We are asked to find the value of $(f \circ g)(-3)$, which is the composition of the functions $f(x)$ and $g(x)$ at the input $x = -3$.

Given: $f(x) = 3x^2 - 7x - 3$ $g(x) = -4x - 10$

The composition of functions $(f \circ g)(x)$ means we first apply $g(x)$, then use the result as the input for $f(x)$. So, we need to find $g(-3)$ first, and then plug that into $f(x)$.

Step 1: Calculate $g(-3)$

$g(-3) = -4(-3) - 10 = 12 - 10 = 2$

Step 2: Calculate $f(g(-3)) = f(2)$

Now, substitute $x = 2$ into the function $f(x)$: $f(2) = 3(2)^2 - 7(2) - 3$ $f(2) = 3(4) - 14 - 3 = 12 - 14 - 3 = -5$

Thus, $(f \circ g)(-3) = -5$.

Would you like more details or have any other questions?

Here are 5 related questions:

1. What is the general process of composing two functions?
2. How would the result change if we composed $g \circ f(x)$ instead of $f \circ g(x)$?
3. How can you verify the result using another method?
4. What happens to the composition of functions if one function is non-invertible?
5. Can you generalize the composition to other polynomial functions?

Tip: Always double-check each function's evaluation step, especially when handling negative inputs and powers.