What are Piecewise Functions?

Date: 05/13/2001 at 16:12:36
From: Stephanie
Subject: What are Piecewise Functions?

How do I define a "Piecewise Function"?  I know it is based on 
expressions between specific intervals, but I do not know how to 
describe this function family.

Date: 05/13/2001 at 21:12:47
From: Doctor Douglas
Subject: Re: What are Piecewise Functions?

Hi Stephanie, and thanks for writing.

By itself, "piecewise" simply means what you wrote above - the "rule" 
for the function depends on a set of intervals. The rules themselves 
are arbitrary:

              sqrt(x)          8 < x
     f(x) =   x^2-log(x)      1 <= x <= 8
              3              -5 <= x < 1
              sqrt(-x)             x < -5

is an example of a piecewise function with four intervals (two of 
which actually extend to either plus or minus infinity).

Sometimes we will add another word after the word piecewise that 
describes how the function behaves within each of these intervals:

            -5         9 < x
     g(x) = +5       -9 <= x <= 9
            -2             x < -9

is an example of a piecewise *constant* function, since on each of the 
three sub-domains, the function is constant.

The absolute value function is the most familiar example of a 
piecewise function (in fact it is piecewise linear):

     h(x) = |x| =  +x      x >= 0
                   -x      x < 0

because on both of the sub-domains (or intervals), the function h(x) 
is linear. On the interval x >= 0, h(x) = 1*x, and on the interval
h(x) = -1*x, so on each interval, the function h(x) is equal to a 
constant times x. The constant is either +1 or -1, depending on which 
interval we are considering.

I hope this explains how mathematicians use the word "piecewise." It 
is a way to generalize the description of how a function behaves. For 
example, the function g(x) above is, strictly speaking, not constant, 
but it IS piecewise constant.

Please write back if you need more explanation about how this works.

- Doctor Douglas, The Math Forum
  http://mathforum.org/dr.math/