Make The Condition Satisfied

Algebra Level 4

Let $P(x)=\ln\dfrac{1-x}{1+x}$ . For some $a$ and $b$ The condition $P(a)+P(b)=P \left (\dfrac{a+b}{1+ab} \right )$ is satisfied.

Given that $a\in (f,g)$ and $b\in (m,n)$ . $f<g$ and $m<n$ .

Find $(g-f)\times(n-m)$

1 solution

Hem Shailabh Sahu
Feb 14, 2015

$P(x)=ln\frac{1-x}{1+x}$

For the function $P(x)$ to be defined $\frac{1-x}{1+x}>0$ .

Therefore,

$-1<x<1$ $\Rightarrow$ $x\in(-1,1)$

This applies for both a & b. Thus,

$a\in(-1,1)$

$b\in(-1,1)$

Here, $(f,g)=(m,n)=(-1,1)$

Therefore,

$(g-f)*(n-m)=(1-(-1))*(1-(-1))=2*2=\boxed{4}$ :D

P.S. Reasons for $x\in(-1,1)$ :

For $x=1$ , $P(x)=ln(0)=Not Defined$

For $x=-1$ , $P(x)=ln\frac{1}{0}=Again Not Defined$

As for $x>1$ , $1-x<0$

And for $x<-1$ , $1+x<0$