Piecewise Functions

Suppose f(x) and g(x) are functions \mathbb{R}\to \mathbb{R}, and consider the piecewise function

p(x)=\begin{cases} f(x) & x<0 \\ g(x) & x>0 \end{cases}.

For no particularly good reason, today I went in search of another way to write this expression. I wasn’t sure exactly what sort of expression I was shooting for, but I hoped I’d know it when I saw it.

I’m want to try to avoid explicitly writing a piecewise function, and somehow use |x| to disguise the piecewise nature of the function. This can be further obfuscated by writing it as \sqrt{x^2}, but I’ll leave it as |x|. Actually, a function I think might be more useful is |x|/x, a favorite from calculus classes. Recall that this is 1 for x positive, and -1 for x negative. Conveniently, the domains of the pieces here are exactly the domains of the pieces I want for p(x).

My first step actually makes p(x) looks worse:

p(x)=\frac{f(x)+g(x)}{2}+\begin{cases} \frac{-g(x)+f(x)}{2} & x<0 \\ \frac{g(x)-f(x)}{2} & x>0 \end{cases}

However, let me write a(x)=(f(x)+g(x))/2 and d(x)=(f(x)-g(x))/2. Then notice that the pieces in the above line are exactly d(x) or -d(x). That is,

p(x)=\begin{cases} a(x)+d(x) & x<0 \\ a(x)-d(x) & x>0 \end{cases}

I can wrap this up using my |x|/x from above to get that sign distinction, arriving at


That’s a reasonable answer I suppose. I’ve written my piecewise function all on one line, I guess, as


With this, I could then shift things horizontally and assume that my pieces where defined on (-\infty,a) and (a,\infty), instead of just splitting at 0. I could also, sort of inductively, get a piecewise function with more than 2 pieces. To do this, just split it into a leftmost non-piecewise function, and then a right-hand piecewise function, and use the ideas above.

I don’t think that this is the answer that makes me the happiest though. But it’s the only idea I’ve got currently. The reason I don’t like this is that f(x) and g(x) need to be defined on all of \mathbb{R}, even though in the original piecewise function they need only be defined on half the line. Also, I’m not sure I’m a huge fan of the handling of more than 2 pieces with the above method.

So, what suggestions have you got for improvement?


One Response to “Piecewise Functions”

  1. A Polar Curve « Sumidiot’s Blog Says:

    […] Sumidiot’s Blog The math fork of sumidiot.blogspot.com? « Piecewise Functions […]

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