User:Egm6322.s12.team2.Xia/RP8.5

=R*8.5 -Even or Odd Function =

Given
Consider the functions $$\displaystyle g_{i}:\,\mathbb{R}\rightarrow\mathbb{R}$$ and the function $$\displaystyle f=\sum_{i}g_{i}\,:\,\mathbb{R}\rightarrow\mathbb{R}$$

(8.5.1)

Find
Show that: 1. If $$\displaystyle g_{i}$$ is odd, then $$\displaystyle f $$ is odd. 2. If $$\displaystyle g_{i}$$ is even, then $$\displaystyle f$$  is even.

Solution
1. If $$\displaystyle g_{i} $$ is odd, we have $$\displaystyle g_{i}(-x)=-g_{i}(x) $$ then $$\displaystyle {\color{blue}f(-x)}=\sum_{i}g_{i}(-x)=g_{1}(-x)+g_{2}(-x)+\cdots+g_{i}(-x)$$ $$\displaystyle =-g_{1}(x)-g_{2}(x)-\cdots-g_{i}(x)=-\sum_{i}g_{i}(x)={\color{blue}-f(x)}$$ thus $$\displaystyle f$$ is odd. 2. If $$\displaystyle g_{i}$$ is even, we have $$\displaystyle g_{i}(-x)=g_{i}(x)$$ then $$\displaystyle {\color{blue}f(-x)}=\sum_{i}g_{i}(-x)=g_{1}(-x)+g_{2}(-x)+\cdots+g_{i}(-x)$$ $$\displaystyle =g_{1}(x)+g_{2}(x)+\cdots+g_{i}(x)=\sum_{i}g_{i}(x)={\color{blue}f(x)}$$ thus $$\displaystyle f$$ is even.