If the denominator is even, the function is not defined for negative x. If the numerator is even, then the function is not one-to-one. For example, x2/3= (x1/3)2 so if x= 8, (8)2/3= 22= 4 and if x= -8, (-8)2/3= (-2)2= 4.
xodd/odd is defined for all x and is one-to-one. xeven/odd is defined for all x and is not one-to-one. xodd/even is defined only for non-negative x and is one-to-one. Since a number is even if and only if it has a factor of 2, with xeven/even you can cancel 2s until you have one of the first three cases.