Given the function \( f(x) = x - 2\cos(x) \), here’s how you can analyze it:

### 1. **Domain**: 
The function is defined for all real values of \(x\) because cosine (\(\cos(x)\)) is defined for all real numbers. Therefore, the domain is:
\[
\text{Domain: } (-\infty, \infty)
\]

### 2. **Range**: 
Since \( \cos(x) \) oscillates between \(-1\) and \(1\), the term \( -2\cos(x) \) oscillates between \(-2\) and \(2\). So, the function \( f(x) = x - 2\cos(x) \) will be slightly shifted depending on the value of \(x\), but as \(x \to \infty\) or \(x \to -\infty\), the effect of \( \cos(x) \) diminishes, and the range will also be all real numbers.

\[
\text{Range: } (-\infty, \infty)
\]

### 3. **Derivative (First derivative)**: 
To find where the function is increasing or decreasing, we differentiate \( f(x) \):
\[
f'(x) = 1 + 2\sin(x)
\]
This derivative tells us about the slope of the function:
- The function is increasing where \( f'(x) > 0 \), which occurs when \( 1 + 2\sin(x) > 0 \).
- The function is decreasing where \( f'(x) < 0 \), which occurs when \( 1 + 2\sin(x) < 0 \).

### 4. **Critical Points**:
Critical points occur when \( f'(x) = 0 \). To find the critical points, solve:
\[
1 + 2\sin(x) = 0
\]
\[
\sin(x) = -\frac{1}{2}
\]
The solutions to this equation are:
\[
x = \arcsin\left(-\frac{1}{2}\right) + 2k\pi \quad \text{or} \quad x = \pi - \arcsin\left(-\frac{1}{2}\right) + 2k\pi, \, k \in \mathbb{Z}
\]
These are the critical points where the function changes from increasing to decreasing or vice versa.

### 5. **Concavity (Second derivative)**: 
To find concavity, compute the second derivative:
\[
f''(x) = 2\cos(x)
\]
- If \( f''(x) > 0 \), the function is concave up.
- If \( f''(x) < 0 \), the function is concave down.

Would you like more details on any specific part of this analysis, or a graphical representation of the function?

Here are some relative questions for further exploration:
1. How do the critical points affect the behavior of the function?
2. What is the behavior of \( f(x) \) as \( x \to \infty \)?
3. What is the significance of the concavity in terms of the function's graph?
4. How can we find the local minima and maxima of this function?
5. How would the function change if the cosine term had a different coefficient?

**Tip**: When differentiating trigonometric functions, remember the chain rule for composite functions, especially if you have more complicated terms inside the trigonometric function.

Explore the function f(x) = x - 2cos(x) with a detailed analysis including its domain, range, derivatives, critical points, and concavity. Understand the behavior of the function and apply key calculus concepts such as the derivative test and concavity test. Suitable for advanced high school students or college-level calculus learners.

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