Piecewise Functions
Pre-Cal Advanced
Ms. Ouseph
Objective for today!
We will use our
knowledge of domain and
transformations to graph
piecewise functions.
A piecewise function is…
A function defined by
different equations on
different domains.
In other words…
Think of it as only
having a “chunk” of a
function!
We’ll harp more about that later…
For now we will explore an everyday
situation where piecewise functions
are applicable. 
arking Garage
Suppose we were parking in a parking garage that
we had to pay for by the hour. The pay structure is
as follows:
• For the first 4 hours, it costs us $3 per hour to park.
• For any number of hours parked after 4 hours, it costs $2 per
hour to park
(These rates are proportional to any amount of time spent parked within the first
three hours; for example if we only stayed for ½ hour, we would pay $1.50.)
OUR GOAL: Create a function which inputs hours
parked and outputs total cost.
In groups, fill out the table below and determine an equation to
calculate the total payment with respect to the hourly rate.
Total Hours Parked (x)
Total Payment (y or f(x))
*For the first 4 hours, it costs us $3/hour to park*
1
2
3
4
Equation 1:
f(1) = 3(1) = $3
f(2) = 3(2) = $6
f(3) = 3(3) = $9
f(4) = 3(4) = $12
y = 3x
*For any number of hours parked after 4 hours, it costs us $2/hour to park*
5
6
7
8
x
Equation 2:
f(5) = 3(4) + 2(5-4) = $14
f(6) = 3(4) + 2(6-4) = $16
f(7) = 3(4) + 2(7-4) = $18
f(8) = 3(4) + 2(8-4) = $20
12 + 2(x – 4)
y = 2x + 4
Equation 1: y
= 3x only works when…
The total number of hours parked (x) are between 0-4
0≤x≤4
Equation 2: y
= 2x + 4 only works when…
The total number of hours parked (x) are between greater than 4
x>4
Combining these two, we have a piecewise function!
f(x) =
{
3x,
2x + 4,
0≤x≤4
x>4
How do we graph a piecewise function??
Using the table we created, we can plot points on a graph to graph
the piecewise function!
y = 3x, 0 ≤ x ≤ 4
x
y
0
1
2
3
4
0
3
6
9
12
y = 2x + 4, x > 4
Open
Closed
x
y
4
5
6
7
8
12
14
16
18
20
Each function must live in its own “neighborhood”
Let's put up a fence to separate these two “neighborhoods.”
x=4
Note: Its OK
for each
neighborhoo
d to lie on
the fence.
They just
can’t cross
over it!
y = 3x
y = 2x + 4
lives here!
lives here!
Parking Hours vs. Payment
Both equations are “on the fence,” but
one point (from y=3x) is closed while
the other (from y=2x+4) is open!
{
0≤x≤4
x>4
Total Payment
f(x) =
3x,
2x + 4,
(0, 0)
(1, 3)
(2, 6)
(3, 9)
(4, 12)
Total Hours Parked
(4, 12)
(5, 14)
(6, 16)
(7, 18)
(8, 20)
This analysis is not meant to explain the
parking situation mathematically. Rather,
it is meant to explain the idea of a
piecewise function by showing you where
one exists in real life.
Now, we will be exploring and graphing
other piecewise functions involving an
array of functions (linear, quadratic, etc.)!
What do we need to know in order to
graph these pieces?
“Fence,” Domain, Points, Open/Closed, etc…
Closed
Open
x
y
x
y
1
0
-1
-2
5
3
1
-1
1
2
3
4
3
2
1
0
Now let’s use these table of values to
graph our piecewise function!
x
y
x
y
1 5
0 3
-1 1
-2 -1
1
2
3
4
3
2
1
0
Domain: (-∞, ∞) Range: (-∞, 5]
Let’s look at a quadratic
piecewise function!
Work with a partner to graph the following function:
x
(0
-1
-2
-3
y
3 ) Closed
2
-1
-6
x
Open ( 0
1
2
3
y
1)
2
3
4
Now let’s use these table of values to
graph our piecewise function!
x
y
x
y
0 3
-1 2
-2 -1
-3 -6
0
1
2
3
1
2
3
4
Domain: (-∞, ∞) Range: (-∞, ∞)
In summary…
• The key to graphing piecewise functions
is:
– Determining the “fence”
• This fence will tell you where each piece of the
function “lives”
– Domain
• Once you know the domain of each function,
its just a matter of creating a table of values to
plot points!
Now that we have explored
various piecewise functions,
you will be given time to finish
the rest of your worksheet.