Determine where each of the following functions are increasing and decreasing.

1. Let . Then .
   Since for all x except x = 0, the function is increasing for all x except x = 0.
   

             
   
   

2. Let . Then .
   Since over the intervals (-π, 0), (π, 2π), and (3π, 4π), the function is increasing over those intervals.
   As over the intervals (0, π), (2π, 3π), and (4π, 5π) the function is decreasing over those intervals.
   

             
   
   

3. Let . Then .
   Since for all x, the function is increasing for all x.
   

             
   
   

4. Let . Then .
   Since for all x > 0, the function is increasing over this interval.
   Note that the ln x is not defined for x ≤ 0.

             

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