We are given two functions $f(x) = \frac{3}{4}x + 10$ and $g(x) = x^2 - 3$. We need to find the value of $f(-2 - g(3))$.

Step 1: Compute $g(3)$

Using the function $g(x) = x^2 - 3$, we substitute $x = 3$:

$g(3) = 3^2 - 3 = 9 - 3 = 6$

Step 2: Substitute $g(3) = 6$ into the expression

We now substitute this result into the expression $f(-2 - g(3))$:

$f(-2 - g(3)) = f(-2 - 6) = f(-8)$

Step 3: Compute $f(-8)$

Using the function $f(x) = \frac{3}{4}x + 10$, we substitute $x = -8$:

$f(-8) = \frac{3}{4}(-8) + 10 = \frac{3 \times -8}{4} + 10 = \frac{-24}{4} + 10 = -6 + 10 = 4$

Final Answer

The value of $f(-2 - g(3))$ is $\boxed{4}$.

Would you like more details on any part of this solution? Here are five related questions to consider:

1. How do you find the value of a function for a specific input?
2. What is the significance of composite functions in mathematics?
3. How would you interpret the geometric meaning of $f(x) = \frac{3}{4}x + 10$?
4. What happens to the function $f(x)$ when you change the slope or intercept?
5. How do you solve for $f(g(x))$ when both functions are given?

Tip: When working with composite functions, always calculate the innermost function first before moving outward to the next layer of the function.