To solve this problem, we need to evaluate two function compositions: $f \circ g (-2)$ and $g \circ f (-2)$. Let’s break it down step by step.

Step 1: Find $f \circ g (-2)$

The notation $f \circ g (-2)$ means $f(g(-2))$. So, we need to first find $g(-2)$ and then substitute that result into the function $f(x)$.

1. Find $g(-2)$: $g(x) = 5 - x^2$ Substituting $x = -2$: $g(-2) = 5 - (-2)^2 = 5 - 4 = 1$

2. Find $f(g(-2)) = f(1)$: The function $f(x) = 2x - 4$. Now, substitute $x = 1$: $f(1) = 2(1) - 4 = 2 - 4 = -2$

Thus, $f \circ g(-2) = -2$.

Step 2: Find $g \circ f (-2)$

The notation $g \circ f (-2)$ means $g(f(-2))$. First, we need to find $f(-2)$, and then substitute that result into the function $g(x)$.

1. Find $f(-2)$: $f(x) = 2x - 4$ Substituting $x = -2$: $f(-2) = 2(-2) - 4 = -4 - 4 = -8$

2. Find $g(f(-2)) = g(-8)$: The function $g(x) = 5 - x^2$. Now, substitute $x = -8$: $g(-8) = 5 - (-8)^2 = 5 - 64 = -59$

Thus, $g \circ f(-2) = -59$.

Final Answers:

1. $f \circ g (-2) = -2$
2. $g \circ f (-2) = -59$

---

Would you like more details or have any questions about the steps? Here are 5 related problems for further practice:

1. Evaluate $f \circ g(3)$ for the same functions.
2. Evaluate $g \circ f(0)$.
3. Find $f(g(x))$ in general form.
4. Find $g(f(x))$ in general form.
5. Solve $f(x) = g(x)$ for $x$.

Tip: When working with function compositions, always apply the inner function first before substituting its result into the outer function.