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To solve the problem of finding \((g \circ f)(-2)\), let's break down the steps:

### Step-by-Step Solution
1. **Understand the Composition of Functions**:
   \((g \circ f)(x)\) means we first apply \(f(x)\) and then apply \(g\) to the result of \(f(x)\). So, we need to find \(f(-2)\) first and then plug that result into \(g\).

2. **Find \(f(-2)\)**:
   Given:
   \[
   f(x) = 2x^2 + 4x + 2
   \]
   Substitute \(x = -2\):
   \[
   f(-2) = 2(-2)^2 + 4(-2) + 2
   \]
   Calculating each term:
   - \((-2)^2 = 4\), so \(2 \cdot 4 = 8\)
   - \(4(-2) = -8\)
   - Adding these results: \(8 - 8 + 2 = 2\)
   
   So, \(f(-2) = 2\).

3. **Apply \(g\) to \(f(-2)\)**:
   Now we use \(g(x) = 2x - 7\) and substitute \(x = f(-2) = 2\):
   \[
   g(2) = 2(2) - 7 = 4 - 7 = -3
   \]

### Final Answer:
\[
(g \circ f)(-2) = -3
\]

Would you like further details on function compositions, or do you have any additional questions?

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Here are five related questions to expand your understanding:
1. How do you compose functions with different types of expressions, such as trigonometric or exponential functions?
2. What is the result of \((f \circ g)(-2)\) using the same functions given here?
3. How can you determine the domain of a composed function?
4. What happens to the range of \(f(x)\) when you compose it with \(g(x)\)?
5. How do you apply the chain rule to compositions of functions in calculus?

**Tip:** When composing functions, always evaluate the inner function first and then apply the outer function to the result.

Given the definitions of f(x) and g(x) below, find the value of (g ∘ f)(-2). f(x) = 2x^2 + 4x + 2, g(x) = 2x - 7.

Calculate the composition of functions (g ∘ f)(-2) where f(x) = 2x^2 + 4x + 2 and g(x) = 2x - 7. Follow a detailed, step-by-step solution that explores function composition and quadratic and linear functions. Suitable for Grades 10-11.

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