Question

Let E be an even function and O be an odd function. Determine the symmetry, if any, of the following function.

E (Dot) O

Answer 1

There should be a dot or a tiny open circle between them but there was no symbol

Answer 2

Alright so we have two functions, \(E(x)\) and \(O(x)\) that are even and odd, respectively.

This means \(E(-x)=E(x)\) and \(O(-x)=-O(x)\).

You want to check whether \((E\circ O)(x)\) is even or odd.
\[\begin{align*}
(E\circ O)(x)&=E(O(x))\\
(E\circ O)(-x)&=E(O(-x))\\
&=E(-O(x))&\text{since }O\text{ is odd}\\
&=E(O(x))&\text{since }E\text{ is even}
\end{align*}\]
Therefore the composition \(E\circ O\) is even.