Surely, a debatable title. But I was in high school, give me a break.

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f(x) and g(x) are odd, so: 

given: -f(x) = f(-x) and -g(x)=g(-x) 

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

Therefore: the sum is odd
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f(x) and g(x) are even, so:

f(-x) = f(x) and g(-x) =g(x)
(f*g)(x)=f(x)*g(x) 
(f*g)(x)=f(-x)*g(-x)
(f*g)(x)=(f*g)(-x) 

Therefore they are even. 

f(x) and g(x) are odd, so:
f(-x)=-f(x) and g(-x)=-g(x)
(f*g)(x)=f(x)*g(x)
(f*g)(-x)=f(-x)*g(-x)
(f*g)(-x)=-f(x)*-g(x) Product of two negative functions is positive.