To find $(g \circ f)(0)$, we calculate the composition of $g(f(0))$. Here are the steps:

Step 1: Calculate $f(0)$

The function $f(x) = x^2 - 3x - 3$. Substitute $x = 0$:

$f(0) = 0^2 - 3(0) - 3 = -3$

So, $f(0) = -3$.

Step 2: Substitute $f(0)$ into $g(x)$

The function $g(x) = -4x - 12$. Substitute $x = -3$:

$g(-3) = -4(-3) - 12 = 12 - 12 = 0$

Final Answer:

$(g \circ f)(0) = g(f(0)) = 0$

Let me know if you would like a detailed explanation of each step or have any questions!

Here are 5 related questions to deepen your understanding:

1. How would $(g \circ f)(x)$ look for a general $x$?
2. What happens if you reverse the order to find $(f \circ g)(0)$?
3. How does the composition of functions change if $g(x)$ is a quadratic function?
4. Can you graph $g(x)$, $f(x)$, and their composition to see their behavior visually?
5. What are the domain and range of $(g \circ f)(x)$?

Tip: Always evaluate the inner function first in function compositions to avoid mistakes!