https://r2.erweima.ai/mathbot/image/1726906195111hno08sn5.jpg

The task is to find the domain (miền giá trị) of the given functions in the problem. Let's solve them step by step.

### a) \( y = \log(1 - 2 \cos x) \)

For the logarithmic function \( \log(z) \), the argument \( z \) must satisfy \( z > 0 \). Hence, we need to solve the inequality:

\[
1 - 2 \cos x > 0
\]

This simplifies to:

\[
\cos x < \frac{1}{2}
\]

The cosine function \( \cos x \) ranges from -1 to 1. Therefore, the solution to \( \cos x < \frac{1}{2} \) means that the values of \( x \) must be within the intervals where \( \cos x \) is less than \( \frac{1}{2} \). This occurs when:

\[
x \in \left( \frac{\pi}{3} + 2k\pi, \frac{5\pi}{3} + 2k\pi \right), \, k \in \mathbb{Z}
\]

So, the domain of \( y \) is defined by these intervals.

### b) \( y = \arcsin \left( \log \frac{x}{10} \right) \)

For the arcsine function \( \arcsin(z) \), the argument \( z \) must satisfy \( -1 \leq z \leq 1 \). Hence, we need to solve:

\[
-1 \leq \log \frac{x}{10} \leq 1
\]

This can be rewritten as:

\[
-1 \leq \log x - \log 10 \leq 1
\]

\[
\log 10^{-1} \leq \log \frac{x}{10} \leq \log 10
\]

Exponentiating both sides:

\[
\frac{1}{10} \leq \frac{x}{10} \leq 10
\]

Multiplying through by 10:

\[
1 \leq x \leq 100
\]

So, the domain for \( y \) is \( x \in [1, 100] \).

### c) \( y = \operatorname{arccot}(\sin x) \)

The arccotangent function is defined for any real number, and since the sine function \( \sin x \) is bounded by \( -1 \leq \sin x \leq 1 \), the domain is all real numbers:

\[
x \in \mathbb{R}
\]

### d) \( y = \arctan(e^x) \)

The arctangent function is defined for all real numbers, and since the exponential function \( e^x \) is always positive and defined for all \( x \), the domain of \( y \) is:

\[
x \in \mathbb{R}
\]

### Summary of Domains:
- a) \( x \in \left( \frac{\pi}{3} + 2k\pi, \frac{5\pi}{3} + 2k\pi \right), \, k \in \mathbb{Z} \)
- b) \( x \in [1, 100] \)
- c) \( x \in \mathbb{R} \)
- d) \( x \in \mathbb{R} \)

Would you like further details or explanations?

### Related Questions:
1. How do you solve inequalities involving trigonometric functions?
2. What is the range of the arcsine function?
3. How is the arccotangent function defined for different inputs?
4. How do you solve logarithmic inequalities?
5. What are the properties of the exponential function?

### Tip:
For solving logarithmic inequalities, always check the domain of the logarithm before attempting to solve!

Bài 3. Tìm miền giá trị của hàm số y = log(1 - 2 cos x), y = arcsin(log(x / 10)), y = arccot(sin x), y = arctan(e^x).

Explore how to find the domain of functions involving logarithms, trigonometric functions, and exponentials. This includes problems like y = log(1 - 2 cos x) and y = arctan(e^x). Learn key concepts from algebra and calculus.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation