Question

If `f(x_1)-f(x_2)=f((x_1-x_2)/(1-x_1x_2))` for `x_1, x_2 in (-1,1)`, then what is `f(x)` equal to? (a) In `((1-x)/(1+x))` (b) In `((2+x)/(1-x))` (c) `tan^(-1) ((1-x)/(1+x))` (d) `tan^(-1) ((1+x)/(1-x))`
A. `ln((1-x)/(1+x))`
B. `ln((2+x)/(1-x))`
C. `tan^(-1)((1-x)/(1+x))`
D. `tan^(-1)((1+x)/(1-x))`

Answer 1

Correct Answer - A
`f(x_(1))-f(x_(2))=f((x_(1)-x_(2))/(1-x_(1)x_(2)))`
`x_(1),x_(2)in(-1,1)`
then `f(x)=log.((1-x))/((1+x))`
`f(x_(1))=log.(1-x_(1))/(1+x_(1))" "f(x_(2))=log.(1-x_(2))/(1+x_(2))`
`f(x_(1))-f(x_(2))=log.(1-x_(1))/(1+x_(1))-log.(1-x_(2))/(1+x_(2))`
`=log.((1-x_(1)))/((1+x_(1)))xx((1+x_(2)))/((1-x_(2)))`
`=log.((1-x_(1)x_(2))-(x_(1)-x_(2)))/((1-x_(1)x_(2))+(x_(1)-x_(2)))`
`=log.(1-((x_(1)-x_(2))/(1-x_(1)x_(2))))/(1+((x_(1)-x_(2))/(1-x_(1)x_(2))))`
`f(x_(1))-f(x_(2))=f((x_(1)-x_(2))/(1-x_(1)x_(2)))`