A) \[f(x)=\frac{{{a}^{x}}+1}{{{a}^{x}}-1}\]
B) \[f(x)=x\left( \frac{{{a}^{x}}-1}{{{a}^{x}}+1} \right)\]
C) \[f(x)=\frac{{{a}^{x}}-{{a}^{-x}}}{{{a}^{x}}+{{a}^{-x}}}\]
D) \[f(x)=\sin x\]
Correct Answer: B
Solution :
In , \[f(-x)=\frac{{{a}^{-x}}+1}{{{a}^{-x}}-1}=\frac{1+{{a}^{x}}}{1-{{a}^{x}}}=-\frac{{{a}^{x}}+1}{{{a}^{x}}-1}=-f(x)\] So, it is an odd function. In , \[f(-x)=(-x)\frac{{{a}^{-x}}-1}{{{a}^{-x}}+1}=-x\frac{1-{{a}^{x}}}{1+{{a}^{x}}}=x\frac{{{a}^{x}}-1}{{{a}^{x}}+1}=f(x)\] So, it is an even function. In , \[f(-x)=-\sin \left[ \log (x+\sqrt{1+{{x}^{2}}}) \right]\] So, it is an odd function. In , \[f(-x)=\sin (-x)=-\sin x=-f(x)\] So, it is an odd function.
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