Check whether the function repeats after a horizontal shift of ( pi )
Verify that ( f(x+pi)=f(x) ) for all ( x ) in its domain
Test several values of ( x ) to see if the equality holds
Confirm that no smaller positive shift works for every ( x )
For trigonometric expressions, simplify using identities to see whether a ( pi )-shift leaves the function unchanged
If the function changes sign after a ( pi )-shift, it does not have period ( pi )
If the function returns to the same value after a ( pi )-shift, it has period ( pi )
If the function is defined piecewise, check the shift condition on every piece
If the function is composed of periodic parts, determine whether the combined period is exactly ( pi )