We are given the functions $f(x) = -3x + 1$ and $g(x) = 3x^2 + 3x - 10$, and we are tasked with finding the value of $(g \circ f)(0)$, which means $g(f(0))$.

Step-by-Step Solution:

1. Find $f(0)$: We first need to evaluate $f(0)$ by substituting $x = 0$ into the function $f(x) = -3x + 1$: $f(0) = -3(0) + 1 = 1$

2. Find $g(f(0)) = g(1)$: Now that we know $f(0) = 1$, we substitute this into $g(x)$, meaning we need to find $g(1)$. Substituting $x = 1$ into $g(x) = 3x^2 + 3x - 10$: $g(1) = 3(1^2) + 3(1) - 10 = 3 + 3 - 10 = -4$

Final Answer:

$(g \circ f)(0) = g(f(0)) = g(1) = -4$

Would you like more details on any step or further explanation?

Here are five questions related to this concept:

1. What is the general definition of function composition?
2. How do you find the composition of two functions?
3. Can function composition be applied to non-linear functions?
4. How can you determine the domain of a composite function?
5. What happens if the inner function's range does not align with the outer function's domain?

Tip: When solving function compositions, always start by evaluating the inner function before applying the result to the outer function.