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STAR3URY
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Homework Statement
Prove that if f(x) is both odd and even (functions) then f(x) must be the constant function 0. Basically prove that no other function other than 0, can be both odd and even.
neutrino said:How do you define an odd function and an even function?
STAR3URY said:even is if f(-x) = f(-x)
An odd function is a type of mathematical function where f(-x) = -f(x) for all values of x. This means that the function is symmetric about the origin and its graph is rotated 180 degrees around the origin.
An even function is a type of mathematical function where f(-x) = f(x) for all values of x. This means that the function is symmetric about the y-axis and its graph remains unchanged when reflected over the y-axis.
If a function is both odd and even, it means that it satisfies both the conditions of an odd and even function. This can occur when the function is f(x) = 0, where f(-x) = -f(x) and f(-x) = f(x). In other words, the function is symmetric about both the origin and the y-axis.
The proof for this statement is based on the definition of an odd and even function. If a function is both odd and even, then f(-x) = -f(x) and f(-x) = f(x). By substituting f(x) = 0 into these equations, we get 0 = -0 and 0 = 0, which are both true. Therefore, if a function is both odd and even, it must be equal to 0.
Yes, a function can be neither odd nor even. This type of function is called an arbitrary function, which means that it does not satisfy the conditions of an odd or even function. In other words, the function is not symmetric about the origin or the y-axis and its graph is not rotated or reflected in any way.