Find the range of each of the following functions: (where {.} and [.] represents fractional part and greatest integer part functions respectively)
`f(x)=3|sin x|-4|cos x|`

Explanation

To find the range of the function \( f(x) = 3|\sin x| - 4|\cos x| \), we will analyze the behavior of the sine and cosine functions, particularly their absolute values.

### Step 1: Determine the ranges of \( |\sin x| \) and \( |\cos x| \)

The absolute values of sine and cosine functions have the following ranges:
- \( |\sin x| \) ranges from 0 to 1.
- \( |\cos x| \) also ranges from 0 to 1.

### Step 2: Express the function in terms of the ranges

Substituting the maximum and minimum values of \( |\sin x| \) and \( |\cos x| \) into the function:
- The maximum value of \( 3|\sin x| \) is \( 3 \times 1 = 3 \).
- The minimum value of \( 4|\cos x| \) is \( 4 \times 0 = 0 \).
- The maximum value of \( 4|\cos x| \) is \( 4 \times 1 = 4 \).
- The minimum value of \( 3|\sin x| \) is \( 3 \times 0 = 0 \).

### Step 3: Analyze the function \( f(x) = 3|\sin x| - 4|\cos x| \)

Now we need to find the minimum and maximum values of \( f(x) \):
- The maximum of \( f(x) \) occurs when \( 3|\sin x| \) is maximized and \( 4|\cos x| \) is minimized. This happens when \( |\sin x| = 1 \) and \( |\cos x| = 0 \):
\[ f(x) = 3(1) - 4(0) = 3 \]

- The minimum of \( f(x) \) occurs when \( 3|\sin x| \) is minimized and \( 4|\cos x| \) is maximized. This happens when \( |\sin x| = 0 \) and \( |\cos x| = 1 \):
\[ f(x) = 3(0) - 4(1) = -4 \]

### Step 4: Conclusion

Thus, the range of the function \( f(x) = 3|\sin x| - 4|\cos x| \) is:
\[ [-4, 3] \]