Find the period of the functions (where [ * ] denotes greatest integer function)
f(x) = `sin 3x + cos^(2) x + |tan x|`

Answer

To find the period of the function \( f(x) = \sin(3x) + \cos^2(x) + |\tan(x)| \), we will analyze the period of each individual component of the function and then determine the overall period.

### Step-by-Step Solution:

1. **Identify the Period of Each Component:**
- **For \( \sin(3x) \):**
- The standard period of \( \sin(x) \) is \( 2\pi \).
- For \( \sin(nx) \), the period is given by \( \frac{2\pi}{n} \).
- Here, \( n = 3 \), so the period of \( \sin(3x) \) is:
\[ \text{Period of } \sin(3x) = \frac{2\pi}{3} \]

2. **For \( \cos^2(x) \):**
- The function \( \cos(x) \) has a period of \( 2\pi \).
- However, \( \cos^2(x) \) can be expressed using the double angle formula:
\[ \cos^2(x) = \frac{1 + \cos(2x)}{2} \]
- The period of \( \cos(2x) \) is \( \frac{2\pi}{2} = \pi \).
- Therefore, the period of \( \cos^2(x) \) is:
\[ \text{Period of } \cos^2(x) = \pi \]

3. **For \( |\tan(x)| \):**
- The function \( \tan(x) \) has a period of \( \pi \).
- The absolute value function does not change the period of \( \tan(x) \).
- Thus, the period of \( |\tan(x)| \) is:
\[ \text{Period of } |\tan(x)| = \pi \]

4. **Determine the Overall Period:**
- Now we have the periods of each component:
- Period of \( \sin(3x) = \frac{2\pi}{3} \)
- Period of \( \cos^2(x) = \pi \)
- Period of \( |\tan(x)| = \pi \)
- To find the overall period of the function \( f(x) \), we need to find the least common multiple (LCM) of the individual periods.
- The periods are \( \frac{2\pi}{3} \) and \( \pi \). We can express \( \pi \) as \( \frac{3\pi}{3} \) for easier calculation.
- Thus, we need to find:
\[ \text{LCM}\left(\frac{2\pi}{3}, \frac{3\pi}{3}\right) \]
- The LCM of the numerators \( 2\pi \) and \( 3\pi \) is \( 6\pi \), and the HCF of the denominators \( 3 \) and \( 3 \) is \( 3 \).
- Therefore, the overall period is:
\[ \text{Period of } f(x) = \frac{6\pi}{3} = 2\pi \]

### Final Answer:
The period of the function \( f(x) = \sin(3x) + \cos^2(x) + |\tan(x)| \) is \( 2\pi \).