# Dual-output Function?

## Main Question or Discussion Point

"Dual-output" Function?

This isn't homework: I am experimenting with factorization.

Does anyone know of a function f(x) which for some value of x returns one value for f(x), but for every other value of x returns some other value?

Example: I'm trying to find a function f(x), where

x = 0, f(x) = 1
x != 0, f(x) = 0

My function is only dealing with non-negative integers, if that helps.

I've already derived a function that does this, but it uses absolute values, which is a nuisance.

Anyone know of anything like this?

For reference, here is my formula:

$$f(x) =\frac{1-\frac{\left|2x-1\right|}{2x-1}}{2} = \frac{\left|4x-2\right| - 4x-2}{8x-4}$$

Last edited:

CRGreathouse
Homework Helper

Example: I'm trying to find a function f(x), where

x = 0, f(x) = 1
x != 0, f(x) = 0

My function is only dealing with non-negative integers, if that helps.
That's the characteristic function of zero (Sloane's http://www.research.att.com/~njas/sequences/A000007 [Broken]). But your post seemed to focus on giving it a closed form. Why?

Last edited by a moderator:
HallsofIvy
Homework Helper

What exactly do you want? When you say "f(0)= 1, f(x)= 0 if x is not 0" you have already defined a function. And if you want a closed form, why is absolute value a "nuisance"?

Thanks to those who have responded, you've helped a lot.

@Vid: The link you supplied helped me solve another problme I was working with, so thanks.

@CRGreathouse: That, too, is of great help. I wasn't sure if 0^0 would be considered defined, but that greatly simplifies my procedure.

@HallsofIvy: Yes, that techincally defines a function, but I was looking for a mathematical equation which would supply that result. Two ways are the one I supplies in ym OP, and f(x) = 0^x.

Also, I called absolute value a nuisance because I am build an equation and I need to then invert it. If there are absolute values, then things get very tricky, by which I mean impossible to solve.

Office_Shredder
Staff Emeritus
Gold Member

It's interesting you decided f(x)=0^x is an equation that gives this, since 0^x has to be defined at x=0 separately anyway, so you haven't really gained anything. And there's no way in hell you're going to invert this sucker

CRGreathouse
Homework Helper

Also, I called absolute value a nuisance because I am build an equation and I need to then invert it. If there are absolute values, then things get very tricky, by which I mean impossible to solve.
This function (regardless of whether you view it as a closed-form equation or not) can't be inverted. f(9) = f(3) = 0, so what would f^-1(0) be?

Terribly sorry, I misspoke/posted. By "invert," I meant not the function, but the final equation that I'm working on.

CRGreathouse
Homework Helper

Terribly sorry, I misspoke/posted. By "invert," I meant not the function, but the final equation that I'm working on.
Yes, but inverting the final equation will involve inverting that special function.

CRGreathouse
Homework Helper

Actually, I've decided that using the characteristic function of 0 essentially allows you to build piecewise functions, so perhaps you can simply invert piecewise.

Ben Niehoff
$$f(n) = \frac{\sin \pi n}{n}$$