1. odd+odd = even:
The result of adding two odd functions is an even function. This is because two odd function images are symmetrical about the origin, and this symmetry is maintained after addition, and an even function symmetrical about the origin is obtained.
2. Odd number x odd number = odd number:
The result of the multiplication of two odd function is still a odd function. This is because the values of odd function on the axis of symmetry are equal, and the values on the axis of symmetry after the multiplication of two odd function are also equal, which keeps the properties of odd function.
3. Odd+even = uncertain:
The result of adding odd function and even function has no definite property. It depends on the specific function form and domain.
4. Odd x even = even:
The result of multiplying an odd function by an even function is an even function. This is because odd function values are equal on the axis of symmetry, and even function values are zero on the axis of symmetry. After multiplication, the value of even function is still zero on the axis of symmetry, and an even function is obtained.
5. Even number+even number = even number:
The result of adding two even functions is an even function. This is because the images of two even functions are symmetrical about the origin, and this symmetry is maintained after addition, and an even function symmetrical about the origin is obtained.
6. Even X Even = Even:
The result of multiplication of two even functions is still an even function. This is because the value of even function on the symmetry axis is zero, and the value on the symmetry axis is still zero after multiplication, which maintains the properties of even function.
It should be noted that the above laws are valid for general functions, and there may be special cases for specific function forms and domains.