The period of the function `f (x) =| sin ((x)/(2))|+| cos x | is `

Options

A. `pi`
B. `2 pi`
C. `pi//2`
D. none

Answer

To find the period of the function \( f(x) = | \sin \left( \frac{x}{2} \right) | + | \cos x | \), we will analyze the individual components of the function and then determine the overall period.

### Step 1: Identify the periods of the individual functions
1. **Period of \( | \sin \left( \frac{x}{2} \right) | \)**:
- The function \( \sin \left( \frac{x}{2} \right) \) has a period of \( 2\pi \) because the standard period of \( \sin x \) is \( 2\pi \), and the factor of \( \frac{1}{2} \) stretches the period by a factor of 2.
- Therefore, the period of \( | \sin \left( \frac{x}{2} \right) | \) is also \( 2\pi \).

2. **Period of \( | \cos x | \)**:
- The function \( \cos x \) has a period of \( 2\pi \).
- Since the absolute value does not change the period, the period of \( | \cos x | \) is also \( 2\pi \).

### Step 2: Determine the overall period of the function
- The overall period of \( f(x) \) will be the least common multiple (LCM) of the periods of its components.
- Both components \( | \sin \left( \frac{x}{2} \right) | \) and \( | \cos x | \) have a period of \( 2\pi \).

### Step 3: Conclusion
- Since both components have the same period, the period of the function \( f(x) = | \sin \left( \frac{x}{2} \right) | + | \cos x | \) is \( 2\pi \).

Thus, the period of the function is \( 2\pi \).

### Final Answer:
The period of the function \( f(x) = | \sin \left( \frac{x}{2} \right) | + | \cos x | \) is \( 2\pi \).

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