We are tasked with finding $(g - f)(-2)$, where:

$f(x) = x^2 - 10 \quad \text{and} \quad g(x) = 5x + 13$

The operation $(g - f)(x)$ is defined as the difference between the functions $g(x)$ and $f(x)$, which is:

$(g - f)(x) = g(x) - f(x)$

Step 1: Set up $(g - f)(x)$

$(g - f)(x) = (5x + 13) - (x^2 - 10)$ Simplifying: $(g - f)(x) = 5x + 13 - x^2 + 10$ $(g - f)(x) = -x^2 + 5x + 23$

Step 2: Evaluate at $x = -2$

Now, substitute $x = -2$ into $(g - f)(x)$: $(g - f)(-2) = -(-2)^2 + 5(-2) + 23$ $(g - f)(-2) = -(4) - 10 + 23$ $(g - f)(-2) = -4 - 10 + 23$ $(g - f)(-2) = 9$

Thus, the answer is: $(g - f)(-2) = 9$

Therefore, the correct choice is A.

Do you want further details or have any questions?

Here are 5 related questions:

1. How do we generally perform function subtraction like $(g - f)(x)$?
2. What happens if $x^2 - 10$ had been $x^2 + 10$?
3. How would the result change if $g(x) = 3x + 15$ instead of $5x + 13$?
4. What is the domain of the function $(g - f)(x)$?
5. How can you check if a result like $(g - f)(-2)$ exists for a given function?

Tip: When performing function subtraction, remember to distribute any negative signs carefully to avoid mistakes.