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.