https://file.aiquickdraw.com/mathbot/file/1724235990940cuo85ofr.jpg

perform the following fundamental operation with solution

To solve the problems in the image, I'll first extract the mathematical expressions, then perform the operations as requested.

### Extracted Problems:

Given two functions \( f(x) \) and \( g(x) \), you need to perform the following operations:

1. \( f(x) + g(x) \)
2. \( f(x) - g(x) \)
3. \( f(x) \cdot g(x) \)
4. \( \frac{f(x)}{g(x)} \)
5. \( g(x) + f(x) \)
6. \( f(x) - g(x) \)
7. \( g(x) - f(x) \)
8. \( g(x) \cdot f(x) \)
9. \( \frac{g(x)}{f(x)} \)
10. \( \frac{f(x)}{g(x)} \)

The specific functions are:
- \( f(x) = 2 \sin x \)
- \( g(x) = \frac{1}{x} \)

### Solutions:

1. **Addition:**
   \[
   f(x) + g(x) = 2\sin x + \frac{1}{x}
   \]
   
2. **Subtraction ( \(f(x) - g(x)\) ):**
   \[
   f(x) - g(x) = 2\sin x - \frac{1}{x}
   \]
   
3. **Multiplication:**
   \[
   f(x) \cdot g(x) = 2\sin x \cdot \frac{1}{x} = \frac{2\sin x}{x}
   \]
   
4. **Division:**
   \[
   \frac{f(x)}{g(x)} = \frac{2\sin x}{\frac{1}{x}} = 2x \sin x
   \]
   
5. **Addition ( \(g(x) + f(x)\) ):**
   \[
   g(x) + f(x) = \frac{1}{x} + 2\sin x
   \]
   
6. **Subtraction ( \(f(x) - g(x)\) ):**
   \[
   f(x) - g(x) = 2\sin x - \frac{1}{x}
   \]
   
7. **Subtraction ( \(g(x) - f(x)\) ):**
   \[
   g(x) - f(x) = \frac{1}{x} - 2\sin x
   \]
   
8. **Multiplication:**
   \[
   g(x) \cdot f(x) = \frac{1}{x} \cdot 2\sin x = \frac{2\sin x}{x}
   \]
   
9. **Division ( \(g(x) \div f(x)\) ):**
   \[
   \frac{g(x)}{f(x)} = \frac{\frac{1}{x}}{2\sin x} = \frac{1}{2x\sin x}
   \]
   
10. **Division ( \(f(x) \div g(x)\) ):**
    \[
    \frac{f(x)}{g(x)} = \frac{2\sin x}{\frac{1}{x}} = 2x\sin x
    \]

Would you like any further details or have any questions? Here are some related questions you might consider:

1. How would the operations change if \( f(x) \) and \( g(x) \) were different functions?
2. Can you compute the derivative of \( f(x) \) and \( g(x) \)?
3. How would the addition \( f(x) + g(x) \) affect the graph of the resulting function?
4. What happens to the quotient \( \frac{f(x)}{g(x)} \) as \( x \) approaches zero?
5. How can you apply these operations in solving real-world problems?

**Tip:** When performing operations with trigonometric and algebraic functions, always consider their domains to avoid undefined expressions.

Explore how to perform fundamental mathematical operations with the functions f(x) = 2sin x and g(x) = 1/x. Learn to add, subtract, multiply, and divide these functions, including detailed step-by-step solutions. Suitable for advanced high school students.

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