F(a), f(b), f[(a+b)/( 1+ab)] are meaningful.

Because f (x) = LG (1-x)/(1+x)

So f (a) = LG (1-a)/(1+a)

f(b)=lg( 1-b)/( 1+b)

f[(a+b)/( 1+ab)]= LG[ 1-(a+b)/( 1+ab)]/ 1+(a+b)/( 1+ab)

f(a)+f(b)

= LG( 1-a)/( 1+a)+LG( 1-b)/( 1+b)

= LG[( 1-a)/( 1+a)( 1-b)/( 1+b)]

= LG( 1-a-b-ab)( 1+a+b+ab)

f[(a+b)/( 1+ab)]

= LG {[ 1-(a+b)/( 1+ab)]/[ 1+(a+b)/( 1+ab)]}

= LG[( 1-a b-a-b)/( 1+ab)]/[( 1+a+b+ab)/( 1+ab)}

= LG( 1-a-b-ab)( 1+a+b+ab)= f(a)+f(b)

That is, f (a)+f (b) = f [(a+b)/(1+ab)]