Question

Which of the following functions are odd functions

• A. $f(x)=x\left(\frac{a^x+a^{-x}}{a^x-a^{-x}}\right)$
• B. $f(x)=\frac{a^x+x}{a^x-x}$
• C. $f(x)=\frac{a^x-1}{a^x+1}$
• D. $f(x)=x \log _2\left(x+\sqrt{x^2+1}\right)$

Answer 1

Answer: (C)

$f(x)=x \frac{a^x+a^{-x}}{a^x-a^{-x}} \Rightarrow f(-x)=f(x) \Rightarrow f$ is not odd.
$f(x)=\frac{a^x+x}{a^x-x} \Rightarrow f(-x)=\frac{1-x a^x}{1+x a^x} \neq-f(x)$
$\Rightarrow f$ is not odd
$f(x)=\frac{a^x-1}{a^x+1} \Rightarrow f(-x)=\frac{1-a^x}{1+a^x}=-f(x) \Rightarrow f$ is odd.
$f(x)=x \log _2 \sqrt{\left(x+\sqrt{x^2+1}\right)} \Rightarrow f(-x) $
$=-x \log \left(-x+\sqrt{x^2+1}\right) \neq f(-x)$