Mathematics - Functions Question with Solution | TestHub

$\mathrm{f}(\mathrm{x})=\frac{\mathrm{x}}{{\mathrm{e}}^{\mathrm{x}}-1}+\frac{\mathrm{x}}{2}+1=\frac{2\mathrm{x}+{\mathrm{xe}}^{\mathrm{x}}-\mathrm{x}}{2\left({\mathrm{e}}^{\mathrm{x}}-1\right)}+1=\frac{\mathrm{x}+{\mathrm{xe}}^{\mathrm{x}}}{2\left({\mathrm{e}}^{\mathrm{x}}-1\right)}+1$

$\mathrm{f}(-\mathrm{x})=\frac{-\mathrm{x}-{\mathrm{xe}}^{-\mathrm{x}}}{2\left({\mathrm{e}}^{-\mathrm{x}}-1\right)}+1=\frac{\mathrm{x}+{\mathrm{xe}}^{\mathrm{x}}}{2\left({\mathrm{e}}^{\mathrm{x}}-1\right)}+1$

$\therefore \mathrm{f}(-\mathrm{x})=\mathrm{f}(\mathrm{x})$ for all x

$\therefore \mathrm{f}(\mathrm{x})$ is an even function.