The function $ f(x)=\frac{x}{e^{x}-1}+\frac{x}{2}+1 $ is

Question

The function $ f(x)=\frac{x}{e^{x}-1}+\frac{x}{2}+1 $ is, an odd function, an even function, neither an odd nor an even function, a periodic function

MARKS App · Accepted Answer

Given, f x = x e x - 1 + x 2 + 1 = 2 x + x e x - x 2 e x - 1 + 1 = x + x e x 2 e x - 1 + 1 f - x = - x - x e - x 2 e - x - 1 + 1 = x + x e x 2 e x - 1 + 1 ∴ f ( - x ) = f ( x ) for all x . ∴ f x is an even function.

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