Download Module 3: Piecewise Defined Functions

Haberman
MTH 111c
Section I: Sets and Functions
Module 3: Piecewise-Defined Functions
Some functions have different rules for different elements of the domain. Since these
functions have different definitions for different pieces of the domain, they are called
piecewise-defined functions.
EXAMPLE: The zoo uses a piecewise-defined function to calculate the entrance fee for
groups. The zoo charges $3 for each adult and $30 for a group of 10 or more.
So if we let n represent the number of people in a group (so n  0 and
n  Z ), then the entrance fee for the group is given by the function
3n if n  10
f (n)  
30 if n  10
Be sure that you understand the piecewise-defined function notation above before moving
on, by making sure that you understand how it communicates the same information
contained in the English description of the entrance fee function.
EXAMPLE: Consider the piecewise-defined function
p whose algebraic rule is given
below:
3  x

p( x )   2
 x2

if x  0
if x  0
if x  0
Since there is a rule that applies to ALL x-values, the domain of p is the set of real
numbers (denoted by R ). We can graph p by graphing each of its pieces on the same
coordinate plane. So we need to graph y  3  x for all x  0 , and we need to plot a
point above x  0 at the height y  2 , i.e., we need to plot the point (0, 2) , and we need
to graph y  x 2 for all x  0 . The graph of y  p ( x ) is given in Figure 1.
Figure 1: Graph of y  p ( x ) .
2
EXAMPLE: Consider the piecewise-defined function
g whose algebraic rule is given
below:
1

 2
g ( x)   3


 n
if 0  x  1
if 1  x  2
if 2  x  3
if n  1  x  n,
n  Z
So for all x-values between 0 and 1 (including 1), the output is 1, while for all x-values
between 1 and 2 (including 2), the output is 2, etc. The graph of g is given in Figure 2.
Figure 2: Graph of y  g ( x ) .
Notice that the domain of g is the set of positive real numbers (denoted by R  ) while the
range is the set of positive integers (denoted Z ).
EXAMPLE: The absolute value function (with which you are hopefully already familiar) can
be studied as a piecewise defined function.
  x if x  0
a( x)  x  
 x if x  0
The graph of the absolute value function a is given in Figure 3.
Figure 3: Graph of y  a( x)  x .
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EXAMPLE: Find an algebraic rule for the function
h graphed in Figure 4.
Figure 4: Graph of y  h( x ) .
SOLUTION:
It should be clear that this function consists of three linear pieces. The easiest piece to find
a rule for is the middle piece, since it is a horizontal line-segment. Clearly this line segment
is represented by the equation y  2 , and this equation applies when 0  x  2 . The leftmost piece is a line with y-intercept (0, 4) and slope 2 (which can be determined by
calculating the ratio of vertical change to horizontal change, i.e., rise over run). Thus, it is
represented by the equation y  2 x  4 and it applies when x  0 . The right-most piece
also has slope 2 (since it is clearly parallel to the left-most piece), so we know its rule looks
like y  2 x  b . To find b we can use the fact that this line passes through the point
(3, 0) :
y  2 x  b and (3, 0)
 0  2(3)  b
 06 b
 b  6
Thus, the right-most piece is represented by equation y  2 x  6 , and it applies when
x  2 . Therefore, an algebraic rule for the function graphed in Figure 4 is
2 x  4 if x  0

h( x )   4
if 0  x  2
2 x  6 if x  2
