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/ |
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