Question

show that if f is any function, then the function O defined by
O(x)=f(x)-f(-x)all over 2 is odd

Answer 1

Aren't there some restrictions on the sort of function f(x) is? Suppose f(x)=4. Then f(-x)=4 also, and (f(x)-f(-x))/2 = (4-4)/2 = 0, and that ISN'T odd. Another one: put f(x)=x², then f(-x)=x², and (f(x)-f(-x))/2=0 again. Same problem: something's wrong here.

Answer 2

Also consider f(x)=sin(x). f(-x)=sin(-x)=-sin(x), so O(x)=(sin(x)+sin(x))/2=sin(x) - and for most values of x that isn't even an integer, let alone an odd one.