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What are even and odd function and what is the difference between them?
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Difference will probably be odd.
Even function is a function that is symmetrical across the y-axis. That is, (x, y) is on the graph if and only if (-x, y) is as well.
Odd function is symmetrical across the origin. That is, (x, y) is on the graph if and only if (-x, -y) is as well.
So here is a question for you: Why is it that there is no name for a graph where (x, y) if and only if (-x, y)?
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
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Thanks ricky, I've still got a few questions though:
To answer your question, would it because it would fail the vertical line test and hence not be a function?
I think I recognise the names now that you've explained them. Are other names for an even function a many-to-one function or a surjective function?
I can't say I've seen that explanation for an odd function before...
And finally, was that first sentence just a joke?
thanks again
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Thanks ricky, I've still got a few questions though:
To answer your question, would it because it would fail the vertical line test and hence not be a function?
I think I recognise the names now that you've explained them. Are other names for an even function a many-to-one function or a surjective function?
I can't say I've seen that explanation for an odd function before...And finally, was that first sentence just a joke?
thanks again
I hope it was a joke and that ricky recognized that you were not talking about "difference" in the sense of subtraction.
The standard definition of "even function" is that f(-x)= f(x) and of "odd function" is that f(-x)= -f(x). It follows from those that the graph of an even function is symmetric about the y-axis and the graph of an odd function is symmetric about the origin.
Neither would "fail the vertical line test": no function can! I think you meant "horizontal line test" for whether a function is one-to-one or not (injective).
Of course, if f(-x) = f(x), then f cannot be one-to-one and so its graph would fail the horizontal line test. However, odd functions need not be one-to-one either: f(x)= x^3- x is an odd function because f(-x)= (-x)^3- (-x)= -x^3+ x= -(x^3- x). The fact that the powers of x are all odd guarentees that. But f(-1)= f(1)= f(0)= 0 so f is certainly not one-to-one.
And "surjective" is NOT the opposite of "injective" (one-to-one). Surjective means "onto"- for every y there exists an x such that f(x)= y. That is not relevant here.
Neither would "fail the vertical line test": no function can! I think you meant "horizontal line test" for whether a function is one-to-one or not (injective).
You missed my question:
So here is a question for you: Why is it that there is no name for a graph where (x, y) if and only if (-x, y)?
And e's response is entirely right. As for the first line of my post, it [i]was[i] a joke, till you guys ruined it. Sheesh.
I think I recognise the names now that you've explained them. Are other names for an even function a many-to-one function or a surjective function?
There aren't really other names, at least not in common use. many-to-one is normally called "noninjective".
So some more questions for you. For the "Is a" questions, the responses should be always, sometimes, or never.
Does there exist a function that is both odd and even?
Is an odd function plus even function either even or odd?
... odd plus odd...?
... even plus even...?
Repeat the last three for composition of functions as well.
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
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I did some experimenting:
No
Even + Odd = Even (always)
Odd + Odd = Odd (always)
Even + Even = Even (always)
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No
Incorrect.
Even + Odd = Even
Nope.
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
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