To find the value of $g(f(-3))$, we first need to evaluate $f(-3)$ using the function $f(x) = -x - 7$, and then substitute that result into the function $g(x) = 2x^2 - 3x + 15$.

Step 1: Evaluate $f(-3)$

Using the function $f(x) = -x - 7$:

$f(-3) = -(-3) - 7 = 3 - 7 = -4$

So, $f(-3) = -4$.

Step 2: Substitute $f(-3)$ into $g(x)$

Now, substitute $f(-3) = -4$ into the function $g(x) = 2x^2 - 3x + 15$:

$g(-4) = 2(-4)^2 - 3(-4) + 15$

Step 3: Simplify $g(-4)$

First, calculate each term:

$2(-4)^2 = 2 \times 16 = 32$ $-3(-4) = 12$

Now sum all terms:

$g(-4) = 32 + 12 + 15 = 59$

Final Answer:

$g(f(-3)) = 59$

Would you like further details or have any questions?

Here are five related questions to expand on this:

1. What is the general approach for solving composite functions like $g(f(x))$?
2. How does the domain of $f(x)$ and $g(x)$ affect the composite function?
3. How would the solution change if $f(x)$ was a different linear function?
4. What happens if you switch the order and compute $f(g(x))$ instead?
5. Can we always compute $g(f(x))$ if $f(x)$ and $g(x)$ are defined differently?

Tip: Always solve the innermost function first when dealing with composite functions.