you could just substitute x and (1 - x) into each of the defined functions in turn, checking whether they're equal, but that's a lot of work for two minutes.

this problem says for ALL x. this means that you can plug in any value of x you want, carry it through, and just check whether it works.
if an answer choice works, it's not definitively the correct answer (because it may have just been a coincidence) - but, if an answer choice is ever wrong, you can cross it out.

let's try plugging in 0 for x
the question becomes:
for which of the following functions is f(0) = f(1)?
a: f(0) = 1, f(1) = 0. nope
b: f(0) = 1, f(1) = 0. nope
c: f(0) = -1, f(1) = 1. nope
d: f(0) = 0, f(1) = 0. looks good
e: f(0) = 0, f(1) = undefined. nope

d wins.