2.5 F15 O’Brien
CA 6th ed HLR
2.5: Piecewise-Defined Functions
I.
Definition of Piecewise-Defined Functions
As the name implies, a piecewise-defined function (also called a piecewise function) is
defined by multiple sub functions. Each sub function applies to a specified interval of the main
function’s domain.

x2 ,

f(x)  
2

 (x  a)  b,
II.
xc
xc
Using the Graph of a Piecewise-Defined Function to Answer Questions
Carefully read the description of the function to determine what each variable represents.
Also note the units for each variable.
Example 1
2.5 #3 p. 139
a.
At t = 0, the pool has 50,000 gallons of water in it. And at t = 5, the pool contains
only 30,000 gallons of water.
b.
On the first and fourth days the water level was constant.
c.
f 2  45,000 gallons
d.
III.
f 4  40,000 gallons
At t = 1, the pool level was 50,000 gallons and at t = 3, the level was 40,000 gallons.
We can write this information in the form of two points (1, 50000) and (3 ,40000).
Now let’s find the slope of the line segment between these two points:
40000  50000 10000

 5000
gal
So the water level is declining by 5000
.
3 1
2
day
Evaluating a Piecewise-Defined Function at a Specified x-value
Determine which piece of the function to plug the specified x into by ascertaining which domain interval
it is within.
Example 2
1
2.5 F15 O’Brien
CA 6th ed HLR
IV.
Graphing a Piecewise-Defined Function
Find three points for each sub function. Be sure that the x-values you chose are in the appropriate
domain interval. To avoid having gaps in your graph, plug each “border” value into each sub function
it applies to. Graph the point for the “or equal to” inequality with a solid dot and the point for the other
inequality with an open dot. To plot overlapping open and closed dots, plot a solid dot surrounded by an
open dot.
Example 3
V.
Finding the Formula for a Piecewise-Defined Function from a Graph
First take note of how many sub functions or pieces the graph contains. Next, determine the domain
interval for each sub function by observing the x-values where its piece begins and ends. Then
determine the defining expression for each sub function by identifying what type of function it is and
whether it has undergone any transformations. From this information, write an appropriate function.
You can check your function by graphing it to see if its graph matches the given graph.
Note:
  2x  1, x  2
To enter a piecewise-defined function such has f(x)  
into a graphing
x 2
 3x  2,
calculator we enter the first defining expression in parentheses followed by its domain interval,
also in parentheses. We then add the second defining expression in parentheses followed by its
domain interval in parentheses. So for the piecewise function listed above, we will enter the
following at y1: (–2x + 1) x  2 + (3x – 2)(x > 2).
Example 4
2.5 #37 p. 141
Our function has two sub functions.
The interval of the first sub function begins at
negative infinity and goes up to, but does not
include x = 1. Thus its domain interval is (, 1) .
The second piece begins at, and includes, x =1
and goes out to positive infinity. Its domain interval
is [1, ) .
The first piece of the graph looks like a “plain vanilla”
cube root function that has not undergone any
transformations. Thus its defining expression will be
3
x and its domain interval is (, 1) .
32 1
 1
2 1 1
If we extended the line to the y-axis, it looks like its y-intercept would be (0, 1). Thus the defining
expression for this sub function is 1x – 1 and its domain interval is [1, ) .
The second piece of the function looks like part of an oblique line. The slope of the line is
Putting all this information together, we generate the following piecewise-defined function:

3 x ,
f(x)  

1x  1,
x 1
x1
2
CA 6th ed HLR
2.5 F15 O’Brien
3
CA 6th ed HLR
2.5 F15 O’Brien
4