Let f(x) = 2x + 1 , g(x) = - 5x ^ 2 - 8 and h(x) = (x + 4)/2

1. Find (f + g)(x)

2. Find (f - g)(x)

3. Find (g - h)(x)

4. Find (g - f)(x)

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

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.

Let f(x) = 2x + 1, g(x) = -5x^2 - 8, and h(x) = (x + 4)/2. Find (f + g)(x), (f - g)(x), (g - h)(x), (g - f)(x), and (h - g)(0).

Explore the process of adding, subtracting, and evaluating functions through the example of f(x) = 2x + 1, g(x) = -5x^2 - 8, and h(x) = (x + 4)/2. Detailed step-by-step solutions are provided, suitable for students in Grades 9-11.

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