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What are the definitions of odd, even and origin symmetric?

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mathman

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It is simply odd ##-\frac{-1}{x}=\frac{1}{x}##.

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HallsofIvy

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A function, f(x), is "even" if f(-x)= f(x) and "odd" if f(-x)= -f(x).

Replacing x with -x in f(x)= -1/x then f(-x)= -1/(-x)= 1/x= -(-1/x)= -f(x).

"Symmetric about the origin" means that if (x, y) is on the graph, so is (-x, -y). With y= -1/x, (x, -1/x) is on the graph and so is (-x, -1/(-x))= (-x, 1/x)= (-x, -(-1/x)).

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For every (x,y) on graph, so is (-x,-y).

or

For every (-x,y) on graph, so is (x,-y).

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jbriggs444

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Both conditions are identical. The first is the proper way of stating the second.

For every (x,y) on graph, so is (-x,-y).

or

For every (-x,y) on graph, so is (x,-y).

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Why is the first way more proper?

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jbriggs444

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When you write "For every (x,y) on graph, so is (-x,-y)", you are invoking a quantifier. In this case it is a Universal quantifier, "for all" (in symbolic form: ##\forall##).Why is the first way more proper?

The typical form of a universal quantifier is "for all <variable[s]> [in range], expression". The first occurrences of x and y in the statement are dummy variables. They exist simply to let the reader know which variables are being quantified over. As such, they should be variable names only, not expressions. The subsequent occurrences of x and y within the expression can be used freely.

If you've done computer programming, a lot of concepts carry over into mathematical discourse. A quantifier opens up a scope in which new variables are declared. The variable list in a quantifier amounts to a declaration of variables applicable to the scope. Rather like formal parameters in a called function. The function header has variable names for the formal parameters, not expressions.

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Mark44

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Because both x and y could be positive or negative. You're tacitly assuming that (x, y) is a point in the first quadrant. For example, if x = -3 and y = 2, then (x, y) is a point in the second quadrant.Why is the first way more proper?

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