Answer is D
F(x)=F(1-X) these two shoud be equal
F(1-x)=(1-x)^2 (1-(1-x))^2= x^2 (1-x)^2
F(x)=x^2 (1-x)^2

yah the answer is D, but i was more specifically asking how to best intepret this question. can you take the question to be asking, in other words, for all functions of x anywhere you see x and replace it with 1-x, will produce an answer that is equal to its function? or better, out of the listed functions, which function will equal its output? is this approach/point of view valid?

Jazzcat has a point.

The question isn't looking for the f(x) equation that gets modified by f(1-x) in order for that new equation to equal 1-x. The question is asking for which equation of f(x) does its modified form f(1-x) (achieved by plugging in (1-x) for x) equal it's original form of f(x).

So that in answer D:
f(1-x) = (-x^2)(1-x)^2 is equal to the original
f(x) = (x^2)(1-x)^2
because for all values of (-x^2) = (x^2).
This solution is true for all values of x by this fact.

By the way, solving answer D (as many have above) by setting
1-x equal to f(1-x) is true for the values of 0 and 1, but not 2. This is certainly not all values of x. The fact that 0 and 1 are shared solutions between this equations happens to be a coincidence and incorrect reasoning to the answer.

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