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.