Question

Let $f:(0,1)\to R$ be a function defined by
$f(x)=\frac{1}{1-e^{-x}}$, and $g(x)=(f(-x)-f(x))$. Consider two statements
(I) $g$ is an increasing function in $(0,1)$
(II) $g$ is one-one in $(0,1)$ Then,

• A. Only (I) is true
• B. Both (I) and (II) are true
• C. Neither (I) nor (II) is true
• D. Only (II) is true

Answer 1

Answer: (B)

$g(x)=f(-x)-f(x)=\frac{1+e^x}{1-e^x}$
$\Rightarrow g^{\prime }(x)=\frac{2e^x}{{\left(1-e^x\right)}^2}>0$
$\Rightarrow g\text{ is increasing in }(0,1)$
$\Rightarrow g\text{ is one-one in }(0,1)$