We are given two functions $f(x) = 3x^2 - 9$ and $g(x) = 5x^3 + 9$, and we are asked to evaluate $h(3)$, where $h = g \circ f$. This means that $h(x) = g(f(x))$, i.e., we first apply $f(x)$, and then apply $g(x)$ to the result of $f(x)$.

Let's calculate $h(3)$ step by step:

Step 1: Compute $f(3)$

Using the function $f(x) = 3x^2 - 9$, substitute $x = 3$:

$f(3) = 3(3)^2 - 9 = 3(9) - 9 = 27 - 9 = 18$

Step 2: Compute $g(f(3)) = g(18)$

Now, using the function $g(x) = 5x^3 + 9$, substitute $x = 18$:

$g(18) = 5(18)^3 + 9$

First, calculate $18^3$:

$18^3 = 18 \times 18 \times 18 = 5832$

Now compute $g(18)$:

$g(18) = 5(5832) + 9 = 29160 + 9 = 29169$

Final Answer:

Therefore, $h(3) = 29169$.

Would you like more details or have any questions? Here are some related questions you might explore:

1. What is $h(2)$ if $h = g \circ f$?
2. How does the composition of two functions affect their domain?
3. Can you simplify $h(x) = g(f(x))$ in general form?
4. What happens if you reverse the composition, $h = f \circ g$?
5. How do transformations of functions affect their graphs?

Tip: When composing functions, always remember to evaluate the inner function first, then apply the outer function.