Floor Function
Function f from [latex]\mathbb{R}[/latex] to [latex]\mathbb{Z}[/latex] such that for every real number x, f(x) is equal to the greatest integer less than or equal to x.
Synonym of greatest integer function.
Notation
- The integer part of x is denoted by [x].
- The relationship defined by f (x) = [x] defines the basic model of the floor function.
Properties
- The general model, which is translated by a step function is defined by the relation f(x) = a[bx] , in which parameter a characterizes a "riser" (or vertical jump) on the graph, and parameter b characterizes the length of a horizontal step or each horizontal segment of the graph.
- Parameter a causes a vertical scale change of value |a| and affects the orientation—or steepness—of the graph depending on whether a is positive or negative.
- Parameter b causes a horizontal scale change of value [latex]\left| \dfrac {1}{b} \right|[/latex] and affects the orientation - or steepness - of the graph, depending on whether b is positive or negative.
- The standard form of the general model is[latex]f(x) = \textrm{a}\left[\textrm{b}\left(x - \textrm{h}\right)\right] + \textrm{k}[/latex] where parameters h and k characterize a horizontal translation and a vertical translation, respectively, of the graph of the basic relation.
Example
The graph of the function defined by the rule f[x] is that of a step function.
The following graph represents the function defined by the relation f[x] = −3[0,5x].
In this case, the direction of the graph was reversed, as parameter a is negative. The graph underwent a scale change of 3 units along the vertical axis and a scale change of |2| units along the horizontal axis.