The Math Forum

Ask Dr. Math - Questions and Answers from our Archives
Associated Topics || Dr. Math Home || Search Dr. Math

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   
Associated Topics:
High School Basic Algebra
High School Functions

Search the Dr. Math Library:

Find items containing (put spaces between keywords):
Click only once for faster results:

[ Choose "whole words" when searching for a word like age.]

all keywords, in any order at least one, that exact phrase
parts of words whole words

Submit your own question to Dr. Math

[Privacy Policy] [Terms of Use]

Math Forum Home || Math Library || Quick Reference || Math Forum Search

Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.