Dual-Output Function: Solutions for Non-Negative Integer x?

[scikidus]

"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:

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

 

[Vid]

$$f(x) = \left\{<br /> \begin{array}{cc}<br /> 1 & x=0\\<br /> 0 & |x| >0<br /> \end{array}$$

http://en.wikipedia.org/wiki/Piecewise

 

[CRGreathouse]

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 ). But your post seemed to focus on giving it a closed form. Why?

 

[HallsofIvy]

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"?

 

[scikidus]

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]

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]

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?

 

[scikidus]

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

 

[CRGreathouse]

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]

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]

If your function only needs to be defined over the integers, you can try

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