To find $(f \circ g)(x)$, we need to compute the composition of functions $f(x)$ and $g(x)$, which means substituting $g(x)$ into $f(x)$.

Given:

• $f(x) = -3x + 12$
• $g(x) = 3x^2 - 6x - 6$

To find $(f \circ g)(x)$, we replace $x$ in $f(x)$ with $g(x)$. So we have: $(f \circ g)(x) = f(g(x)) = f(3x^2 - 6x - 6)$

Substitute $g(x)$ into $f(x)$: $f(g(x)) = -3(3x^2 - 6x - 6) + 12$

Now distribute the $-3$: $f(g(x)) = -9x^2 + 18x + 18 + 12$

Combine like terms: $f(g(x)) = -9x^2 + 18x + 30$

Thus, the composition $(f \circ g)(x)$ is: $(f \circ g)(x) = -9x^2 + 18x + 30$

Would you like more details or have any questions about this? Here are five related questions to explore further:

1. How do you find the composition of functions in general?
2. What is the difference between $(f \circ g)(x)$ and $(g \circ f)(x)$?
3. Can you determine the degree of a polynomial after composition?
4. What happens when both functions are quadratic, not linear like $f(x)$?
5. How can we determine if the composition of two functions is one-to-one?

Tip: Always simplify the expression after substitution to make sure the function is in standard polynomial form.