Odd or Even? -1/x: Origin Symmetric?

[FortranMan]

Is the function -1/x an odd or even function? Is it origin symmetric? For a function to be origin symmetric, must it lie in the 1st and 3rd quadrant or can it lie in the 2nd and 4th quadrant? I suspect it is odd and origin symmetric, but I don't know if I am missing some fine math rule/definition regarding symmetry.

 

[PeroK]

Is the function -1/x an odd or even function? Is it origin symmetric? For a function to be origin symmetric, must it lie in the 1st and 3rd quadrant or can it lie in the 2nd and 4th quadrant? I suspect it is odd and origin symmetric, but I don't know if I am missing some fine math rule/definition regarding symmetry.

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

 

[mathman]

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

 

[HallsofIvy]

As Perok suggested, this is about knowing the definitions.
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)).

 

[FortranMan]

So to answer the question about the symmetry of -1/x, a function is origin symmetric if EITHER

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

 

[jbriggs444]

So to answer the question about the symmetry of -1/x, a function is origin symmetric if EITHER

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

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

 

[FortranMan]

Why is the first way more proper?

 

[jbriggs444]

Why is the first way more proper?

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: $\forall$).

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.

 

[Mark44]

Why is the first way more proper?

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.