Problem Solving

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)

Answer is d. Please explain.

Your best bet here would be to plug in:

f(x)=f(1-x)

A.) f(x)=1-x
f(1-x)=1-1+x=x
x is not equal to 1-x.

B.) f(x)=1-x^2
f(1-x)=1-(1-2x+x^2)=1-1+2x-x^2=2x-x^2
2x-x^2 is not equal to 1-x^2

C.) f(x)= x^2-(1-x)^(2)---> x^2-(1-2x+x^2)=x^2-1+2x-x^2
f(1-x)= 1-2x+x^2-x^2=1-2x
1-2x is not equal to 2x-1

D.) f(x)=x^2*(1-x)^(2)
f(1-x)=(1-2x+x^2)(x^2)= (1-x)^(2)*x^2
THIS IS OUR ANSWER. D

for D you don't even need to calculate anything, just substitute:

(d) f(1-x)=(1-x)^2 * (1-(1-x))^2 = (1-x)^2 * x^2