To solve for $(f \circ g)(-3)$, we need to evaluate the composition of the functions $f(x)$ and $g(x)$. The composition $f \circ g$ means that we will first evaluate $g(-3)$, and then use this result as the input to the function $f(x)$.

Step 1: Evaluate $g(-3)$

We are given the function $g(x) = -4x - 15$. Substituting $x = -3$ into $g(x)$: $g(-3) = -4(-3) - 15 = 12 - 15 = -3$

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

We are given the function $f(x) = 2x^2 - 4x + 7$. Now, substitute $x = -3$ into $f(x)$: $f(-3) = 2(-3)^2 - 4(-3) + 7 = 2(9) + 12 + 7 = 18 + 12 + 7 = 37$

Final Answer:

$(f \circ g)(-3) = f(-3) = 37$

Would you like further clarification on any of these steps?

Related Questions:

1. How do you compute the composition of functions in general?
2. What is the difference between the composition of functions and multiplication of functions?
3. How do you find the domain of a composed function like $(f \circ g)(x)$?
4. How does the process change when the functions involved are not polynomials?
5. What happens if the input for the composition is not within the domain of the inner function?

Tip: Always work step by step with composition of functions: solve for the inner function first before substituting into the outer function.