The problem asks for the value of $f(g(3)) - g(f(7))$ given the following functions:

$f(x) = x^2 - 1 \quad \text{and} \quad g(x) = x + 5$

Step-by-Step Solution:

1. Find $f(g(3))$:

   • First, calculate $g(3)$: $g(3) = 3 + 5 = 8$
   • Now calculate $f(g(3)) = f(8)$: $f(8) = 8^2 - 1 = 64 - 1 = 63$ Thus, $f(g(3)) = 63$.

2. Find $g(f(7))$:

   • First, calculate $f(7)$: $f(7) = 7^2 - 1 = 49 - 1 = 48$
   • Now calculate $g(f(7)) = g(48)$: $g(48) = 48 + 5 = 53$ Thus, $g(f(7)) = 53$.

3. Subtract $g(f(7))$ from $f(g(3))$: $f(g(3)) - g(f(7)) = 63 - 53 = 10$

Final Answer:

The correct value is 10, which corresponds to option D.

Would you like further clarification or have any other questions on this problem?

Here are some related questions for further practice:

1. What is the general rule for combining functions like $f(g(x))$?
2. How can you interpret the composition of functions in real-world applications?
3. Can you find the value of $f(g(x)) - g(f(x))$ for other given functions?
4. How do you handle different types of functions (linear, quadratic) when performing function composition?
5. What is the difference between $f(g(x))$ and $g(f(x))$?

Tip: When solving function compositions, always perform the inner function first, and then apply the outer function.