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Let's solve the problem step by step.

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

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

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

### Step 2: Calculate \( g(f(1)) \)
Now that we know \( f(1) = 2 \), we substitute \( x = 2 \) into the function \( g(x) \):

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

Calculate each term:

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

Simplify the expression:

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

So, the value of \( g(f(1)) \) is \( \boxed{7} \).

Would you like any further details or have any questions?

### Related Questions:
1. What is the value of \( g(f(2)) \)?
2. How would the answer change if \( f(x) \) were \( -2x + 5 \) instead?
3. If \( g(x) \) were a linear function, how would you calculate \( g(f(1)) \)?
4. What does the composition of functions mean in this context?
5. How can you verify the solution to ensure it's correct?

### Tip:
When dealing with compositions of functions like \( g(f(x)) \), always start by finding the inner function's value first and then substitute 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 a function composition problem where f(x) = -2x + 4 and g(x) = 3x^2 - 7x + 9. The solution involves calculating g(f(1)) using key algebraic concepts. Suitable for students in Grades 9-12.

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