F(x) = (x - [x] )^2

Question

x is between -3 and 3. [x] denotes the greatest integer not greater than x.

1. Sketch the graph, find the range.

2. Is F(x) a periodic function of x?

3. Does x = F-1(y) exist?

Answer 1

1. The range is [0,1) that is, 0≤F(x)<1 since F(x)=0 when x is an integer and F(x+1)=F(x) for all x. For example, if x=0.7, F(0.7)=(0.7-0)²=0.49 and F(1.7)=(1.7-1)²=0.7²=0.49. The graph looks like f(x)=x² for x in [0,1) with the pattern repeated for each integer value of x.
2. F(x) is a periodic function because F(x+1)=F(x).
3. There can be no inverse because F(x) is not unique. In the example above, F(x)=0.49 has infinitely many values for x because of the cyclic nature of the function: x=0.7+n where n is an integer, positive or negative. More importantly, F(x) is not continuous.

The dotted line at y=1 shows the upper limit of F(x). Sorry for the poor quality of the drawing.