Question

Determine whether the following is even or odd

$f\left(x\right)=x\left(\frac{a^x+1}{a^x-1}\right)$

Answer 1

Given, $f\left(x\right)=\frac{x(a^x+1)}{a^x-1}$
Substituting $x=-x$ we get
$f(-x)$ $=\frac{-x(a^{-x}+1)}{a^{-x}-1}$
$=\frac{-x(1+a^x)}{1-a^x}$
$=\frac{x(a^x+1)}{a^x-1}$
$=f\left(x\right)$
Hence $f\left(x\right)$ is even