Question

If $f:R\to R$ be an odd function and $f\left(e^x\right)=e^{f\left(x\right)}$ for all $x\in R,$ then

• A. $f\left(\frac{1}{e}\right)<f\left(1\right)<f\left(e\right)$
• B. $f\left(\frac{1}{e}\right)>f\left(1\right)>f\left(e\right)$
• C. $f\left(1\right)<f\left(\frac{1}{e}\right)<f\left(e\right)$
• D. $f\left(1\right)>f\left(\frac{1}{e}\right)>f\left(e\right)$

Answer 1

The correct option is A $f\left(\frac{1}{e}\right)<f\left(1\right)<f\left(e\right)$

$\because f\left(x\right)$ is an odd function
$\therefore f\left(x\right)+f(-x)=0$
Put $x=0\Rightarrow f\left(0\right)=0$

Also, $f\left(e^x\right)=e^{f\left(x\right)}$
Put $x=0\Rightarrow f\left(1\right)=1$
Put $x=1\Rightarrow f\left(e\right)=e$
Put $x=-1\Rightarrow f\left(\frac{1}{e}\right)=\frac{1}{e}$

We know that,
$\frac{1}{e}<1<e$
$\Rightarrow f\left(\frac{1}{e}\right)<f\left(1\right)<f\left(e\right)$