The document explains piecewise functions, detailing their definition, evaluation, and real-life applications. It provides examples to illustrate how to evaluate such functions based on specified domains, along with graphical representations. Furthermore, it discusses several real-life scenarios where piecewise functions could be applied, such as mobile plan charges, transportation fares, and medical costs.

1. W
ISE
F
U
N
C
T
I
O
N
S
These are functions that are
defined differently on different
parts of the domain.

2. Definition:
Piecewise Function –a function
defined by two or more functions
over a specified domain.

3. Evaluate f(x) when x=0, x=2, x=4








2
,
1
2
2
,
2
)
(
x
if
x
x
if
x
x
f
• First you have to figure out which equation to use
• You NEVER use both
X=0
This one fits
Into the top
equation
So:
0+2=2
f(0)=2
X=2
This one fits here
So:
2(2) + 1 = 5
f(2) = 5
X=4
This one fits here
So:
2(4) + 1 = 9
f(4) = 9

4. Your turn:
f(x) =
2x + 1, x  0
2x + 2, x  0
Evaluate the following:
f(-2) = -3
f(0) = 2
f(5) = 12
f(1) = 4

5. One more:
f(x) =
3x - 2, x  -2
-x , -2  x  1
x2
– 7x, x  1
Evaluate the following:
f(-2) = 2
f(-4) = -14
f(3) = -12
f(1) = -6

6. Graph:
1, 2
( )
1, 2
x if x
f x
x if x
  

  

Point of Discontinuity

7. Step Functions
















4
3
,
4
3
2
,
3
2
1
,
2
1
0
,
1
)
(
x
if
x
if
x
if
x
if
x
f

8. 















4
3
,
4
3
2
,
3
2
1
,
2
1
0
,
1
)
(
x
if
x
if
x
if
x
if
x
f

9. Graph :























0
1
,
4
1
2
,
3
2
3
,
2
3
4
,
1
)
(
x
if
x
if
x
if
x
if
x
f

11. Graphing Piecewise Functions:
f(x) =
x2
+ 1 , x  0
x – 1 , x  0
Determine the shapes of the graphs.
Parabola and Line
Determine the boundaries of each graph.
Graph the parabola
where x is less than
zero. 





Graph the line
where x is greater
than or equal to
zero. 







12.  







0
,
0
,
2
x
x
x
x
x
f
This means for x’s less than
0, put them in f(x) = -x but for
x’s greater than or equal to
0, put them in f(x) = x2
What does the graph
of f(x) = -x look like?
Remember y = f(x) so let’s
graph y = - x which is a line of
slope –1 and y-intercept 0.
Since we are only
supposed to graph this for
x< 0, we’ll stop the graph
at x = 0.
What does the graph
of f(x) = x2
look like?
Since we are only
supposed to graph this for
x  0, we’ll only keep the
right half of the graph.
Remember y = f(x) so lets
graph y = x2
which is a square
function (parabola)
This then is the graph
for the piecewise
function given above.

13. 





3x + 2, x  -2
-x , -2  x  1
x2
– 2, x  1
f(x) =
Graphing Piecewise Functions:
Determine the shapes of the graphs.
Line, Line, Parabola
Determine the boundaries of each graph.










 


14.  














0
,
5
0
,
3
0
3
,
5
2
x
x
x
x
x
x
f
For x values between
–3 and 0 graph the
line y = 2x + 5.
Since you know the graph is a
piece of a line, you can just plug in
each end value to get the
endpoints. f(-3) = -1 and f(0) = 5
For x = 0 the function
value is supposed to
be –3 so plot the
point (0, -3)
For x > 0 the function
is supposed to be
along the line y = - 5x.
Since you know this graph is a
piece of a line, you can just plug in
0 to see where to start the line and
then count a – 5 slope.
So this the graph of
the piecewise
function
solid dot for
"or equal to"
open dot
since not
"or equal to"

15. Real-life applications using Piecewise Functions
• Example1. A user is charged monthly for a particular mobile
plan, which includes 100 free text messages. Messages in
excess of 100 are charged ₱1.00 each. Represent the amount
a consumer pays each month as a function of the number of
messages m sent in a month.
Solution: Let represent the amount paid by the
consumer each month. It can be expressed by
the piecewise function

16. Real-life applications using Piecewise Functions
• Example 2. Due to community quarantine, a tricycle
ride costs ₱40.00 for the first 2 kilometers, and each
additional integer kilometer adds ₱5.00 to the fare.
Use a piecewise function to represent the tricycle’s
fare in terms of the distance in kilometers.
Solution: The input value is the distance and the
output is the cost of the tricycle fare. If F(x) represents
the fare as a function of distance, the function can be
represented as follows:
𝐹 ( 𝑥 )=
{ 40if 0< 𝑥 ≤ 2
40+5 (𝑥 −2)if 𝑥>2

17. Real-life applications using Piecewise Functions
Example 3. Getting treated for corona virus comes
with an expensive price tag. If you wanted to go to a
private hospital, the fee is based on the length of time
you have stayed. (figures not drawn to scale)
 Over 5 and up to 10 days costs ₱10,000.00
 Over 10 days costs ₱10,000 plus ₱1,500.00 per day
Which we can write as: