The output of the ,
f(x)=[[x]], is the greatest less than or equal to the input value
x. To help us understand how this function works we will first look at a few examples.
f(1.2)f(3.9)f(-2.5)=[[1.2]]=1=[[3.9]]=3=[[-2.5]]=-3
We can think of it as rounding down the input value to the nearest integer. This results in a function that is divided into multiple , where the function is within an interval.
Interval |
f(x)=[[x]]
|
-4≤x<-3 |
-4
|
-3≤x<-2 |
-3
|
-2≤x<-1 |
-2
|
-1≤x<0 |
-1
|
0≤x<1 |
0
|
1≤x<2 |
1
|
2≤x<3 |
2
|
3≤x<4 |
3
|
Let's for -4≤x<4.
Recall that a closed dot corresponds to an end value that is included and an open dot means that the end value is not included.
Is It a Piecewise Function?
A is any function defined by two or more equations, each applying to a different part of the domain. To show that the greatest integer function is a piecewise function, we can write the equation as constant functions defined on intervals.
f(x)=⎩⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎧-4,-3,-2,-1,0,1,2,3,-4≤x<-3-3≤x<-2-2≤x<-1-1≤x<00≤x<11≤x<22≤x<33≤x<4.
Is It a Step Function?
A , on the other hand, is a special type of piecewise function characterized by a series of horizontal line segments. Let's once more study the graph of the greatest integer function.
The graph is a series of horizontal line segments. Therefore, the function f(x)=[[x]] is a step function.