Answer
If they are both even, their sum is even.
If they are both odd, their sum is odd.
(proofs in the step-by-step section)
Work Step by Step
$f(x)$ is even if $f(-x)=f(x).$ If so, it is symmetric relative to the y-axis.
$f(x)$ is odd if $f(-x)=-f(x).$ If so, it is symmetric relative to the origin.
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1. Suppose that both f and g are even. Then
$(g+f)(-x)=g(-x)+f(-x)=$ ... both are even ,so
$=g(x)+f(x)$
$=(g+f)(x).$
The sum of g and f is also even.
2. Suppose that both f and g are odd. Then
$(g+f)(-x)=g(-x)+f(-x)=$ ... both are odd ,so
$=-g(x)+[-f(x)]$
$=-[g(x)+f(x)]$
$=-(g+f)(x).$
The sum of g and f is also odd.
