Let's solve each of the problems step by step.

Given Functions:

• $f(x) = 2x + 1$
• $g(x) = -5x^2 - 8$
• $h(x) = \frac{x + 4}{2}$

1. Find $(f + g)(x)$

$(f + g)(x) = f(x) + g(x) = (2x + 1) + (-5x^2 - 8)$ Simplifying: $(f + g)(x) = -5x^2 + 2x + (1 - 8) = -5x^2 + 2x - 7$

2. Find $(f - g)(x)$

$(f - g)(x) = f(x) - g(x) = (2x + 1) - (-5x^2 - 8)$ Simplifying: $(f - g)(x) = 2x + 1 + 5x^2 + 8 = 5x^2 + 2x + 9$

3. Find $(g - h)(x)$

$(g - h)(x) = g(x) - h(x) = (-5x^2 - 8) - \frac{x + 4}{2}$ To subtract $h(x)$, first rewrite it: $h(x) = \frac{x + 4}{2} = \frac{1}{2}x + 2$ Now subtract: $(g - h)(x) = -5x^2 - 8 - \left(\frac{1}{2}x + 2\right)$ Simplifying: $(g - h)(x) = -5x^2 - \frac{1}{2}x - 10$

4. Find $(g - f)(x)$

$(g - f)(x) = g(x) - f(x) = (-5x^2 - 8) - (2x + 1)$ Simplifying: $(g - f)(x) = -5x^2 - 2x - 9$

5. Find $(h - g)(0)$

First, evaluate $h(0)$ and $g(0)$: $h(0) = \frac{0 + 4}{2} = 2$ $g(0) = -5(0)^2 - 8 = -8$ Now subtract: $(h - g)(0) = h(0) - g(0) = 2 - (-8) = 2 + 8 = 10$

Final Answers:

1. $(f + g)(x) = -5x^2 + 2x - 7$
2. $(f - g)(x) = 5x^2 + 2x + 9$
3. $(g - h)(x) = -5x^2 - \frac{1}{2}x - 10$
4. $(g - f)(x) = -5x^2 - 2x - 9$
5. $(h - g)(0) = 10$

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

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Here are 5 related questions for practice:

1. Find $(f \cdot g)(x)$.
2. Find $(f / g)(x)$, assuming $g(x) \neq 0$.
3. Evaluate $(h + f)(-2)$.
4. Determine the roots of $(g + h)(x) = 0$.
5. Find the derivative of $f(x)$, $g(x)$, and $h(x)$.

Tip: When combining functions, always simplify step by step to avoid mistakes.