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Let's go through the problem step by step to determine the domain of each function. The given functions are:

1. \( f(x) = 2 \sin(2x) \)
2. \( h(x) = \cos(2x) \)
3. \( m(x) = \operatorname{arccot}(x) \), where \( y \in (0, \pi) \)

We need to determine the domain of the following expressions:

### Part (a) \( (f + h)(x) \)

- **\( f(x) = 2 \sin(2x) \):** The sine function is defined for all real numbers, so \( f(x) \) is defined for all \( x \in \mathbb{R} \).

- **\( h(x) = \cos(2x) \):** The cosine function is also defined for all real numbers, so \( h(x) \) is defined for all \( x \in \mathbb{R} \).

- **\( (f + h)(x) = 2 \sin(2x) + \cos(2x) \):** Since both functions are defined for all \( x \in \mathbb{R} \), the sum \( f(x) + h(x) \) is also defined for all \( x \in \mathbb{R} \).

**Domain of \( (f + h)(x) \):** \( x \in \mathbb{R} \).

### Part (b) \( 3m(x) \)

- **\( m(x) = \operatorname{arccot}(x) \):** The arccotangent function \( \operatorname{arccot}(x) \) is defined for all real numbers \( x \). Its range is given as \( y \in (0, \pi) \), but this does not affect the domain.

- **\( 3m(x) = 3 \operatorname{arccot}(x) \):** Since \( m(x) \) is defined for all \( x \in \mathbb{R} \), the function \( 3m(x) \) is also defined for all \( x \in \mathbb{R} \).

**Domain of \( 3m(x) \):** \( x \in \mathbb{R} \).

### Part (c) \( \left(\frac{f}{h}\right)(x) \)

- **\( \frac{f(x)}{h(x)} = \frac{2 \sin(2x)}{\cos(2x)} \):** This fraction is defined as long as the denominator \( h(x) = \cos(2x) \neq 0 \).

  - \( \cos(2x) = 0 \) when \( 2x = \frac{\pi}{2} + n\pi \) for any integer \( n \).
  - Solving for \( x \), we get \( x = \frac{\pi}{4} + \frac{n\pi}{2} \).

Therefore, the values of \( x \) where the function is undefined are \( x = \frac{\pi}{4} + \frac{n\pi}{2} \) for any integer \( n \).

**Domain of \( \left(\frac{f}{h}\right)(x) \):** \( x \in \mathbb{R} \setminus \left\{ \frac{\pi}{4} + \frac{n\pi}{2} \mid n \in \mathbb{Z} \right\} \).

### Summary of Domains
- **(a) \( (f + h)(x) \):** \( x \in \mathbb{R} \)
- **(b) \( 3m(x) \):** \( x \in \mathbb{R} \)
- **(c) \( \left(\frac{f}{h}\right)(x) \):** \( x \in \mathbb{R} \setminus \left\{ \frac{\pi}{4} + \frac{n\pi}{2} \mid n \in \mathbb{Z} \right\} \)

Would you like further details on any of these steps, or do you have any questions?

### Related Questions:
1. What are the properties of the arccotangent function?
2. How would the domain change if the range of \( m(x) \) were different?
3. Can the sine and cosine functions ever be undefined?
4. How do you derive the values for which \( \cos(2x) = 0 \)?
5. What happens if we change the trigonometric functions in \( f(x) \) and \( h(x) \)?

**Tip:** When finding the domain of a fraction, always check when the denominator is zero to find points of discontinuity.

Determine the domain of the following expressions given the functions f(x) = 2 sin(2x), h(x) = cos(2x), and m(x) = arccot(x) where y ∈ (0, π):
(a) (f + h)(x)
(b) 3m(x)
(c) (f/h)(x)

Explore how to determine the domain of trigonometric and inverse trigonometric functions such as f(x) = 2 sin(2x), h(x) = cos(2x), and m(x) = arccot(x). This step-by-step guide covers the principles behind combining functions, identifying domain restrictions, and solving for undefined values in trigonometric expressions.

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