The function you've provided is \( y = 1 - \cos\left(\frac{x}{2}\right) \).

### Analysis:

1. **Domain**: The domain of the function is all real numbers, \( x \in \mathbb{R} \), since cosine is defined for all real numbers.

2. **Range**: The cosine function, \( \cos\left(\frac{x}{2}\right) \), ranges from -1 to 1. Therefore, the range of \( y = 1 - \cos\left(\frac{x}{2}\right) \) will be:
   \[
   y \in [0, 2]
   \]
   This is because \( y \) will be maximized when \( \cos\left(\frac{x}{2}\right) = -1 \) and minimized when \( \cos\left(\frac{x}{2}\right) = 1 \).

3. **Period**: The period of the cosine function \( \cos\left(\frac{x}{2}\right) \) is \( 4\pi \) since the period of \( \cos(x) \) is \( 2\pi \) and the coefficient inside the cosine function is \( \frac{1}{2} \).

4. **Amplitude**: The amplitude of the original cosine function is 1, but since the cosine is subtracted from 1, it doesn't have a traditional amplitude. Instead, we observe that the oscillation occurs between the values of 0 and 2.

5. **Graph**: The graph of \( y = 1 - \cos\left(\frac{x}{2}\right) \) will be a vertically shifted and reflected version of the standard cosine graph. The graph starts at \( y = 0 \), reaches a maximum of 2 at \( x = 2\pi \), and returns to 0 at \( x = 4\pi \).

Would you like to see a graph or need further details?

### Related Questions:
1. How does the function \( y = 1 - \cos\left(\frac{x}{2}\right) \) compare to the standard cosine function in terms of graph transformation?
2. What are the intercepts of the function \( y = 1 - \cos\left(\frac{x}{2}\right) \)?
3. How would the function change if the coefficient inside the cosine were altered, e.g., \( \cos\left(\frac{x}{3}\right) \)?
4. Can you find the derivative of \( y = 1 - \cos\left(\frac{x}{2}\right) \)?
5. How does this function behave as \( x \) approaches infinity?

**Tip:** The function \( y = 1 - \cos\left(\frac{x}{2}\right) \) can be analyzed using trigonometric identities and transformations to understand its behavior better, particularly when solving related problems in calculus.

Explore the function y = 1 - cos(x/2) with detailed analysis of its domain, range, period, and graph. This trigonometric function demonstrates transformations and is relevant for high school students studying advanced functions and trigonometry.

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