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You have provided an exercise involving two functions \( f(x) \) and \( g(x) \), and you're asked to determine \( (f \circ g)(x) \) and \( (g \circ f)(x) \).

The functions are defined as:
- \( f(x) = 2x + 3 \)
- \( g(x) = x^2 - 4 \)

Let's work on both compositions:

1. **\( (f \circ g)(x) \)**:
   - This means \( f(g(x)) \), or applying \( g(x) \) first and then \( f \).
   - \( g(x) = x^2 - 4 \), so \( f(g(x)) = f(x^2 - 4) \).
   - Now apply \( f(x) = 2x + 3 \) to \( g(x) \):
     \[
     f(x^2 - 4) = 2(x^2 - 4) + 3 = 2x^2 - 8 + 3 = 2x^2 - 5.
     \]
   - Therefore, \( (f \circ g)(x) = 2x^2 - 5 \).

2. **\( (g \circ f)(x) \)**:
   - This means \( g(f(x)) \), or applying \( f(x) \) first and then \( g \).
   - \( f(x) = 2x + 3 \), so \( g(f(x)) = g(2x + 3) \).
   - Now apply \( g(x) = x^2 - 4 \) to \( f(x) \):
     \[
     g(2x + 3) = (2x + 3)^2 - 4.
     \]
   - Expanding \( (2x + 3)^2 \):
     \[
     (2x + 3)^2 = 4x^2 + 12x + 9.
     \]
   - Therefore, \( g(2x + 3) = 4x^2 + 12x + 9 - 4 = 4x^2 + 12x + 5 \).
   - So, \( (g \circ f)(x) = 4x^2 + 12x + 5 \).

### Summary:
- \( (f \circ g)(x) = 2x^2 - 5 \)
- \( (g \circ f)(x) = 4x^2 + 12x + 5 \)

Would you like further details or explanations?

Here are 5 related questions to expand your understanding:
1. How do you find the domain of composite functions like \( (f \circ g)(x) \)?
2. What are the key differences between \( f \circ g \) and \( g \circ f \)?
3. Can you graph the composite functions \( f(g(x)) \) and \( g(f(x)) \)?
4. How do you check if two functions are inverses using composition?
5. What is the geometric interpretation of function composition?

**Tip:** Always substitute the inner function first when composing two functions, as this ensures the correct application of both functions.

Given the functions f(x) = 2x + 3 and g(x) = x^2 - 4, determine the compositions (f ∘ g)(x) and (g ∘ f)(x).

Learn how to determine the function compositions (f ∘ g)(x) and (g ∘ f)(x) with detailed step-by-step solutions. The problem explores algebraic manipulation and function composition with polynomials, suitable for students in Grades 9-12.

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