Question

If the real valued function $f\left(x\right)=\frac{a^x-1}{x^n(a^x+1)}$ is even and $n\in N$, then least possible value of $n$ is

Answer 1

$\because f\left(x\right)$ is even
$\therefore f\left(x\right)=f(-x)$
$\Rightarrow \frac{a^x-1}{x^n(a^x+1)}=\frac{a^{-x}-1}{(-x{)}^n(a^{-x}+1)}$
$\Rightarrow \frac{a^x-1}{x^n(a^x+1)}=-\frac{(a^x-1)}{(-x{)}^n(a^x+1)}$
$\Rightarrow \frac{1}{x^n}=-\frac{1}{(-x{)}^n}$
$\therefore$ least possible value for $n=1$