RE: How to ...? Piecewise Function Definition

• To: mathgroup at smc.vnet.net
• Subject: [mg36337] RE: [mg36309] How to ...? Piecewise Function Definition
• From: "David Park" <djmp at earthlink.net>
• Date: Mon, 2 Sep 2002 04:08:45 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```PasKo,

You could look up "piecewise functions" in the Master Index in Help.
Unfortunately, the book does not have a unified discussion of methods for
defining piecewise functions.

The first method is to use a set of conditional definitions. For
completeness I made f[x] = 0 if one of the conditional definitions is not
met.

f[x_] /; 0 < x < 6 := x^2
f[x_] /; x >= 6 := x + 1
f[x_] := 0

Plot[f[x], {x, 0, 10}];

The second method is to use a Which statement.

Clear[f];
f[x_] :=
Which[
0 < x < 6, x^2,
x >= 6, x + 1,
True, 0]

Plot[f[x], {x, 0, 10}];

The above methods are fine in many cases, but if you want to perform
functions on f, such as differentiation or integration, you should use the
UnitStep function. So our third method is...

Clear[f]
f[x_] := x^2(UnitStep[x] - UnitStep[x - 6]) + (x + 1)UnitStep[x - 6]

Plot[f'[x], {x, 0, 10}];

Plot[Integrate[f[y], {y, 0, x}] // Evaluate, {x, 0, 10}];

David Park

To: mathgroup at smc.vnet.net

Dear MathGroup,
I'd like to use Mathematica 4.0 to write a function having different
expressions in different domain's intervals.
Let's say:

F[x_]=    x^2   if 0<x<6
x+1   if x>=6

I know It's a stupid syntax problem, but I really do not know how/where to
search the solution on the Mathematica Book