Greatest Integer/Absolute Value Functions
Students will be able to find greatest
integers and absolute values and
graph the both functions.
Greatest Integer
• Another special function that we will be
studying is the greatest integer function.
The greatest integer function of a real
number x, represented by [x], is the greatest
integer that is less than or equal to x.
• For example:
[4.25] = 4
FHS
[6] = 6
[5.99] = 5
Functions
[-2.3] = -3
2
•
Determine whether each statement below is
true or false for all real numbers x and y.
1. [x] + [y] = [x + y]
if x = 4.2 and y = 3.1, then x + y = 7.3
[4.2] + [3.1] = [7.3]
4 + 3 = 7 Is this correct?
if x = 4.7 and y = 3.9, then x + y = 8.6
[4.7] + [3.9] = [8.6]
4 + 3 = 8 Is this correct?
FHS
Functions
3
Graph
The greatest integer function is sometimes
called a step function, because of the shape
of its graph.
y
y = [x]
2
Graph y = [x]
x
-2
FHS
Functions
4
Graph
What happens when we change the function?
First multiply the function by 2.
y
y = [x]
2
Graph y =2[x]
x
On calculator:
y = 2int(X)
FHS
-2
Functions
5
Graph
What happens when we change the function?
Next multiply the independent variable by 2.
y
y = [x]
2
Graph y =[2x]
x
On calculator:
y = int(2X)
FHS
-2
Functions
6
Absolute Value
• All integers are composed of two parts – the
size and the direction. For example, +5 is
five units in the positive direction; –5 is five
units in the negative direction.
• The absolute value {written like this: 5 }of
a number gives the size of the number
without the direction. For example, 5 = 5
and 5 = 5. The answer is always positive.
FHS
Functions
7
Absolute Value
• Graphing the absolute
value function.
Graph: y  x
y
6
4
x
-4
-2
0
2
4
FHS
y
4
2
0
2
4
2
x
-5
5
-2
-4
-6
Functions
8