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The image contains three function problems based on the given functions \( f(x) = -2x + 6 \) and \( g(x) = -2x^2 + 9x + 3 \). Here's how we approach each problem:

### a. \((f + g)(x)\)
This is the sum of the two functions:
\[
(f + g)(x) = f(x) + g(x)
\]
Substitute the expressions for \( f(x) \) and \( g(x) \):
\[
(f + g)(x) = (-2x + 6) + (-2x^2 + 9x + 3)
\]
Now, combine like terms:
\[
(f + g)(x) = -2x^2 + (-2x + 9x) + (6 + 3) = -2x^2 + 7x + 9
\]

### b. \((f - g)(x)\)
This is the difference between the two functions:
\[
(f - g)(x) = f(x) - g(x)
\]
Substitute the expressions for \( f(x) \) and \( g(x) \):
\[
(f - g)(x) = (-2x + 6) - (-2x^2 + 9x + 3)
\]
Distribute the negative sign:
\[
(f - g)(x) = -2x + 6 + 2x^2 - 9x - 3
\]
Now, combine like terms:
\[
(f - g)(x) = 2x^2 + (-2x - 9x) + (6 - 3) = 2x^2 - 11x + 3
\]

### c. \((g - f)(x)\)
This is the difference with the order reversed:
\[
(g - f)(x) = g(x) - f(x)
\]
Substitute the expressions for \( g(x) \) and \( f(x) \):
\[
(g - f)(x) = (-2x^2 + 9x + 3) - (-2x + 6)
\]
Distribute the negative sign:
\[
(g - f)(x) = -2x^2 + 9x + 3 + 2x - 6
\]
Now, combine like terms:
\[
(g - f)(x) = -2x^2 + (9x + 2x) + (3 - 6) = -2x^2 + 11x - 3
\]

### Final answers:
a. \((f + g)(x) = -2x^2 + 7x + 9\)

b. \((f - g)(x) = 2x^2 - 11x + 3\)

c. \((g - f)(x) = -2x^2 + 11x - 3\)

Would you like further details on any of these operations or have any other questions?

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Here are 5 related questions:
1. How do you find the sum of two functions in general?
2. What happens when you subtract functions in reverse order?
3. How would these operations change if \( f(x) \) and \( g(x) \) had higher powers of \( x \)?
4. How do you multiply two functions, say \( f(x) \) and \( g(x) \)?
5. What is the meaning of the graph of \( f(x) + g(x) \) compared to their individual graphs?

**Tip:** Always carefully distribute negative signs when subtracting functions, especially when parentheses are involved!

Given the functions f(x) = -2x + 6 and g(x) = -2x^2 + 9x + 3, determine the following: (f + g)(x), (f - g)(x), and (g - f)(x).

Learn how to solve function operations with given functions f(x) = -2x + 6 and g(x) = -2x^2 + 9x + 3. This solution covers adding and subtracting polynomials step-by-step, applying key algebraic concepts for high school students in Grades 9-12.

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