For which of the following functions f is f(x)=f(1-x) for all x?
A.f(x)=1-x
B.f(x)=1-x^2
C.f(x)=x^2-(1-x)^2
D.f(x)=x^2(1-x)^2
E.f(x)=x/1-x
A. f(x)=1-x
We know the value of f(x) so let's calculate f(1-x). For that you just reimplace x by 1-x
f(1-x)=1-(1-x)=x
We can clearly see that in answer A, f(x) and f(1-x) are different
B.f(x)=1-x^2
Let's calculate f(1-x) for question B
f(1-x)= 1-(1-x)^2 = x(x-2)
They are not the same...
C.f(x)=x^2-(1-x)^2
Let's calculate f(1-x) for question C
f(1-x)= (1-x)^2 - (1-(1-x))^2= 1-2x+x^2-x^2=1-2x
They are not the same
D.f(x)=x^2(1-x)^2
f(1-x) = (1-x)^2 (1-(1-x))^2
= (1-x)^2 (1-1+x)^2
= (1-x)^2 (x)^2=x^2(1-x)^2 =f(x)
For question D we have: f(x)=f(1-x).
