https://r2.erweima.ai/mathbot/image/1726562463720qd8662if.jpg

In this image, the task is to find the domain of the following functions:

**a) \( y = \sqrt{2 \, \text{arccot} \, x - \pi} \)**

For this function to be defined, the expression inside the square root must be non-negative because the square root is only defined for non-negative numbers in the set of real numbers. Additionally, the arccotangent function is defined for all real values of \( x \).

Steps:
1. \( 2 \, \text{arccot} \, x - \pi \geq 0 \)
2. Solve this inequality for \( x \).

**b) \( y = \arcsin \left( \frac{2x}{1 + x} \right) \)**

The arcsine function, \( \arcsin(z) \), is defined only for \( z \in [-1, 1] \). Thus, the expression \( \frac{2x}{1+x} \) must lie between -1 and 1.

Steps:
1. Solve the inequality \( -1 \leq \frac{2x}{1 + x} \leq 1 \) to find the domain of \( x \).

**c) \( y = \frac{\sqrt{x}}{\sin(\pi x)} \)**

For this function, two conditions must be met:
1. \( \sqrt{x} \) is defined for \( x \geq 0 \).
2. The sine function, \( \sin(\pi x) \), must not be zero to avoid division by zero.

Steps:
1. Solve for \( x \geq 0 \).
2. Solve for \( \sin(\pi x) \neq 0 \), which occurs when \( x \neq n \) where \( n \) is any integer.

**d) \( y = \arccos(\sin x) \)**

The arccosine function, \( \arccos(z) \), is defined for \( z \in [-1, 1] \), and since the sine function always takes values in the interval \( [-1, 1] \), \( \arccos(\sin x) \) is defined for all real \( x \).

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Would you like me to solve each of these domains in detail?

Here are 5 related questions:
1. What are the properties of the arccotangent and arcsine functions?
2. How do you solve inequalities involving trigonometric functions?
3. What happens when sine or cosine functions cause division by zero?
4. Can the domain of a function change if it's composed with a square root?
5. How can we analyze piecewise functions involving trigonometric inverses?

**Tip**: When working with trigonometric functions and their inverses, always start by considering the range of values for which they are defined.

Tìm tập xác định của các hàm số:

a) y = √(2 arccot x - π)
b) y = arcsin(2x / (1 + x))
c) y = √x / sin(πx)
d) y = arccos(sin(x))

Learn how to determine the domains of functions involving trigonometric and inverse trigonometric functions. This problem set includes domain analysis for square roots, arccotangent, arcsine, and conditions for avoiding division by zero.

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