Given algebraic proofs forodd and even functions:

1. even times = even; odd times odd = even; even times odd = odd;
2. the derivative of an even function is odd; the derivative of an odd function is even.

Even functionsand odd functionsare functions which satisfy particular harmony relations, with respect to taking cumulative inverses.

However, if end up with the exact same function that started with (that is if f(-x)=f(x) so all of the signs are the same). However, If end up with the exact contrary of what started with (that is. if f(-x)=-f(x), so all of the signs are switched)

The given odd and even functions are

1. even times = even; odd times odd = even, even times odd = odd
2. the derivative of an even function is odd; the derivative of an odd function is even.

Let h(x) = f(x)g(x).

If f and g are both even then:

h(-x) = f(-x)g(-x)

=f(x)g(x)

=h(x)

If is even and is odd:

h(-x) = f(-x)g(-x)

=f(x)(-g(x))

= -f(x)g(x)

= -h(x)

If both and are odd:

h(-x) = f(-x)g(-x)

=(-f(x))(-g(x))

=f(x)g(x)

=h(x)

If the function is even then

Here, the eveness of the function is used.

If the function is odd then

Here, the oddness of the function is used.

Thus, the odd and even function are algebraically proved.