https://file.aiquickdraw.com/mathbot/file/1724260512700corglheq.jpg

Let's solve the problem step by step. The image contains three sets of functions \( f(x) \) and \( g(x) \). We need to find the following for each pair:

1. \( (f + g)(x) = f(x) + g(x) \)
2. \( (f - g)(x) = f(x) - g(x) \)
3. \( (fg)(x) = f(x) \cdot g(x) \)
4. \( \frac{f}{g}(x) = \frac{f(x)}{g(x)} \)

Additionally, we need to determine the domain for each function operation. Let's start solving for each pair of functions.

### 1. \( f(x) = 2x + 3 \) and \( g(x) = x - 1 \)
- **Sum**: 
  \[
  (f + g)(x) = (2x + 3) + (x - 1) = 3x + 2
  \]
  - **Domain**: \( (-\infty, \infty) \) (since both \( f(x) \) and \( g(x) \) are linear functions)

- **Difference**: 
  \[
  (f - g)(x) = (2x + 3) - (x - 1) = x + 4
  \]
  - **Domain**: \( (-\infty, \infty) \)

- **Product**: 
  \[
  (fg)(x) = (2x + 3)(x - 1) = 2x^2 - 2x + 3x - 3 = 2x^2 + x - 3
  \]
  - **Domain**: \( (-\infty, \infty) \)

- **Quotient**: 
  \[
  \frac{f}{g}(x) = \frac{2x + 3}{x - 1}
  \]
  - **Domain**: \( x \neq 1 \) (since division by zero is undefined)

### 2. \( f(x) = x - 6 \) and \( g(x) = 5x^2 \)
- **Sum**: 
  \[
  (f + g)(x) = (x - 6) + 5x^2 = 5x^2 + x - 6
  \]
  - **Domain**: \( (-\infty, \infty) \)

- **Difference**: 
  \[
  (f - g)(x) = (x - 6) - 5x^2 = -5x^2 + x - 6
  \]
  - **Domain**: \( (-\infty, \infty) \)

- **Product**: 
  \[
  (fg)(x) = (x - 6)(5x^2) = 5x^3 - 30x^2
  \]
  - **Domain**: \( (-\infty, \infty) \)

- **Quotient**: 
  \[
  \frac{f}{g}(x) = \frac{x - 6}{5x^2}
  \]
  - **Domain**: \( x \neq 0 \) (since division by zero is undefined)

### 3. \( f(x) = x^2 - x - 3 \) and \( g(x) = x + 1 \)
- **Sum**: 
  \[
  (f + g)(x) = (x^2 - x - 3) + (x + 1) = x^2 - 2x - 2
  \]
  - **Domain**: \( (-\infty, \infty) \)

- **Difference**: 
  \[
  (f - g)(x) = (x^2 - x - 3) - (x + 1) = x^2 - 2x - 4
  \]
  - **Domain**: \( (-\infty, \infty) \)

- **Product**: 
  \[
  (fg)(x) = (x^2 - x - 3)(x + 1) = x^3 + x^2 - x^2 - x - 3x - 3 = x^3 - 4x - 3
  \]
  - **Domain**: \( (-\infty, \infty) \)

- **Quotient**: 
  \[
  \frac{f}{g}(x) = \frac{x^2 - x - 3}{x + 1}
  \]
  - **Domain**: \( x \neq -1 \) (since division by zero is undefined)

### Summary of Domains:
1. For the first pair: \( (-\infty, \infty) \) except \( x \neq 1 \) for the quotient.
2. For the second pair: \( (-\infty, \infty) \) except \( x \neq 0 \) for the quotient.
3. For the third pair: \( (-\infty, \infty) \) except \( x \neq -1 \) for the quotient.

Would you like further details or explanations on any of these steps?

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Here are five related questions for practice:

1. How would the domains change if the functions involved square roots?
2. What would be the result of \( f(g(x)) \) for any of the given pairs?
3. How do you determine the domain of a function composed of two other functions?
4. What is the significance of the domain in the context of function operations?
5. How would you find the inverse of the sum of two functions?

**Tip:** Always carefully check for restrictions in the domain, especially when performing division or working with square roots.

Learn how to solve functions step by step: sum, difference, product, and quotient. This problem covers key algebraic concepts, including operations on functions and their domains. Suitable for high school students.

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