Question

if f(x)=log 1+x/1-x then show that f(x)+f(y)=f(x+y/1+xy)

Answer 1

here is your answer .

Answer 2

Solution :-

f(x) = ln (1-x/1+x)

Given that f(x) +f (y) = f(x+y /1 +xy)

ln (1-x/1+x) + ln (1-y/1+y) = ln (1 -(x+y)/1 +x+y)

ln((1-x)× (1-y)/(1+x)× (1+y)) = ln (1-(x+y)/1+x+y)

Removing log

And. Then on solving

We get

2xy(x+y) = 0

Hence xy = 0 or x+y = 0

For xy = 0, solution is but obvious

For x+y = 0, x = -y

Putting in the given equation

ln((1+x)× (1-x)/(1-x)(1+x)) = ln1

LHS and RHS contradicts

Hence

x = 0 = y

Also

1-x >0

x 1

hence

x belongs to (0, 1)

================

@GauravSaxena01