Answer
odd
Work Step by Step
We know that if a function is odd, then $f(-x)=-f(x).$
We know that if a function is even, then $f(-x)=f(x).$
Hence, we plug in $-x$ into $x$ to see what happens.
\begin{align*}
f(-x)&=\dfrac{(-x)^2+2}{3(-x)}\\&=\dfrac{x^2+2}{-3x}\\&=-\dfrac{x^2+2}{3x}\\&=-f(x)
\end{align*}
Therefore, the function is odd.