To solve $(f \circ g)(-3)$, we need to first understand that $(f \circ g)(x)$ means $f(g(x))$, meaning we substitute $g(x)$ into $f(x)$.

Step 1: Find $g(-3)$

The function $g(x)$ is given as: $g(x) = -4x - 15$ Substitute $x = -3$ into $g(x)$: $g(-3) = -4(-3) - 15 = 12 - 15 = -3$

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

Now, use $g(-3) = -3$ and substitute $-3$ into the function $f(x)$. The function $f(x)$ is given as: $f(x) = 2x^2 - 4x + 7$ Substitute $x = -3$ into $f(x)$: $f(-3) = 2(-3)^2 - 4(-3) + 7 = 2(9) + 12 + 7 = 18 + 12 + 7 = 37$

Final Answer:

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

Would you like more details on how function composition works or have any other questions?

Here are five related questions for further practice:

1. Find $(f \circ g)(2)$ using the same $f(x)$ and $g(x)$.
2. If $f(x) = 3x^2 - x + 5$ and $g(x) = 2x - 1$, find $(f \circ g)(1)$.
3. What is $(g \circ f)(-3)$ using the same functions $f(x)$ and $g(x)$?
4. Find the inverse function of $g(x) = -4x - 15$.
5. What is the domain of $(f \circ g)(x)$?

Tip: In function composition, always start by evaluating the innermost function before applying the outer function.