Given the definitions of f, of, xf(x) and g, of, xg(x) below, find the value of g, of, f, of, 2, .g(f(2)).
f, of, x, equals, 2, x, squared, minus, x, minus, 9
f(x)=
2x 
2
 −x−9
g, of, x, equals, 4, x, plus, 1
g(x)=
4x+1

We are given the functions \( f(x) \) and \( g(x) \):

- \( f(x) = 2x^2 - x - 9 \)
- \( g(x) = 4x + 1 \)

To find \( g(f(2)) \), we will follow these steps:

### Step 1: Find \( f(2) \)
Substitute \( x = 2 \) into the function \( f(x) \):
\[
f(2) = 2(2)^2 - (2) - 9
\]
\[
f(2) = 2(4) - 2 - 9 = 8 - 2 - 9 = -3
\]

### Step 2: Find \( g(f(2)) \) or \( g(-3) \)
Now substitute \( f(2) = -3 \) into \( g(x) \):
\[
g(-3) = 4(-3) + 1 = -12 + 1 = -11
\]

### Final Answer:
The value of \( g(f(2)) \) is \( \boxed{-11} \).

Would you like more details on any of these steps, or do you have any other questions?

Here are 5 related questions for practice:
1. Find \( g(f(3)) \) using the same functions.
2. If \( f(x) = 2x^2 - x - 9 \), what is \( f(-1) \)?
3. If \( g(x) = 4x + 1 \), what is \( g(5) \)?
4. Solve for \( x \) if \( f(x) = 0 \).
5. What is the composition \( f(g(x)) \)?

**Tip:** When evaluating compositions like \( g(f(x)) \), always start by finding the inner function first and then apply the outer function.

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

Learn how to solve a function composition problem involving the quadratic function f(x) = 2x^2 - x - 9 and the linear function g(x) = 4x + 1. This step-by-step solution demonstrates how to find g(f(2)), focusing on key algebraic concepts suitable for Grades 9-12.

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