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The problem asks to find the value of \( g(f(1)) \) given the functions \( f(x) = -2x + 4 \) and \( g(x) = 3x^2 - 7x + 9 \).

### Step 1: Find \( f(1) \)
Substitute \( x = 1 \) into the function \( f(x) \):

\[
f(1) = -2(1) + 4 = -2 + 4 = 2
\]

### Step 2: Find \( g(f(1)) = g(2) \)
Now, substitute \( x = 2 \) into the function \( g(x) \):

\[
g(2) = 3(2)^2 - 7(2) + 9
\]

\[
g(2) = 3(4) - 14 + 9 = 12 - 14 + 9 = 7
\]

### Final Answer:
The value of \( g(f(1)) \) is \( \boxed{7} \).

Would you like more details on any part of this solution, or do you have any questions?

Here are five related questions for further exploration:
1. What is the value of \( f(g(1)) \) given the same functions?
2. How would the value of \( g(f(x)) \) change if \( f(x) \) were a different linear function?
3. Can you find the general expression for \( g(f(x)) \)?
4. What is the derivative of the composite function \( g(f(x)) \) with respect to \( x \)?
5. How would you solve for \( x \) if \( g(f(x)) = 0 \)?

**Tip:** When solving composite functions, always start by finding the inner function's value before substituting it into the outer function.

Given the definitions of f(x) and g(x) below, find the value of g(f(1)). f(x) = -2x + 4, g(x) = 3x^2 - 7x + 9.

Learn how to solve the composite function g(f(1)) where f(x) = -2x + 4 and g(x) = 3x^2 - 7x + 9. This problem involves key algebraic concepts and composite functions, making it suitable for students in Grades 9-10.

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