Exercise 1.1.87

Question

If $f$ and $g$ are both even functions, is $f +g$ even? If $f$ and $g$ are both odd functions, is $f+g$ odd? What if $f$ is even and $g$ is odd? Justify your answers.

Khagan Gulusoy · Accepted Answer

Recall that the sum $f+g$ of two functions $f:X 	o \mathbb{R}$ and $g:X 	o \mathbb{R}$ is defined by
$$\forall x\in X:\quad(f+g)(x)=f(x)+g(x).$$
\begin{itemize}
    \item 	extit{(if $f$ and $g$ are both even functions, is $f +g$ even)} We have, for all $x \in X$:
    $$(f+g)(-x)=f(-x)+g(-x)=f(x)+g(x)=(f+g)(x).$$
    Thus, $f+g$ is even.
    \item 	extit{(if $f$ and $g$ are both odd functions, is $f +g$ odd)} We have, for all $x\in X$:
    $$(f+g)(-x)=f(-x)+g(-x)=-f(x)-g(x)=-(f(x)+g(x))=-(f+g)(x).$$
    Thus, $f+g$ is odd.
\end{itemize}

Exercise 1.1.87

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Exercise 1.1.87

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