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.