Question

Let f:[0, 1] `rarr` [0, 1] be defined by `f(x) = (1-x)/(1+x),0lexle1 and g:[0,1]rarr[0,1]` be defined by `g(x)=4x(1-x),0lexle1`
Determine the functions fog and gof.
Note that [0,1] stands for the set of all real members x that satisfy the condition `0lexle1`.

Answer 1

`(fog)x=f{g(x)}=f{4x(1-x)}" "[because g(x)=4x(1-x)]`
`=(1-4x(1-x))/(1+4x(1-x))" "[becausef(x)=(1-x)/(1+x)]`
`=(1-4x+4x^(2))/(1+4x-4x^(2))=((2x-1)^(2))/(1+4x-4x^(2))`
and `(gof)x=g{f(x)}=g{(1-x)/(1+x)}" "[becausef(x)=(1-x)/(1+x)]`
`=4((1-x)/(1+x))(1-(1-x)/(1+x))=4((1-x)/(1+x))((2x)/(1+x))`
`=(8x(1-x))/((1+x)^(2))`