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

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• FortranMan
In summary: But (-x, -y) is a point in the first quadrant, which is not the same quadrant as (x, y). This logic applies to both conditions.In summary, the function -1/x is both an odd function and origin symmetric. A function is considered odd if f(-x) = -f(x), and origin symmetric if for every (x, y) on the graph, so is (-x, -y). Both of these conditions are met for the function -1/x. Additionally, it does not matter which quadrant the function lies in for it to be origin symmetric, as long as both conditions are satisfied.
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.

FortranMan said:
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?

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

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)).

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).

FortranMan said:
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.

Why is the first way more proper?

FortranMan said:
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.

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FortranMan said:
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.

jbriggs444

## 1. What is the concept of "Odd or Even?"

The concept of "Odd or Even" refers to whether a number is divisible by 2 or not. If a number is divisible by 2, it is considered an even number. If a number is not divisible by 2, it is considered an odd number.

## 2. What does "-1/x: Origin Symmetric" mean?

"-1/x: Origin Symmetric" is a mathematical function that describes a curve that is symmetric about the origin (0,0). This means that if you were to fold the curve along the y-axis, the two sides would overlap perfectly.

## 3. How does the "-1/x: Origin Symmetric" function relate to "Odd or Even?"

The "-1/x: Origin Symmetric" function is used to determine if a number is odd or even. If the function produces a positive value, the number is considered odd. If the function produces a negative value, the number is considered even.

## 4. Can the "-1/x: Origin Symmetric" function be used for all numbers?

No, the "-1/x: Origin Symmetric" function can only be used for real numbers. It cannot be used for complex numbers or imaginary numbers.

## 5. Are there any real-life applications of the "-1/x: Origin Symmetric" function?

Yes, the "-1/x: Origin Symmetric" function has applications in physics, particularly in the study of electric fields and gravitational fields. It is also used in economics to model demand and supply curves.

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