Since the parent function y=[ [ x ] ] is a step function, the given function f(x)= 12[ [ x ] ] is also a step function.
Graph:
Domain: All real numbers. Range: All integer multiples of 0.5.
Practice makes perfect
We want to graph the given function, then find the domain and range.
f(x)= 1/2 [ [ x ] ]
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. Remembering this rule, let's make a table of values for the given function.
x
1/2[ [ x ] ]
f(x)=1/2 [ [ x ] ]
- 3
1/2[ [ -3 ] ]
- 1.5
- 2.5
1/2[ [ -2.5 ] ]
- 1.5
- 2
1/2[ [ -2 ] ]
- 1
- 1.5
1/2[ [ -1.5 ] ]
- 1
- 1
1/2[ [ -1 ] ]
- 0.5
- 0.5
1/2[ [ -0.5 ] ]
- 0.5
0
1/2[ [ 0 ] ]
0
0.5
1/2[ [ 0.5 ] ]
0
1
1/2[ [ 1 ] ]
0.5
1.5
1/2[ [ 1.5 ] ]
0.5
2
1/2[ [ 2 ] ]
1
2.5
1/2[ [ 2.5 ] ]
1
We can plot these points on a coordinate plane. Since the parent function y=[ [ x ] ] is a step function, the given function is also a step function. This means that the right endpoint of each segment will be an "empty point."
We can see in the graph that the y-variable takes all integer multiples of 0.5, 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 integer multiples of0.5
