Assignment: Answer 1, 3, 5, 7, 9. Just refer to a and b. No need to find the domain.
 

a. (f + g)(x)

b. (f - g)(x)



1. f(x) = x + 1 g(x) = 2x - 3

2. f(x) = 3x - 1 g(x) = 4 - 2x

3. f(x) = x ^ 2 + 1 g(x) = x - 3

4. f(x) = 2x ^ 2 - 3x g(x) = x ^ 2 - 1

5. f(x) = x - 2 g(x) = 1/x

6. f(x) = x ^ 2 g(x) = 1/(2x)

7. f(x) = 3x ^ 2 + 1 g(x) = x - 3

8. f(x) = x ^ 2; g(x) = 2x ^ 2

9. f(x) = 2/x g(x) = 1/x

10. f(x) = x g(x) = sqrt(2x)

Sure, let's solve the required problems for \( (f+g)(x) \) and \( (f-g)(x) \).

### Problem 1
Given:
\( f(x) = x + 1 \)
\( g(x) = 2x - 3 \)

a. \( (f + g)(x) \)
\[ (f + g)(x) = f(x) + g(x) = (x + 1) + (2x - 3) = 3x - 2 \]

b. \( (f - g)(x) \)
\[ (f - g)(x) = f(x) - g(x) = (x + 1) - (2x - 3) = x + 1 - 2x + 3 = -x + 4 \]

### Problem 3
Given:
\( f(x) = x^2 + 1 \)
\( g(x) = x - 3 \)

a. \( (f + g)(x) \)
\[ (f + g)(x) = f(x) + g(x) = (x^2 + 1) + (x - 3) = x^2 + x - 2 \]

b. \( (f - g)(x) \)
\[ (f - g)(x) = f(x) - g(x) = (x^2 + 1) - (x - 3) = x^2 + 1 - x + 3 = x^2 - x + 4 \]

### Problem 5
Given:
\( f(x) = x - 2 \)
\( g(x) = \frac{1}{x} \)

a. \( (f + g)(x) \)
\[ (f + g)(x) = f(x) + g(x) = (x - 2) + \frac{1}{x} = x - 2 + \frac{1}{x} \]

b. \( (f - g)(x) \)
\[ (f - g)(x) = f(x) - g(x) = (x - 2) - \frac{1}{x} = x - 2 - \frac{1}{x} \]

### Problem 7
Given:
\( f(x) = 3x^2 + 1 \)
\( g(x) = x - 3 \)

a. \( (f + g)(x) \)
\[ (f + g)(x) = f(x) + g(x) = (3x^2 + 1) + (x - 3) = 3x^2 + x - 2 \]

b. \( (f - g)(x) \)
\[ (f - g)(x) = f(x) - g(x) = (3x^2 + 1) - (x - 3) = 3x^2 + 1 - x + 3 = 3x^2 - x + 4 \]

### Problem 9
Given:
\( f(x) = \frac{2}{x} \)
\( g(x) = \frac{1}{x} \)

a. \( (f + g)(x) \)
\[ (f + g)(x) = f(x) + g(x) = \frac{2}{x} + \frac{1}{x} = \frac{2 + 1}{x} = \frac{3}{x} \]

b. \( (f - g)(x) \)
\[ (f - g)(x) = f(x) - g(x) = \frac{2}{x} - \frac{1}{x} = \frac{2 - 1}{x} = \frac{1}{x} \]

Do you want any further details or have any questions?

Here are some related questions for further practice:

1. What is the result of \( (f \cdot g)(x) \) for \( f(x) = x + 1 \) and \( g(x) = 2x - 3 \)?
2. How would you find \( \left(\frac{f}{g}\right)(x) \) for \( f(x) = 3x - 1 \) and \( g(x) = 4 - 2x \)?
3. Determine \( (f \cdot g)(x) \) for \( f(x) = x^2 + 1 \) and \( g(x) = x - 3 \).
4. Calculate \( \left(\frac{f}{g}\right)(x) \) for \( f(x) = 2x^2 - 3x \) and \( g(x) = x^2 - 1 \).
5. What is the composition \( f(g(x)) \) for \( f(x) = x - 2 \) and \( g(x) = \frac{1}{x} \)?
6. Find \( g(f(x)) \) for \( f(x) = x^2 \) and \( g(x) = \frac{1}{2x} \).
7. Determine the derivative of \( (f + g)(x) \) for \( f(x) = 3x^2 + 1 \) and \( g(x) = x - 3 \).
8. What is the integral of \( (f - g)(x) \) for \( f(x) = 2/x \) and \( g(x) = 1/x \)?

**Tip:** When combining functions, make sure to correctly apply the distributive property and combine like terms.

Explore how to solve function addition and subtraction problems step-by-step with examples. This page covers adding and subtracting functions such as (f + g)(x) and (f - g)(x), using specific function pairs like f(x) = x + 1 and g(x) = 2x - 3. Suitable for high school students learning about functions and algebraic operations.

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