Find the fundamental period of the following function:
`f(x)=sin 3x+cos^(2)x+|tanx|`

Explanation

To find the fundamental period of the function \( f(x) = \sin(3x) + \cos^2(x) + |\tan(x)| \), we will analyze the periods of each component of the function separately and then find the least common multiple (LCM) of these periods.

### Step 1: Determine the period of \( \sin(3x) \)

The standard period of the sine function \( \sin(x) \) is \( 2\pi \). When the function is modified to \( \sin(kx) \), the period becomes:

\[ \text{Period of } \sin(kx) = \frac{2\pi}{|k|} \]

For \( \sin(3x) \):

\[ \text{Period of } \sin(3x) = \frac{2\pi}{3} \]

### Step 2: Determine the period of \( \cos^2(x) \)

The cosine function \( \cos(x) \) has a period of \( 2\pi \). However, since we are dealing with \( \cos^2(x) \), we need to find its period. The function \( \cos^2(x) \) can be rewritten using the double angle formula:

\[ \cos^2(x) = \frac{1 + \cos(2x)}{2} \]

The period of \( \cos(2x) \) is \( \frac{2\pi}{2} = \pi \). Thus, the period of \( \cos^2(x) \) is:

\[ \text{Period of } \cos^2(x) = \pi \]

### Step 3: Determine the period of \( |\tan(x)| \)

The tangent function \( \tan(x) \) has a period of \( \pi \). Since the absolute value does not change the period, the period of \( |\tan(x)| \) remains:

\[ \text{Period of } |\tan(x)| = \pi \]

### Step 4: Find the least common multiple (LCM) of the periods

Now we have the periods of each component:

- \( \text{Period of } \sin(3x) = \frac{2\pi}{3} \)
- \( \text{Period of } \cos^2(x) = \pi \)
- \( \text{Period of } |\tan(x)| = \pi \)

To find the fundamental period of the function \( f(x) \), we need to find the LCM of these periods:

1. Convert \( \pi \) to a fraction for LCM calculation:
- \( \pi = \frac{3\pi}{3} \)
- \( \pi = \frac{3\pi}{3} \)

2. Now we have:
- \( \frac{2\pi}{3}, \frac{3\pi}{3}, \frac{3\pi}{3} \)

3. The LCM of the numerators \( 2, 3, 3 \) is \( 6 \).

4. Therefore, the LCM of the periods is:

\[ \text{LCM} = \frac{6\pi}{3} = 2\pi \]

### Conclusion

The fundamental period of the function \( f(x) = \sin(3x) + \cos^2(x) + |\tan(x)| \) is:

\[ \boxed{2\pi} \]