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The function
f(x) is even
g(x) is neither even nor odd
h(x) is odd
Steps:
for an even function it holds that f(-x) = f(x):
f(-x) = (-1)^6 x^6 - (-1)^4 x^ 4 = x^6 - x^4 = f(x) => f is even
for an odd h(x) it holds that h(-x) = -h(x):
It is easy to show that g(x) does not match any of the two possibilities above.