Use 
f(x) = 2x − 4
 and 
g(x) = 5 − x2
 to evaluate the expression. find f o g (-2) and g o f (-2)

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

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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.

Use f(x) = 2x − 4 and g(x) = 5 − x² to evaluate the expression. Find f o g (-2) and g o f (-2)

Learn how to evaluate function compositions f o g(-2) and g o f(-2) using the functions f(x) = 2x - 4 and g(x) = 5 - x². This problem is ideal for students in Grades 9-11, focusing on function composition and algebraic manipulation.

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