Answer
Given below.
Work Step by Step
We know that an even function equates $ f\left( -x \right)=f\left( x \right)$, while an odd function satisfies $ f\left( -x \right)=-f\left( x \right)$.
If $f,g$ are even functions, then the sum is defined by
$(f+g) (-x)= f(-x)+g(-x)=[f(x)] + [g(x)]= (f+g) (x)$.
That is, the sum is an even function.
Similarly; if $f,g$ are odd functions, then the sum is defined by
$(f+g) (-x)= f(-x)+g(-x)=[-f(x)] + [-g(x)]= -[f(x)+g(x)]= -(f+g) (x)$.
That is, the sum is an odd function.
