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Case 1 both

*f*and

*g*are even:

I know ##f(x) = f(-x) ## and ##g(x)=g(-x) ## for the domain of the function. I can reason by substitution that

##f(x)+g(x)=f(-x)+g(-x) ##

##(f+g)(x)=(f+g)(-x) ##

##(f+g)(x) ## is even. So far so good.

Case 2 both

*f*and

*g*are odd:

I found that if ##f(x)=-g(x)+c ## then

##(f+g)(x)=c ## which is even.

Otherwise I think that ##(f+g)(x) ## would be odd though I don't know how to assert that.

Case 3

*f*is even and

*g*is odd:

I think that other than is special case where one or both of our functions are zero for all

*x*in the domain ##(f+g)(x) ## would neither be even or odd. I don't know how to prove this.

I know that the notions of even an odd is defined in terms of sets rather than algebraically like I did here. I think if I understood sets better I might have more of a handle on this, I don't know. Hints?