Piecewise Function

Date: 10/23/2001 at 23:13:33
From: Amie Szymanski
Subject: Algebra/trigonometry

How do you graph the Piecewise Function? For example:

f(x) = 1/2x+3/2, if x < 1

       -x+3, IF X _> 1

I don't understand how to graph it if they don't give you a value 
for X. How do you find X?

Date: 10/23/2001 at 23:30:05
From: Doctor Peterson
Subject: Re: Algebra/trigonometry

Hi, Amie.

You don't have to find x, any more than you do when you graph y = x+1. 
Remember, x is the independent variable; you can choose ANY value for 
x, and graph the corresponding point (x,y).

What is probably confusing you is that now x appears not only in 
expressions, but also in "if" clauses. What this means is that, once 
you have chosen a value of x as usual, you have to decide which 
expression to use for f(x). For example, for x = 0, since 0 < 1, we 
use the first expression, and find that y is 1/2(0)+3/2 = 3/2, and we 
plot the point (0,3/2). For x = 2, we have to use the second 
expression (since 2 >= 1), and find that y is -2+3 = 1. So we plot 
(2,1).

Note that when x = 1, we are right at the boundary between the two 
"pieces" of the function. Because of the "x >= 1", we have to use the 
second expression and get y = -1+3 = 2. It can be helpful, however, to 
see what the first expression gives for x = 1, since that is where the 
left piece of the function "wants" to go. Then y = 1/2(1)+3/2 = 2. 
This happens to be the same as we got for the actual value of y at 
x = 1; that means the two pieces meet at (1,2), and the function is 
continuous.

To draw the graph, you can lightly draw the graphs of both pieces, 
which will be lines with slopes 1/2 and -1 respectively. Then draw 
lightly a vertical line at x = 1, where the two pieces meet. Then 
darken the part of the first line to the left of this, and the part 
of the right line to the right. That will be the graph of the 
piecewise-defined function. Do you see why?

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