Question

Prove that the function $f\left(x\right)=x\frac{e^x+1}{e^x-1}$ is an even function.

Answer 1

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

Now, $f(-x)$

$=(-x)\frac{e^{-x}+1}{e^{-x}-1}$

$=(-x)\frac{1+e^x}{1-e^x}$

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

$\forall x$

So, the function $f\left(x\right)$ is an even function.