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?

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.