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 $x = 4\pi$.

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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.