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.