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lLovePhysics
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How is y=0 an odd function when it isn't symmetric to the origin?
Is y=0 an Odd Function Despite Not Being Symmetric About the Origin?
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In summary, y=0 is considered an odd function because it follows the definition of an odd function, despite not being symmetric to the origin. It is also possible for a function to be both odd and even.
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benorin
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Sure it is. Consider f(x)=0 and recall the definition of an odd function.
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CompuChip
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Note here, that for any function f in general, it's not necessary that f is either odd or even; nor that it cannot be both. You've found an example of the latter 
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HallsofIvy
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Did you mean "odd" and "even"? f(x)= 0 is clearly a closed function, certainly not open!CompuChip said:Note here, that for any function f in general, it's not necessary that f is either closed or open; nor that it cannot be both. You've found an example of the latter
It is definitely both even and odd.
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CompuChip
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Heh, just read a topology thread. I was thinking "odd" and "even", but I wrote "open" and "closed". Still a bit sleepy probably :O
I'll change my post
I'll change my post
Related to Is y=0 an Odd Function Despite Not Being Symmetric About the Origin?
1. What is an odd function?
An odd function is a type of mathematical function where the output of the function changes sign when the input is changed to its negative value. In other words, if f(x) is an odd function, then f(-x) = -f(x).
2. How is y=0 an odd function?
When y=0, the output of the function is always zero regardless of the input. This means that for any value of x, f(-x) = f(x) = 0. Since the sign of 0 remains the same when the input is changed to its negative value, y=0 can be considered an odd function.
3. What is the significance of y=0 being an odd function?
The significance of y=0 being an odd function is that it helps us identify symmetry in graphs. An odd function has a specific type of symmetry called origin symmetry, which means that the graph is symmetric about the origin (0,0). This can be seen by the fact that when the function is reflected about the origin, the graph remains unchanged.
4. How can we prove that y=0 is an odd function?
To prove that y=0 is an odd function, we can use the definition of an odd function and show that f(-x) = -f(x). Since y=0 is a constant function, f(-x) = 0 and -f(x) = 0, which means that f(-x) = -f(x) is true for all values of x. Therefore, y=0 is an odd function.
5. Can a function be both even and odd?
No, a function cannot be both even and odd. A function is considered even if f(x) = f(-x) for all values of x, while a function is considered odd if f(-x) = -f(x) for all values of x. These two conditions cannot be satisfied simultaneously, so a function cannot be both even and odd.
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