If `(f(x))^2*f((1-x)/(1+x))=64 x AA in D_f` then

Options

A. `4x^(2//3)((1+x)/(1-x))^(1//3)`
B. `x^(1//3)((1-x)/(1+x))^(1//3)`
C. `x^(1//3)((1-x)/(1+x))^(1//3)`
D. `x((1+x)/(1-x))^(1//3)`

Answer

A

Explanation

`(f(x))^(2)f((1-x)/(1+x))=64x " (1)" `
Putting `(1-x)/(1+x)=y or x=(1-y)/(1+y),` We get
`{f((1-y)/(1+y))}^(2)*f(y)=64((1-y)/(1+y))`
` or f(x)*{f((1-x)/(1+x))}^(2)=64((1-x)/(1+x)) " (2)" `
Squaring (1) and dividing by (2),
`(f(x)^(4){f((1-x)/(1+x))}^(2))/(f(x){f((1-x)/(1+x))}^(2))=((64x)^(2))/(64((1-x)/(1+x)))`
`or {f(x)}^(3)=64x^(2)((1+x)/(1-x))`
` or f(x)=4x^(2//3)((1+x)/(1-x))^(1//3)`
` :. f(9//7)= -4(9//7)^(2//3)(2)`