# [SOLVED]make a function even and odd

#### dwsmith

##### Well-known member
There was a question but I figured it out.
$$g(\theta) = \begin{cases} \theta, & 0\leq\theta\leq\pi\\ \theta+\pi, & -\pi\leq\theta < 0 \end{cases}$$
So $g_e=\frac{g(\theta)+g(-\theta)}{2}$ and $g_o=\frac{g(\theta)-g(-\theta)}{2}$
\begin{alignat}{3}
g_e & = & \frac{\begin{cases}
\theta, & 0\leq\theta\leq\pi\\
\theta+\pi, & -\pi\leq\theta < 0
\end{cases}+\begin{cases}
-\theta, & 0\leq -\theta\leq\pi\\
-\theta+\pi, & -\pi\leq -\theta < 0
\end{cases}}{2}\\
& = & \frac{\begin{cases}
\theta, & 0\leq\theta\leq\pi\\
\theta+\pi, & -\pi\leq\theta < 0
\end{cases}+\begin{cases}
-\theta, & 0\geq \theta\geq -\pi\\
-\theta+\pi, & \pi\geq \theta > 0
\end{cases}}{2}\\
& = & \frac{\pi}{2}
\end{alignat}
For $g_o$, we have
\begin{alignat*}{3}
g_o & = & \frac{\begin{cases}
\theta, & 0\leq\theta\leq\pi\\
\theta + \pi, & -\pi\leq\theta < 0
\end{cases} -
\begin{cases}
-\theta, & 0\leq -\theta\leq\pi\\
-\theta + \pi, & -\pi\leq -\theta < 0
\end{cases}}{2}\\
& = & \frac{\begin{cases}
\theta, & 0\leq\theta\leq\pi\\
\theta + \pi, & -\pi\leq\theta < 0
\end{cases} +
\begin{cases}
\theta, & 0\geq \theta\geq -\pi\\
\theta - \pi, & \pi\geq \theta > 0
\end{cases}}{2}\\
& = & \theta +
\begin{cases} -\frac{\pi}{2}, & 0\leq\theta\leq\pi\\
\frac{\pi}{2}, & -\pi\leq\theta < 0
\end{cases}
\end{alignat*}

#### HallsofIvy

##### Well-known member
MHB Math Helper
It should be emphasized that you are NOT "making" the function "even" or "odd", you are separating it into its "even" and "odd' parts.

For any function, f(x), $$f_e(x)=\frac{f(x)+ f(-x)}{2}$$ is an even function, $$f_o(x)= \frac{f(x)- f(-x)}{2}$$ is an odd function, and $$f(x)= f_e(x)+ f_o(x)$$