Question

Let $f:(0,1) \rightarrow R$ 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{2 e ^{ 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)$