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?

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!