How does the domain of \( g(x) = \sqrt{3x} \) affect the domain of \( f(g(x)) \)?
5. Could you generalize a method to evaluate composite functions for any \( f(x) \) and \( g(x) \)?

### Tip:
When dealing with composite functions, always solve the inner function first, then substitute its result into the outer function for efficient computation.

Evaluate the function f(g(x)) for x = 12, where f(x) = x^3 + 4x and g(x) = √(3x).

Learn how to solve the function composition f(g(x)) for x = 12, with f(x) = x^3 + 4x and g(x) = √(3x). This problem covers key algebraic concepts and is suitable for students in Grades 10-12.

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