Daily Check
Solve and graph the following equations.
1. 2  4 x  5
2. 4  x  12
Math II
UNIT QUESTION: How are absolute
value equations similar to piecewise
functions?
Standard: MM2A1
Today’s Question:
How do we graph piecewise
functions?
Standard: MM2A1.a,b
2.5 Piecewise Functions
• 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.
 2 x  1, if x  1
f x   
 3 x  1, if x  1
•One equation gives the value of f(x)
when x ≤ 1
•And the other when x>1
Evaluate f(x) when x=0, x=2, x=4
 x  2, if x  2
f ( x)  
 2 x  1, if x  2
•First you have to figure out which equation to use
•You NEVER use both
X=0
So:
This one fits
Into the top
0+2=2
equation
f(0)=2
X=2
This
So: one fits here
2(2) + 1 = 5
f(2) = 5
X=4
So:one fits here
This
2(4) + 1 = 9
f(4) = 9
Graph:
 x  , if x  1
f ( x)  
  x  3, if x  1
1
2
3
2
•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)
3
1
x

, if x  1
2
2
f ( x)  
  x  3, if x  1

x=1 is the breaking
point of the graph.
To the left is the top
equation.
To the right is the
bottom equation.
Graph:
 x  1, if x  2
f ( x)  
  x  1, if x  2
Point of Discontinuity
Step Functions
 1, if
 2, if
f ( x)  
if
,
3

 4, if
0  x 1
1 x  2
2 x3
3 x  4
 1, if 0  x  1
 2, if 1  x  2
f ( x)  
 3, if 2  x  3
 4, if 3  x  4
Graph :
 1, if  4  x  3
 2, if  3  x  2
f ( x)  
3
,
if

2

x


1

 4, if  1  x  0
Special Step Functions
Two particular kinds of step functions are called ceiling functions
( f (x)=  x  and floor functions ( f (x)=  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.
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].
Class work
Textbook pg. 51 #1-8
HW Assignment
Textbook p. 52 #1-6;
and p. 53 #26