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.