Q 93. - Precalculus Enhanced with Graphing Utilities

Question

Can a function be both even and odd? Explain.

Step-by-Step Solution

Answer

The function can be both even and odd if $f\left(x\right)=0$ for all $x$.

Step 1. Define even and odd function

A function is odd if $f(-x)=-f\left(x\right)\dots \left(1\right)$ and a function is even if $f(-x)=f\left(x\right)\dots \left(2\right)$

Step 2. Explain when the function be both even and odd.

From the eqution $\left(2\right)$, the value of $f\left(x\right)$ is as follows:

 $f\left(x\right)=-f(-x)\dots \left(3\right)$

Add equation $\left(1\right)$ and $\left(3\right)$.

$\begin{array}{rcl}f\left(x\right)+f\left(x\right)& =& f(-x)+(-f(-x\left)\right)\\ 2f\left(x\right)& =& f(-x)-f(-x)\\ 2f\left(x\right)& =& 0\\ & \Rightarrow & f\left(x\right)=0\end{array}$

Hence, the function can be both even and odd if $f\left(x\right)=0$ for all $x$.