Question

Which one of the following is an even function?

• A. $ f(x)=\frac{{{a}^{x}}+{{a}^{-x}}}{{{a}^{x}}-{{a}^{-x}}} $
• B. $ f(x)=\frac{{{a}^{x}}+1}{{{a}^{x}}-1} $
• C. $ f(x)=x\frac{{{a}^{x}}-1}{{{a}^{x}}+1} $
• D. $ f(x)={{\log }_{e}}(x+\sqrt{{{x}^{2}}+1}) $

Answer 1

Answer: (C)

If $ f(-x)=f(x), $ then $ f(x) $ is an even function
$ \therefore $ Putting $ x=-x $ in $ x\frac{{{a}^{x}}-1}{{{a}^{x}}+1} $
$ \therefore $ $ f(-x)=-x.\frac{{{a}^{-x}}-1}{{{a}^{-x}}+1} $
$ =-x.\frac{{{a}^{-x}}(1-{{a}^{x}})}{{{a}^{-x}}(1+{{a}^{x}})} $
$ =-x\frac{(1-{{a}^{x}})}{(1+{{a}^{x}})} $
$ =x\left( \frac{{{a}^{x}}-1}{{{a}^{x}}+1} \right)=f(x) $
$ \therefore $ $ x\frac{{{a}^{x}}-1}{{{a}^{x}}+1} $ is an even function.