We want to graph the given function, then find the domain and range.
f(x)= [ [ 2x ] ]
To do so, we need to recall that the symbol [ [ x ] ] means the greatest integer less than or equal to x. For example, [ [ 2.1 ] ] =2, [ [ 2.9 ] ] =2, and [ [ - 1.5 ] ] =- 2. Let's make a table of values for the given function.
x
[ [ 2x ] ]
f(x)=[ [ 2x ] ]
- 2.5
[ [ 2( -2.5) ] ]
- 5
- 2
[ [ 2( -2) ] ]
-4
- 1.5
[ [ 2( -1.5) ] ]
-3
- 1
[ [ 2( -1) ] ]
-2
- 0.5
[ [ 2( -0.5) ] ]
-1
0
[ [ 2( 0) ] ]
0
0.5
[ [ 2( 0.5) ] ]
1
1
[ [ 2( 1) ] ]
2
1.5
[ [ 2( 1.5) ] ]
3
2
[ [ 2( 2) ] ]
4
2.5
[ [ 2( 2.5) ] ]
5
We can plot these points on a coordinate plane. Since f(x)=[ [ 2x ] ] is a step function, the right endpoint of each segment will be an "empty point."
We can see in the graph that the y-variable takes all integers, and that there are no restrictions for the x-variable. With this in mind, we can write the domain and range.
Domain:& All real numbers
Range:& All integers
