Hi,
In the quiz, I came across the following question:
If I take the following case:
f(x) = x^2 which is an even function
g(x) = -x^2 which is an even function
f(x) + g(x) = 0 , which is neither an even nor an odd function, correct?
Can you please clarify how this answer is true?
0 is both an even and an odd function, no?
0 = f(x) = f(-x) = -f(x)
Also instead of examples, you can just do:
h(x) = f(x) + g(x) = f(-x) + g(-x) = h(-x)
Yes it it both even and odd, I understand. But doesn’t the question become a bit hazy then if it is both? It’s like the Schrodinger’s cat for 0
Is ETS likely to ask such boundary questions?
Uhhh there’s no boundary here though. You just don’t care that 0 can also be an odd function. In our context, we added two even functions and got back a function that is even as well.
To restate the definition for you, a function is even iff:
f(x) = f(-x) \implies f(x) - f(-x) = 0
0 satisfies this definition and so it is an even function. Nothing was stopping it from being an odd function as well.
It’s not related imo. Schrondinger asserted that the cat can’t be both alive and dead at the same time, but that’s not the case here with 0 being both even and odd which is more in line with quantum states being able to simultaneously be in multiple states at once.
