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Given the functions \( f(x) = \frac{3}{4}x + 10 \) and \( g(x) = x^2 - 3 \), we need to find the value of \( f(-2 - g(3)) \).

Let's break this down step-by-step:

1. **Calculate \( g(3) \):**
   \[
   g(3) = 3^2 - 3 = 9 - 3 = 6
   \]

2. **Substitute \( g(3) \) into \( -2 - g(3) \):**
   \[
   -2 - g(3) = -2 - 6 = -8
   \]

3. **Now, calculate \( f(-8) \):**
   \[
   f(-8) = \frac{3}{4} \times (-8) + 10
   \]
   \[
   f(-8) = \frac{3 \times (-8)}{4} + 10 = \frac{-24}{4} + 10 = -6 + 10 = 4
   \]

So, the value of \( f(-2 - g(3)) \) is **4**.

Would you like a detailed explanation of any part of this solution? Here are some related questions to explore further:

1. What is the significance of composite functions in mathematics?
2. How do you find the composition of two functions?
3. Can you calculate the inverse of \( f(x) \) or \( g(x) \)?
4. What happens if we change the input value of \( g(x) \) in this problem?
5. How does the slope of the linear function \( f(x) \) affect its graph?

**Tip:** Always substitute and simplify in small steps to avoid errors when dealing with composite functions.

Let f(x) = (3/4)x + 10, g(x) = x^2 - 3. What is the value of f(-2 - g(3))?

Solve the composite function f(-2 - g(3)) where f(x) = (3/4)x + 10 and g(x) = x^2 - 3. Learn how to break down and calculate composite functions step-by-step. Suitable for students in Grades 9-12.

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