Hi ,
I can see this problem as substitution.
As mention F(x) = F(1-x). Let us take any value of X and evaluate it.
let x = 1.
ie 1-x = 0.
A. f(X) = 1-X
f(x)=f(1) = 1 - 1 = 0
f(1-x) = f(1-1) = f(0) = 1-0 = 1
This is not a correct anser.
B. f(X) = 1-X^2
f(x)=f(1) = 1 - 1^2 = 1 - 1 = 0 .
f(1-x) = f(1-1)= f(0) = 1 - 0^2 = 1 - 0 = 1 .
This is not a correct anser.
C. f(x) = x^2-(1-x)^2
f(x)=f(1) = 1^2 - (1 - 1)^2 = 1 - 0 = 1 .
f(1-x) = f(1-1)= f(0) = 0^2 - (1 - 0)^2 = 0 - 1 = -1 .
This is not a correct anser.
D. f(x) = x^2(1-x)^2
f(x)=f(1) = 1^2 *(1 - 1)^2 = 1 * 0 = 0 .
f(1-x) = f(1-1)= f(0) = 0^2 *(1 - 0)^2 = 0 * 1 = 0 .
This may be correct answer. Let us evaluate last option.
E. f(x) = x/(1-x)
f(x)=f(1) = 1 /(1 - 1) = 1 / 0 = infinity .
f(1-x) = f(1-1)= f(0) = 0/ (1 - 0) = 0 / 1 = 0 .
This is not a correct answer.
We only left with Option D.
Thanks & Regards,
Shrikant
