Question

The curve $y={\log }_e\left(\frac{1-x}{1+x}\right)$ is

• A. Symmetric about $x$-axis
• B. Symmetric about $y$-axis
• C. Symmetric about origin
• D. Symmetric in all four quadrant

Answer 1

The correct option is C Symmetric about origin

Let $y=f\left(x\right)={\log }_e\left(\frac{1-x}{1+x}\right)$
Now, $f(-x)={\log }_e\left(\frac{1+x}{1-x}\right)=-{\log }_e\left(\frac{1-x}{1+x}\right)=-f\left(x\right)$
$\therefore f\left(x\right)$ is an odd function
Hence the curve is symmetric about origin.
$\Rightarrow f(x,-y)\ne f(x,y)$
$\therefore$ Not symmetric about $x-$axis
$\Rightarrow f(-x,y)\ne f(x,y)$
$\therefore$ Not symmetric about $y-$axis
and,
by interchanging $x$ and $y$ , same function is not appeared again, so it is not symmetric in all quadrants.