Determine weather the given function is odd , even or neither:
$\frac{1}{3 x + 5 x^{- n}}$

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Solution

$\begin{matrix} E v e n f u n c t i o n : A f u n c t i o n i s e v e n i s f (- x) = f (x) f o r a l l x \in R o d d f u n c t i n : A f u n c t i o n i s o d d i f f (- x) = - f (x) f o r a l l x \in R f (- x) : \frac{1}{- 3 x + 5 {(- x)}^{- n}} - f (x) : \frac{- 1}{3 x + 5 x^{- n}} f (- x) \neq f (x), t h e r e f o r e \frac{1}{3 x + 5 x^{- n}} i s n o t e v e n f u c t i o n f (- x) \neq f (x), t h e r e f o r e \frac{1}{3 x + 5 x^{- n}} i s n o t o d d f u c t i o n N e i t h e r e v e n n o r o d d . \end{matrix}$

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