This document discusses piecewise functions and step functions. Piecewise functions are represented by a combination of equations, with each equation corresponding to a part of the domain. Step functions use separate equations for different intervals of the domain, resulting in a graph that makes distinct jumps at the boundaries between intervals. Examples are provided of evaluating piecewise functions and graphing both piecewise and step functions. Special types of step functions called ceiling functions and floor functions are also introduced, which round non-integer values up or down to the nearest integer.

1. Piecewise Functions

2. • Up to now, we’ve been looking at
functions represented by a single
equation.
• In real life, however, functions are
represented by a combination of
equations, each corresponding to
a part of the domain.
• These are called piecewise
functions.

3. ( )



>+
≤−
=
1,13
1,12
xifx
xifx
xf
•One equation gives the value of f(x)
when x ≤ 1
•And the other when x>1

4. Evaluate f(x) when x=0, x=2, x=4



≥+
<+
=
2,12
2,2
)(
xifx
xifx
xf
•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 hereSo:
2(2) + 1 = 5
f(2) = 5
X=4
This one fits hereSo:
2(4) + 1 = 9
f(4) = 9

5. Graph:



≥+−
<+
=
1,3
1,
)( 2
3
2
1
xifx
xifx
xf
•For all x’s < 1, use the top graph (to the left of 1)
•For all x’s ≥ 1, use the bottom graph (to the
•right of 1)

6. 1
2
3
, 1
2( )
3, 1
x if x
f x
x if x

+ <
= 
− + ≥

x=1 is the breaking
point of the graph.
To the left is the top
equation.
To the right is the
bottom equation.

7. Graph:
1, 2
( )
1, 2
x if x
f x
x if x
 − >
= 
− + ≤
Point of Discontinuity

8. Step Functions






<≤
<≤
<≤
<≤
=
43,4
32,3
21,2
10,1
)(
xif
xif
xif
xif
xf

9. 





<≤
<≤
<≤
<≤
=
43,4
32,3
21,2
10,1
)(
xif
xif
xif
xif
xf

10. Graph :






<≤−
−<≤−
−<≤−
−<≤−
=
01,4
12,3
23,2
34,1
)(
xif
xif
xif
xif
xf

12. Special Step Functions
Two particular kinds of step functions are called ceiling functions
( f (x)= and floor functions ( f (x)= ).
In a ceiling function, all nonintegers are rounded up to the nearest
integer.
An example of a ceiling function is when a phone service
company charges by the number of minutes used and always
rounds up to the nearest integer of minutes.
x   x  

13. Special Step Functions
In a floor function, all nonintegers are rounded down to the
nearest integer.
The way we usually count our age is an example of a floor
function since we round our age down to the nearest year and do
not add a year to our age until we have passed our birthday.
The floor function is the same thing as the greatest integer
function which can be written as f (x)=[x].